diff options
| author | Enrico Tassi | 2018-04-20 10:54:22 +0200 |
|---|---|---|
| committer | Enrico Tassi | 2018-04-20 10:54:22 +0200 |
| commit | ed05182cece6bb3706e09b2ce14af4a41a2e8141 (patch) | |
| tree | e850d7314b6372d0476cf2ffaf7d3830721db7b1 /docs/htmldoc/mathcomp.fingroup.gproduct.html | |
| parent | 3d196f44681fb3b23ff8a79fbd44e12308680531 (diff) | |
generate the documentation for 1.7
Diffstat (limited to 'docs/htmldoc/mathcomp.fingroup.gproduct.html')
| -rw-r--r-- | docs/htmldoc/mathcomp.fingroup.gproduct.html | 1313 |
1 files changed, 1313 insertions, 0 deletions
diff --git a/docs/htmldoc/mathcomp.fingroup.gproduct.html b/docs/htmldoc/mathcomp.fingroup.gproduct.html new file mode 100644 index 0000000..7670d1a --- /dev/null +++ b/docs/htmldoc/mathcomp.fingroup.gproduct.html @@ -0,0 +1,1313 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" +"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml"> +<head> +<meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> +<link href="coqdoc.css" rel="stylesheet" type="text/css" /> +<title>mathcomp.fingroup.gproduct</title> +</head> + +<body> + +<div id="page"> + +<div id="header"> +</div> + +<div id="main"> + +<h1 class="libtitle">Library mathcomp.fingroup.gproduct</h1> + +<div class="code"> +<span class="comment">(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. <br/> + Distributed under the terms of CeCILL-B. *)</span><br/> +<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="mathcomp.ssreflect.ssreflect.html#"><span class="id" title="library">mathcomp.ssreflect.ssreflect</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Partial, semidirect, central, and direct products. + ++ Internal products, with A, B : {set gT}, are partial operations : + partial_product A B == A * B if A is a group normalised by the group B, + and the empty set otherwise. + A ><| B == A * B if this is a semi-direct product (i.e., if A + is normalised by B and intersects it trivially). + A \* B == A * B if this is a central product ( [A, B] = 1). + A \x B == A * B if this is a direct product. + [complements to K in G] == set of groups H s.t. K * H = G and K :&: H = 1. + [splits G, over K] == [complements to K in G] is not empty. + remgr A B x == the right remainder in B of x mod A, i.e., + some element of (A :* x) :&: B. + divgr A B x == the "division" in B of x by A: for all x, + x = divgr A B x * remgr A B x. + ++ External products : + pairg1, pair1g == the isomorphisms aT1 -> aT1 * aT2, aT2 -> aT1 * aT2. + (aT1 * aT2 has a direct product group structure.) + sdprod_by to == the semidirect product defined by to : groupAction H K. + This is a finGroupType; the actual semidirect product is + the total set [set: sdprod_by to] on that type. + sdpair[12] to == the isomorphisms injecting K and H into + sdprod_by to = sdpair1 to @* K ><| sdpair2 to @* H. + External central products (with identified centers) will be defined later + in file center.v. + ++ Morphisms on product groups: + pprodm nAB fJ fAB == the morphism extending fA and fB on A <*> B when + nAB : B \subset 'N(A), + fJ : {in A & B, morph_actj fA fB}, and + fAB : {in A :&: B, fA =1 fB}. + sdprodm defG fJ == the morphism extending fA and fB on G, given + defG : A ><| B = G and + fJ : {in A & B, morph_act 'J 'J fA fB}. + xsdprodm fHKact == the total morphism on sdprod_by to induced by + fH : {morphism H >-> rT}, fK : {morphism K >-> rT}, + with to : groupAction K H, + given fHKact : morph_act to 'J fH fK. + cprodm defG cAB fAB == the morphism extending fA and fB on G, when + defG : A \* B = G, + cAB : fB @* B \subset 'C(fB @* A), + and fAB : {in A :&: B, fA =1 fB}. + dprodm defG cAB == the morphism extending fA and fB on G, when + defG : A \x B = G and + cAB : fA @* B \subset 'C(fA @* A) + mulgm (x, y) == x * y; mulgm is an isomorphism from setX A B to G + iff A \x B = G . +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Set Implicit Arguments</span>.<br/> + +<br/> +<span class="id" title="keyword">Import</span> <span class="id" title="var">GroupScope</span>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Defs"><span class="id" title="section">Defs</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="Defs.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#Defs.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="partial_product"><span class="id" title="definition">partial_product</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :=<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#set0"><span class="id" title="definition">set0</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="semidirect_product"><span class="id" title="definition">semidirect_product</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :=<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> 1%<span class="id" title="var">G</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#partial_product"><span class="id" title="definition">partial_product</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#set0"><span class="id" title="definition">set0</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="central_product"><span class="id" title="definition">central_product</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :=<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#partial_product"><span class="id" title="definition">partial_product</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#set0"><span class="id" title="definition">set0</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="direct_product"><span class="id" title="definition">direct_product</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :=<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> 1%<span class="id" title="var">G</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#central_product"><span class="id" title="definition">central_product</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#set0"><span class="id" title="definition">set0</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="complements_to_in"><span class="id" title="definition">complements_to_in</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :=<br/> + <a class="idref" href="mathcomp.ssreflect.finset.html#92573f9b19c03e948cd1a21ac092cb5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#92573f9b19c03e948cd1a21ac092cb5a"><span class="id" title="notation">set</span></a> <span class="id" title="var">K</span> <a class="idref" href="mathcomp.ssreflect.finset.html#92573f9b19c03e948cd1a21ac092cb5a"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#Defs.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#92573f9b19c03e948cd1a21ac092cb5a"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1 <a class="idref" href="mathcomp.ssreflect.finset.html#92573f9b19c03e948cd1a21ac092cb5a"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#92573f9b19c03e948cd1a21ac092cb5a"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="splits_over"><span class="id" title="definition">splits_over</span></a> <span class="id" title="var">B</span> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.fingroup.gproduct.html#complements_to_in"><span class="id" title="definition">complements_to_in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#set0"><span class="id" title="definition">set0</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Product remainder functions -- right variant only. +</div> +<div class="code"> +<span class="id" title="keyword">Definition</span> <a name="remgr"><span class="id" title="definition">remgr</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#repr"><span class="id" title="definition">repr</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#291276ea06db0b00a2747a79d012bbe0"><span class="id" title="notation">:*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>).<br/> +<span class="id" title="keyword">Definition</span> <a name="divgr"><span class="id" title="definition">divgr</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#remgr"><span class="id" title="definition">remgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">)^-1</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#Defs"><span class="id" title="section">Defs</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="pprod"><span class="id" title="abbreviation">pprod</span></a> := (<a class="idref" href="mathcomp.fingroup.gproduct.html#partial_product"><span class="id" title="definition">partial_product</span></a> <span class="id" title="var">_</span>).<br/> +<span class="id" title="keyword">Notation</span> <a name="sdprod"><span class="id" title="abbreviation">sdprod</span></a> := (<a class="idref" href="mathcomp.fingroup.gproduct.html#semidirect_product"><span class="id" title="definition">semidirect_product</span></a> <span class="id" title="var">_</span>).<br/> +<span class="id" title="keyword">Notation</span> <a name="cprod"><span class="id" title="abbreviation">cprod</span></a> := (<a class="idref" href="mathcomp.fingroup.gproduct.html#central_product"><span class="id" title="definition">central_product</span></a> <span class="id" title="var">_</span>).<br/> +<span class="id" title="keyword">Notation</span> <a name="dprod"><span class="id" title="abbreviation">dprod</span></a> := (<a class="idref" href="mathcomp.fingroup.gproduct.html#direct_product"><span class="id" title="definition">direct_product</span></a> <span class="id" title="var">_</span>).<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">"</span></a>G ><| H" := (<a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod"><span class="id" title="abbreviation">sdprod</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span>)%<span class="id" title="var">g</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 40, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>).<br/> +<span class="id" title="keyword">Notation</span> <a name="9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">"</span></a>G \* H" := (<a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span>)%<span class="id" title="var">g</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 40, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>).<br/> +<span class="id" title="keyword">Notation</span> <a name="3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">"</span></a>G \x H" := (<a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span>)%<span class="id" title="var">g</span> (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 40, <span class="id" title="tactic">left</span> <span class="id" title="keyword">associativity</span>).<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">"</span></a>[ 'complements' 'to' A 'in' B ]" := (<a class="idref" href="mathcomp.fingroup.gproduct.html#complements_to_in"><span class="id" title="definition">complements_to_in</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'complements' 'to' A 'in' B ]") : <span class="id" title="var">group_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="0a7352dd0cab58f1c154ca74b1d45ebb"><span class="id" title="notation">"</span></a>[ 'splits' B , 'over' A ]" := (<a class="idref" href="mathcomp.fingroup.gproduct.html#splits_over"><span class="id" title="definition">splits_over</span></a> <span class="id" title="var">B</span> <span class="id" title="var">A</span>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'splits' B , 'over' A ]") : <span class="id" title="var">group_scope</span>.<br/> + +<br/> +</div> + +<div class="doc"> + Prenex Implicits remgl divgl. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Section</span> <a name="InternalProd"><span class="id" title="section">InternalProd</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="InternalProd.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">C</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> <span class="id" title="var">L</span> <span class="id" title="var">M</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> + +<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pprod1g"><span class="id" title="lemma">pprod1g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> 1 <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pprodg1"><span class="id" title="lemma">pprodg1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_id"><span class="id" title="definition">right_id</span></a> 1 <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a>.<br/> + +<br/> +<span class="id" title="keyword">CoInductive</span> <a name="are_groups"><span class="id" title="inductive">are_groups</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <span class="id" title="keyword">Prop</span> := <a name="AreGroups"><span class="id" title="constructor">AreGroups</span></a> <span class="id" title="var">K</span> <span class="id" title="var">H</span> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> & <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="group_not0"><span class="id" title="lemma">group_not0</span></a> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#set0"><span class="id" title="definition">set0</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="mulg0"><span class="id" title="lemma">mulg0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_zero"><span class="id" title="definition">right_zero</span></a> (@<a class="idref" href="mathcomp.ssreflect.finset.html#set0"><span class="id" title="definition">set0</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a>) <a class="idref" href="mathcomp.fingroup.fingroup.html#mulg"><span class="id" title="definition">mulg</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="mul0g"><span class="id" title="lemma">mul0g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_zero"><span class="id" title="definition">left_zero</span></a> (@<a class="idref" href="mathcomp.ssreflect.finset.html#set0"><span class="id" title="definition">set0</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a>) <a class="idref" href="mathcomp.fingroup.fingroup.html#mulg"><span class="id" title="definition">mulg</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pprodP"><span class="id" title="lemma">pprodP</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#are_groups"><span class="id" title="inductive">are_groups</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pprodE"><span class="id" title="lemma">pprodE</span></a> <span class="id" title="var">K</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pprodEY"><span class="id" title="lemma">pprodEY</span></a> <span class="id" title="var">K</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pprodW"><span class="id" title="lemma">pprodW</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pprodWC"><span class="id" title="lemma">pprodWC</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pprodWY"><span class="id" title="lemma">pprodWY</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pprodJ"><span class="id" title="lemma">pprodJ</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a>) (<a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +</div> + +<div class="doc"> + Properties of the remainders +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="remgrMl"><span class="id" title="lemma">remgrMl</span></a> <span class="id" title="var">K</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#remgr"><span class="id" title="definition">remgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#remgr"><span class="id" title="definition">remgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="remgrP"><span class="id" title="lemma">remgrP</span></a> <span class="id" title="var">K</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#remgr"><span class="id" title="definition">remgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#291276ea06db0b00a2747a79d012bbe0"><span class="id" title="notation">:*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="remgr1"><span class="id" title="lemma">remgr1</span></a> <span class="id" title="var">K</span> <span class="id" title="var">H</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#remgr"><span class="id" title="definition">remgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="divgr_eq"><span class="id" title="lemma">divgr_eq</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#divgr"><span class="id" title="definition">divgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#remgr"><span class="id" title="definition">remgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="divgrMl"><span class="id" title="lemma">divgrMl</span></a> <span class="id" title="var">K</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#divgr"><span class="id" title="definition">divgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#divgr"><span class="id" title="definition">divgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="divgr_id"><span class="id" title="lemma">divgr_id</span></a> <span class="id" title="var">K</span> <span class="id" title="var">H</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#divgr"><span class="id" title="definition">divgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="mem_remgr"><span class="id" title="lemma">mem_remgr</span></a> <span class="id" title="var">K</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#remgr"><span class="id" title="definition">remgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="mem_divgr"><span class="id" title="lemma">mem_divgr</span></a> <span class="id" title="var">K</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#divgr"><span class="id" title="definition">divgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="InternalProd.DisjointRem"><span class="id" title="section">DisjointRem</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="InternalProd.DisjointRem.K"><span class="id" title="variable">K</span></a> <a name="InternalProd.DisjointRem.H"><span class="id" title="variable">H</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="InternalProd.DisjointRem.tiKH"><span class="id" title="variable">tiKH</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="remgr_id"><span class="id" title="lemma">remgr_id</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#remgr"><span class="id" title="definition">remgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="remgrMid"><span class="id" title="lemma">remgrMid</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#remgr"><span class="id" title="definition">remgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem.H"><span class="id" title="variable">H</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="divgrMid"><span class="id" title="lemma">divgrMid</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#divgr"><span class="id" title="definition">divgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem.H"><span class="id" title="variable">H</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.DisjointRem"><span class="id" title="section">DisjointRem</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Intersection of a centraliser with a disjoint product. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="subcent_TImulg"><span class="id" title="lemma">subcent_TImulg</span></a> <span class="id" title="var">K</span> <span class="id" title="var">H</span> <span class="id" title="var">A</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">C_K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">C_H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#82eaaf0a77a7ca85d3ca7dcf4463ae79"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#82eaaf0a77a7ca85d3ca7dcf4463ae79"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#82eaaf0a77a7ca85d3ca7dcf4463ae79"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#82eaaf0a77a7ca85d3ca7dcf4463ae79"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#82eaaf0a77a7ca85d3ca7dcf4463ae79"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Complements, and splitting. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="complP"><span class="id" title="lemma">complP</span></a> <span class="id" title="var">H</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>) (<a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">complements</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">to</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">]</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="splitsP"><span class="id" title="lemma">splitsP</span></a> <span class="id" title="var">B</span> <span class="id" title="var">A</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">H</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">complements</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">to</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">]</span></a>) <a class="idref" href="mathcomp.fingroup.gproduct.html#0a7352dd0cab58f1c154ca74b1d45ebb"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#0a7352dd0cab58f1c154ca74b1d45ebb"><span class="id" title="notation">splits</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#0a7352dd0cab58f1c154ca74b1d45ebb"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#0a7352dd0cab58f1c154ca74b1d45ebb"><span class="id" title="notation">over</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#0a7352dd0cab58f1c154ca74b1d45ebb"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="complgC"><span class="id" title="lemma">complgC</span></a> <span class="id" title="var">H</span> <span class="id" title="var">K</span> <span class="id" title="var">G</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">complements</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">to</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">complements</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">to</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">]</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="InternalProd.NormalComplement"><span class="id" title="section">NormalComplement</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="InternalProd.NormalComplement.K"><span class="id" title="variable">K</span></a> <a name="InternalProd.NormalComplement.H"><span class="id" title="variable">H</span></a> <a name="InternalProd.NormalComplement.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="InternalProd.NormalComplement.complH_K"><span class="id" title="variable">complH_K</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">complements</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">to</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="remgrM"><span class="id" title="lemma">remgrM</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#remgr"><span class="id" title="definition">remgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="divgrM"><span class="id" title="lemma">divgrM</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#divgr"><span class="id" title="definition">divgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.NormalComplement"><span class="id" title="section">NormalComplement</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Semi-direct product +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod1g"><span class="id" title="lemma">sdprod1g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> 1 <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod"><span class="id" title="abbreviation">sdprod</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodg1"><span class="id" title="lemma">sdprodg1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_id"><span class="id" title="definition">right_id</span></a> 1 <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod"><span class="id" title="abbreviation">sdprod</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodP"><span class="id" title="lemma">sdprodP</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#are_groups"><span class="id" title="inductive">are_groups</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodE"><span class="id" title="lemma">sdprodE</span></a> <span class="id" title="var">K</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodEY"><span class="id" title="lemma">sdprodEY</span></a> <span class="id" title="var">K</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodWpp"><span class="id" title="lemma">sdprodWpp</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodW"><span class="id" title="lemma">sdprodW</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodWC"><span class="id" title="lemma">sdprodWC</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodWY"><span class="id" title="lemma">sdprodWY</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodJ"><span class="id" title="lemma">sdprodJ</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_context"><span class="id" title="lemma">sdprod_context</span></a> <span class="id" title="var">G</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#c7fe7fb0f694e91a7e258ff78a0390ef"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#c7fe7fb0f694e91a7e258ff78a0390ef"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#c7fe7fb0f694e91a7e258ff78a0390ef"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#c7fe7fb0f694e91a7e258ff78a0390ef"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#c7fe7fb0f694e91a7e258ff78a0390ef"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#c7fe7fb0f694e91a7e258ff78a0390ef"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_compl"><span class="id" title="lemma">sdprod_compl</span></a> <span class="id" title="var">G</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">complements</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">to</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_normal_complP"><span class="id" title="lemma">sdprod_normal_complP</span></a> <span class="id" title="var">G</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">complements</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">to</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#44af7d5c4de43cd8d378a56e587bd9b2"><span class="id" title="notation">]</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_card"><span class="id" title="lemma">sdprod_card</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_isom"><span class="id" title="lemma">sdprod_isom</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">{</span></a><span class="id" title="var">nAB</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a>) (<a class="idref" href="mathcomp.fingroup.morphism.html#restrm"><span class="id" title="definition">restrm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#nAB"><span class="id" title="variable">nAB</span></a> (<a class="idref" href="mathcomp.fingroup.quotient.html#coset"><span class="id" title="definition">coset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a>))<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#72ca3fac4636a1b19c963b12162882cf"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_isog"><span class="id" title="lemma">sdprod_isog</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_subr"><span class="id" title="lemma">sdprod_subr</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">M</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="index_sdprod"><span class="id" title="lemma">index_sdprod</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="index_sdprodr"><span class="id" title="lemma">index_sdprodr</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">M</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#f65ecb5148d1ef5a9c551827b20e9bfa"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="quotient_sdprodr_isom"><span class="id" title="lemma">quotient_sdprodr_isom</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">M</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">{</span></a><span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#coset_of"><span class="id" title="record">coset_of</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a>)<a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">|</span></a><br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a>) (<a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">)</span></a>) <a class="idref" href="mathcomp.fingroup.gproduct.html#f"><span class="id" title="variable">f</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">&</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">L</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#602b9943a639fb973abed6e2c7854421"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="quotient_sdprodr_isog"><span class="id" title="lemma">quotient_sdprodr_isog</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">M</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_modl"><span class="id" title="lemma">sdprod_modl</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_modr"><span class="id" title="lemma">sdprod_modr</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="subcent_sdprod"><span class="id" title="lemma">subcent_sdprod</span></a> <span class="id" title="var">B</span> <span class="id" title="var">C</span> <span class="id" title="var">G</span> <span class="id" title="var">A</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">C_B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">C_C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_recl"><span class="id" title="lemma">sdprod_recl</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> <span class="id" title="var">K1</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K1"><span class="id" title="variable">K1</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#K1"><span class="id" title="variable">K1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">G1</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G1"><span class="id" title="variable">G1</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G1"><span class="id" title="variable">G1</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K1"><span class="id" title="variable">K1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G1"><span class="id" title="variable">G1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_recr"><span class="id" title="lemma">sdprod_recr</span></a> <span class="id" title="var">n</span> <span class="id" title="var">G</span> <span class="id" title="var">K</span> <span class="id" title="var">H</span> <span class="id" title="var">H1</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#n"><span class="id" title="variable">n</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#219b95257a323aaee1742e9bec4975d7"><span class="id" title="notation">proper</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">G1</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G1"><span class="id" title="variable">G1</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#989c98e7ddd65d5bf37c334ff2076de8"><span class="id" title="notation"><</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#n"><span class="id" title="variable">n</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G1"><span class="id" title="variable">G1</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G1"><span class="id" title="variable">G1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="mem_sdprod"><span class="id" title="lemma">mem_sdprod</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">&</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">u</span> <span class="id" title="var">t</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Central product +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprod1g"><span class="id" title="lemma">cprod1g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> 1 <a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodg1"><span class="id" title="lemma">cprodg1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_id"><span class="id" title="definition">right_id</span></a> 1 <a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodP"><span class="id" title="lemma">cprodP</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#are_groups"><span class="id" title="inductive">are_groups</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodE"><span class="id" title="lemma">cprodE</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodEY"><span class="id" title="lemma">cprodEY</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodWpp"><span class="id" title="lemma">cprodWpp</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodW"><span class="id" title="lemma">cprodW</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodWC"><span class="id" title="lemma">cprodWC</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodWY"><span class="id" title="lemma">cprodWY</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodJ"><span class="id" title="lemma">cprodJ</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprod_normal2"><span class="id" title="lemma">cprod_normal2</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="bigcprodW"><span class="id" title="lemma">bigcprodW</span></a> <span class="id" title="var">I</span> (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">P</span> <span class="id" title="var">F</span> <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#47eadcc9475f6ce09c2ad11c889b5183"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#47eadcc9475f6ce09c2ad11c889b5183"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#47eadcc9475f6ce09c2ad11c889b5183"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#47eadcc9475f6ce09c2ad11c889b5183"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#47eadcc9475f6ce09c2ad11c889b5183"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#47eadcc9475f6ce09c2ad11c889b5183"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="bigcprodWY"><span class="id" title="lemma">bigcprodWY</span></a> <span class="id" title="var">I</span> (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">P</span> <span class="id" title="var">F</span> <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation"><<</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#eac6f811013bd0926df74c1fd1d5e66c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#eac6f811013bd0926df74c1fd1d5e66c"><span class="id" title="notation">bigcup_</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#eac6f811013bd0926df74c1fd1d5e66c"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.finset.html#eac6f811013bd0926df74c1fd1d5e66c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#eac6f811013bd0926df74c1fd1d5e66c"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#eac6f811013bd0926df74c1fd1d5e66c"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="triv_cprod"><span class="id" title="lemma">triv_cprod</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprod_ntriv"><span class="id" title="lemma">cprod_ntriv</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">if</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">then</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#0348819abaa88c2cd747e8fa60dde7ae"><span class="id" title="notation">else</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#set0"><span class="id" title="definition">set0</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="trivg0"><span class="id" title="lemma">trivg0</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a>@<a class="idref" href="mathcomp.ssreflect.finset.html#set0"><span class="id" title="definition">set0</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="group0"><span class="id" title="lemma">group0</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> (@<a class="idref" href="mathcomp.ssreflect.finset.html#set0"><span class="id" title="definition">set0</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprod0g"><span class="id" title="lemma">cprod0g</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#set0"><span class="id" title="definition">set0</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#set0"><span class="id" title="definition">set0</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodC"><span class="id" title="lemma">cprodC</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#commutative"><span class="id" title="definition">commutative</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodA"><span class="id" title="lemma">cprodA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cprod_law</span> := <a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.Law"><span class="id" title="constructor">Monoid.Law</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodA"><span class="id" title="lemma">cprodA</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprod1g"><span class="id" title="lemma">cprod1g</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodg1"><span class="id" title="lemma">cprodg1</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cprod_abelaw</span> := <a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.ComLaw"><span class="id" title="constructor">Monoid.ComLaw</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodC"><span class="id" title="lemma">cprodC</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprod_modl"><span class="id" title="lemma">cprod_modl</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprod_modr"><span class="id" title="lemma">cprod_modr</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="bigcprodYP"><span class="id" title="lemma">bigcprodYP</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span> <span class="id" title="var">j</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a>)<br/> + (<a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">G</span>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="bigcprodEY"><span class="id" title="lemma">bigcprodEY</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#a8ebd13d7d023f5f44ac87addefd929b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a8ebd13d7d023f5f44ac87addefd929b"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a8ebd13d7d023f5f44ac87addefd929b"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#a8ebd13d7d023f5f44ac87addefd929b"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#a8ebd13d7d023f5f44ac87addefd929b"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a8ebd13d7d023f5f44ac87addefd929b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">G</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="perm_bigcprod"><span class="id" title="lemma">perm_bigcprod</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.eqType"><span class="id" title="abbreviation">eqType</span></a>) <span class="id" title="var">r1</span> <span class="id" title="var">r2</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r1"><span class="id" title="variable">r1</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#30705c25db0a97e8b1b08168f9199b27"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r1"><span class="id" title="variable">r1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.ssreflect.seq.html#perm_eq"><span class="id" title="definition">perm_eq</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r1"><span class="id" title="variable">r1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r2"><span class="id" title="variable">r2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.fingroup.fingroup.html#644e2d6a4fd9b4978c70acde75bf082b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#644e2d6a4fd9b4978c70acde75bf082b"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#644e2d6a4fd9b4978c70acde75bf082b"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#644e2d6a4fd9b4978c70acde75bf082b"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r1"><span class="id" title="variable">r1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#644e2d6a4fd9b4978c70acde75bf082b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#644e2d6a4fd9b4978c70acde75bf082b"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#644e2d6a4fd9b4978c70acde75bf082b"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#644e2d6a4fd9b4978c70acde75bf082b"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#644e2d6a4fd9b4978c70acde75bf082b"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r2"><span class="id" title="variable">r2</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#644e2d6a4fd9b4978c70acde75bf082b"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="reindex_bigcprod"><span class="id" title="lemma">reindex_bigcprod</span></a> (<span class="id" title="var">I</span> <span class="id" title="var">J</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">h</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#J"><span class="id" title="variable">J</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">P</span> (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#59b0bfb23a642e2d3d15d13178fc2b18"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#59b0bfb23a642e2d3d15d13178fc2b18"><span class="id" title="notation">on</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#SimplPred"><span class="id" title="definition">SimplPred</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#59b0bfb23a642e2d3d15d13178fc2b18"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#59b0bfb23a642e2d3d15d13178fc2b18"><span class="id" title="notation">bijective</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#h"><span class="id" title="variable">h</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#59b0bfb23a642e2d3d15d13178fc2b18"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#SimplPred"><span class="id" title="definition">SimplPred</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a>)<a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#h"><span class="id" title="variable">h</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a>).<br/> + +<br/> +</div> + +<div class="doc"> + Direct product +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprod1g"><span class="id" title="lemma">dprod1g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> 1 <a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodg1"><span class="id" title="lemma">dprodg1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#right_id"><span class="id" title="definition">right_id</span></a> 1 <a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodP"><span class="id" title="lemma">dprodP</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#are_groups"><span class="id" title="inductive">are_groups</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodE"><span class="id" title="lemma">dprodE</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodEY"><span class="id" title="lemma">dprodEY</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodEcp"><span class="id" title="lemma">dprodEcp</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodEsd"><span class="id" title="lemma">dprodEsd</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodWcp"><span class="id" title="lemma">dprodWcp</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodWsd"><span class="id" title="lemma">dprodWsd</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodW"><span class="id" title="lemma">dprodW</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodWC"><span class="id" title="lemma">dprodWC</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodWY"><span class="id" title="lemma">dprodWY</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprod_card_dprod"><span class="id" title="lemma">cprod_card_dprod</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodJ"><span class="id" title="lemma">dprodJ</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprod_normal2"><span class="id" title="lemma">dprod_normal2</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodYP"><span class="id" title="lemma">dprodYP</span></a> <span class="id" title="var">K</span> <span class="id" title="var">H</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>) (<a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#1db838ba797020f3b39c07ed7167bc93"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#adb8044960c962a921cca1bd48aae97d"><span class="id" title="notation">^#</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodC"><span class="id" title="lemma">dprodC</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#commutative"><span class="id" title="definition">commutative</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodWsdC"><span class="id" title="lemma">dprodWsdC</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodA"><span class="id" title="lemma">dprodA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">dprod_law</span> := <a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.Law"><span class="id" title="constructor">Monoid.Law</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodA"><span class="id" title="lemma">dprodA</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprod1g"><span class="id" title="lemma">dprod1g</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodg1"><span class="id" title="lemma">dprodg1</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">dprod_abelaw</span> := <a class="idref" href="mathcomp.ssreflect.bigop.html#Monoid.ComLaw"><span class="id" title="constructor">Monoid.ComLaw</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodC"><span class="id" title="lemma">dprodC</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="bigdprodWcp"><span class="id" title="lemma">bigdprodWcp</span></a> <span class="id" title="var">I</span> (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">P</span> <span class="id" title="var">F</span> <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="bigdprodW"><span class="id" title="lemma">bigdprodW</span></a> <span class="id" title="var">I</span> (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">P</span> <span class="id" title="var">F</span> <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#47eadcc9475f6ce09c2ad11c889b5183"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#47eadcc9475f6ce09c2ad11c889b5183"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#47eadcc9475f6ce09c2ad11c889b5183"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#47eadcc9475f6ce09c2ad11c889b5183"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#47eadcc9475f6ce09c2ad11c889b5183"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#47eadcc9475f6ce09c2ad11c889b5183"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="bigdprodWY"><span class="id" title="lemma">bigdprodWY</span></a> <span class="id" title="var">I</span> (<span class="id" title="var">r</span> : <a class="idref" href="mathcomp.ssreflect.seq.html#seq"><span class="id" title="abbreviation">seq</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">P</span> <span class="id" title="var">F</span> <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation"><<</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#eac6f811013bd0926df74c1fd1d5e66c"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#eac6f811013bd0926df74c1fd1d5e66c"><span class="id" title="notation">bigcup_</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#eac6f811013bd0926df74c1fd1d5e66c"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.finset.html#eac6f811013bd0926df74c1fd1d5e66c"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#eac6f811013bd0926df74c1fd1d5e66c"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#eac6f811013bd0926df74c1fd1d5e66c"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="bigdprodYP"><span class="id" title="lemma">bigdprodYP</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">F</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + (<a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">(</span></a><span class="id" title="var">j</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a>)%<span class="id" title="var">G</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#1db838ba797020f3b39c07ed7167bc93"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#adb8044960c962a921cca1bd48aae97d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#adb8044960c962a921cca1bd48aae97d"><span class="id" title="notation">)^#</span></a>)<br/> + (<a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#af9c2fa5885e183f37430995d7cdd9d7"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a>)%<span class="id" title="var">G</span>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprod_modl"><span class="id" title="lemma">dprod_modl</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprod_modr"><span class="id" title="lemma">dprod_modr</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="subcent_dprod"><span class="id" title="lemma">subcent_dprod</span></a> <span class="id" title="var">B</span> <span class="id" title="var">C</span> <span class="id" title="var">G</span> <span class="id" title="var">A</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">C_B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">C_C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">C_G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprod_card"><span class="id" title="lemma">dprod_card</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#697e4695610f677ae98a52af81f779d2"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="bigdprod_card"><span class="id" title="lemma">bigdprod_card</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">E</span> <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + (<a class="idref" href="mathcomp.ssreflect.bigop.html#9d27735af9c069a15e48cb2f0aad6a15"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9d27735af9c069a15e48cb2f0aad6a15"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9d27735af9c069a15e48cb2f0aad6a15"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#9d27735af9c069a15e48cb2f0aad6a15"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#9d27735af9c069a15e48cb2f0aad6a15"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9d27735af9c069a15e48cb2f0aad6a15"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#E"><span class="id" title="variable">E</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>)%<span class="id" title="var">N</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="bigcprod_card_dprod"><span class="id" title="lemma">bigcprod_card_dprod</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#9d27735af9c069a15e48cb2f0aad6a15"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9d27735af9c069a15e48cb2f0aad6a15"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9d27735af9c069a15e48cb2f0aad6a15"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#9d27735af9c069a15e48cb2f0aad6a15"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#9d27735af9c069a15e48cb2f0aad6a15"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#9d27735af9c069a15e48cb2f0aad6a15"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.ssrnat.html#9b077c369e19739ef880736ba34623ff"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="bigcprod_coprime_dprod"><span class="id" title="lemma">bigcprod_coprime_dprod</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span> <span class="id" title="var">j</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="mem_dprod"><span class="id" title="lemma">mem_dprod</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">y</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">z</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">[/\</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">&</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">u</span> <span class="id" title="var">t</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#z"><span class="id" title="variable">z</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ca13a32469ebe56c9f4cc99d00e8eeba"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="mem_bigdprod"><span class="id" title="lemma">mem_bigdprod</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">F</span> <span class="id" title="var">G</span> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">c</span><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">[/\</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">&</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">e</span>, <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">prod_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#12c1d637d10a949e2e4e04c0b0b1b2dd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <span class="id" title="keyword">∀</span> <span class="id" title="var">i</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#c"><span class="id" title="variable">c</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#962a3cb7af009aedac7986e261646bd1"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#InternalProd"><span class="id" title="section">InternalProd</span></a>.<br/> + +<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="MorphimInternalProd"><span class="id" title="section">MorphimInternalProd</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="MorphimInternalProd.gT"><span class="id" title="variable">gT</span></a> <a name="MorphimInternalProd.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="MorphimInternalProd.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="MorphimInternalProd.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="MorphimInternalProd.OneProd"><span class="id" title="section">OneProd</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a> <a name="MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a> <a name="MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="MorphimInternalProd.OneProd.sGD"><span class="id" title="variable">sGD</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.D"><span class="id" title="variable">D</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_pprod"><span class="id" title="lemma">morphim_pprod</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a>) (<a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_coprime_sdprod"><span class="id" title="lemma">morphim_coprime_sdprod</span></a> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_sdprod"><span class="id" title="lemma">injm_sdprod</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_cprod"><span class="id" title="lemma">morphim_cprod</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_dprod"><span class="id" title="lemma">injm_dprod</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_coprime_dprod"><span class="id" title="lemma">morphim_coprime_dprod</span></a> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.OneProd"><span class="id" title="section">OneProd</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">G</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_bigcprod"><span class="id" title="lemma">morphim_bigcprod</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#cprod"><span class="id" title="abbreviation">cprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_bigdprod"><span class="id" title="lemma">injm_bigdprod</span></a> <span class="id" title="var">I</span> <span class="id" title="var">r</span> (<span class="id" title="var">P</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#pred"><span class="id" title="definition">pred</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a>) (<span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation"><-</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#r"><span class="id" title="variable">r</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#52c4d552b36d01307b4a33177122d4d1"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_coprime_bigdprod"><span class="id" title="lemma">morphim_coprime_bigdprod</span></a> (<span class="id" title="var">I</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.finType"><span class="id" title="abbreviation">finType</span></a>) <span class="id" title="var">P</span> (<span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#I"><span class="id" title="variable">I</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">i</span> <span class="id" title="var">j</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#b1eeadc2feabc7422252baa895418c7b"><span class="id" title="notation">!=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#j"><span class="id" title="variable">j</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">big</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#dprod"><span class="id" title="abbreviation">dprod</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">/</span></a>1<a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">_</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">(</span></a><span class="id" title="var">i</span> <a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a><a class="idref" href="mathcomp.ssreflect.bigop.html#8850ee6edf9a388b1213678f3d3ee856"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#i"><span class="id" title="variable">i</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#MorphimInternalProd"><span class="id" title="section">MorphimInternalProd</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="QuotientInternalProd"><span class="id" title="section">QuotientInternalProd</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="QuotientInternalProd.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="QuotientInternalProd.G"><span class="id" title="variable">G</span></a> <a name="QuotientInternalProd.K"><span class="id" title="variable">K</span></a> <a name="QuotientInternalProd.H"><span class="id" title="variable">H</span></a> <a name="QuotientInternalProd.M"><span class="id" title="variable">M</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="QuotientInternalProd.nMG"><span class="id" title="variable">nMG</span></a>: <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="quotient_pprod"><span class="id" title="lemma">quotient_pprod</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pprod"><span class="id" title="abbreviation">pprod</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.M"><span class="id" title="variable">M</span></a>) (<a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.M"><span class="id" title="variable">M</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.M"><span class="id" title="variable">M</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="quotient_coprime_sdprod"><span class="id" title="lemma">quotient_coprime_sdprod</span></a> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.M"><span class="id" title="variable">M</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="quotient_cprod"><span class="id" title="lemma">quotient_cprod</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.M"><span class="id" title="variable">M</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="quotient_coprime_dprod"><span class="id" title="lemma">quotient_coprime_dprod</span></a> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.ssreflect.div.html#coprime"><span class="id" title="definition">coprime</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.M"><span class="id" title="variable">M</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.quotient.html#c7768147d2d560601601fbf95706ddcc"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd.M"><span class="id" title="variable">M</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#QuotientInternalProd"><span class="id" title="section">QuotientInternalProd</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ExternalDirProd"><span class="id" title="section">ExternalDirProd</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a> <a name="ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="extprod_mulg"><span class="id" title="definition">extprod_mulg</span></a> (<span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="extprod_invg"><span class="id" title="definition">extprod_invg</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a>) := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="extprod_mul1g"><span class="id" title="lemma">extprod_mul1g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#extprod_mulg"><span class="id" title="definition">extprod_mulg</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="extprod_mulVg"><span class="id" title="lemma">extprod_mulVg</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_inverse"><span class="id" title="definition">left_inverse</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#extprod_invg"><span class="id" title="definition">extprod_invg</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#extprod_mulg"><span class="id" title="definition">extprod_mulg</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="extprod_mulgA"><span class="id" title="lemma">extprod_mulgA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#extprod_mulg"><span class="id" title="definition">extprod_mulg</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="extprod_groupMixin"><span class="id" title="definition">extprod_groupMixin</span></a> :=<br/> + <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Mixin"><span class="id" title="definition">FinGroup.Mixin</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#extprod_mulgA"><span class="id" title="lemma">extprod_mulgA</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#extprod_mul1g"><span class="id" title="lemma">extprod_mul1g</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#extprod_mulVg"><span class="id" title="lemma">extprod_mulVg</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">extprod_baseFinGroupType</span> :=<br/> + <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.BaseFinGroupType"><span class="id" title="abbreviation">BaseFinGroupType</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a>) <a class="idref" href="mathcomp.fingroup.gproduct.html#extprod_groupMixin"><span class="id" title="definition">extprod_groupMixin</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">prod_group</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.FinGroupType"><span class="id" title="abbreviation">FinGroupType</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#extprod_mulVg"><span class="id" title="lemma">extprod_mulVg</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="group_setX"><span class="id" title="lemma">group_setX</span></a> (<span class="id" title="var">H1</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">H2</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.fingroup.fingroup.html#group_set"><span class="id" title="definition">group_set</span></a> (<a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">setX_group</span> <span class="id" title="var">H1</span> <span class="id" title="var">H2</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#Group"><span class="id" title="constructor">Group</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#group_setX"><span class="id" title="lemma">group_setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="pairg1"><span class="id" title="definition">pairg1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="pair1g"><span class="id" title="definition">pair1g</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pairg1_morphM"><span class="id" title="lemma">pairg1_morphM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pairg1"><span class="id" title="definition">pairg1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">pairg1_morphism</span> := @<a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#in2W"><span class="id" title="lemma">in2W</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pairg1_morphM"><span class="id" title="lemma">pairg1_morphM</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pair1g_morphM"><span class="id" title="lemma">pair1g_morphM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pair1g"><span class="id" title="definition">pair1g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">pair1g_morphism</span> := @<a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#in2W"><span class="id" title="lemma">in2W</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pair1g_morphM"><span class="id" title="lemma">pair1g_morphM</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="fst_morphM"><span class="id" title="lemma">fst_morphM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">(</span></a>@<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#fst"><span class="id" title="definition">fst</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="snd_morphM"><span class="id" title="lemma">snd_morphM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">(</span></a>@<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#snd"><span class="id" title="definition">snd</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">fst_morphism</span> := @<a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#in2W"><span class="id" title="lemma">in2W</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#fst_morphM"><span class="id" title="lemma">fst_morphM</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">snd_morphism</span> := @<a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#in2W"><span class="id" title="lemma">in2W</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#snd_morphM"><span class="id" title="lemma">snd_morphM</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_pair1g"><span class="id" title="lemma">injm_pair1g</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pair1g"><span class="id" title="definition">pair1g</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_pairg1"><span class="id" title="lemma">injm_pairg1</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pairg1"><span class="id" title="definition">pairg1</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_pairg1"><span class="id" title="lemma">morphim_pairg1</span></a> (<span class="id" title="var">H1</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.fingroup.gproduct.html#pairg1"><span class="id" title="definition">pairg1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_pair1g"><span class="id" title="lemma">morphim_pair1g</span></a> (<span class="id" title="var">H2</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.fingroup.gproduct.html#pair1g"><span class="id" title="definition">pair1g</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> 1 <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_fstX"><span class="id" title="lemma">morphim_fstX</span></a> (<span class="id" title="var">H1</span>: <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">H2</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">of</span></a> <span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_sndX"><span class="id" title="lemma">morphim_sndX</span></a> (<span class="id" title="var">H1</span>: <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">H2</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) : <br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">of</span></a> <span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="setX_prod"><span class="id" title="lemma">setX_prod</span></a> (<span class="id" title="var">H1</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">H2</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) :<br/> + <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> 1 <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> 1 <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="setX_dprod"><span class="id" title="lemma">setX_dprod</span></a> (<span class="id" title="var">H1</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">H2</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) :<br/> + <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> 1 <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> 1 <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isog_setX1"><span class="id" title="lemma">isog_setX1</span></a> (<span class="id" title="var">H1</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.fingroup.morphism.html#isog"><span class="id" title="definition">isog</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> (<a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> 1).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isog_set1X"><span class="id" title="lemma">isog_set1X</span></a> (<span class="id" title="var">H2</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.fingroup.morphism.html#isog"><span class="id" title="definition">isog</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a> (<a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> 1 <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="setX_gen"><span class="id" title="lemma">setX_gen</span></a> (<span class="id" title="var">H1</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT1"><span class="id" title="variable">gT1</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">H2</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd.gT2"><span class="id" title="variable">gT2</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) :<br/> + 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">>></span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">>></span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">>></span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalDirProd"><span class="id" title="section">ExternalDirProd</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ExternalSDirProd"><span class="id" title="section">ExternalSDirProd</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="ExternalSDirProd.aT"><span class="id" title="variable">aT</span></a> <a name="ExternalSDirProd.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>) (<a name="ExternalSDirProd.D"><span class="id" title="variable">D</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="ExternalSDirProd.R"><span class="id" title="variable">R</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>).<br/> + +<br/> +</div> + +<div class="doc"> + The pair (a, x) denotes the product sdpair2 a * sdpair1 x +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Inductive</span> <a name="sdprod_by"><span class="id" title="inductive">sdprod_by</span></a> (<span class="id" title="var">to</span> : <a class="idref" href="mathcomp.fingroup.action.html#groupAction"><span class="id" title="record">groupAction</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.R"><span class="id" title="variable">R</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#predArgType"><span class="id" title="definition">predArgType</span></a> :=<br/> + <a name="SdPair"><span class="id" title="constructor">SdPair</span></a> (<span class="id" title="var">ax</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.rT"><span class="id" title="variable">rT</span></a>) <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#ax"><span class="id" title="variable">ax</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Coercion</span> <span class="id" title="var">pair_of_sd</span> <span class="id" title="var">to</span> (<span class="id" title="var">u</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_by"><span class="id" title="inductive">sdprod_by</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#to"><span class="id" title="variable">to</span></a>) := <span class="id" title="keyword">let</span>: <a class="idref" href="mathcomp.fingroup.gproduct.html#SdPair"><span class="id" title="constructor">SdPair</span></a> <span class="id" title="var">ax</span> <span class="id" title="var">_</span> := <a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a> <span class="id" title="tactic">in</span> <span class="id" title="var">ax</span>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="ExternalSDirProd.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#groupAction"><span class="id" title="record">groupAction</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="sdT"><span class="id" title="abbreviation">sdT</span></a> := (<a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_by"><span class="id" title="inductive">sdprod_by</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.to"><span class="id" title="variable">to</span></a>).<br/> +<span class="id" title="keyword">Notation</span> <a name="sdval"><span class="id" title="abbreviation">sdval</span></a> := (@<a class="idref" href="mathcomp.fingroup.gproduct.html#pair_of_sd"><span class="id" title="definition">pair_of_sd</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.to"><span class="id" title="variable">to</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sdprod_subType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#341c160c3e7b20d967b85d1852a7f89f"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#341c160c3e7b20d967b85d1852a7f89f"><span class="id" title="notation">subType</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#341c160c3e7b20d967b85d1852a7f89f"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdval"><span class="id" title="abbreviation">sdval</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#341c160c3e7b20d967b85d1852a7f89f"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="sdprod_eqMixin"><span class="id" title="definition">sdprod_eqMixin</span></a> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">eqMixin</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#4bc2d2dce12edef0fb9c71d4a902ae5d"><span class="id" title="notation"><:]</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sdprod_eqType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.eqtype.html#Equality.Exports.EqType"><span class="id" title="abbreviation">EqType</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_eqMixin"><span class="id" title="definition">sdprod_eqMixin</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="sdprod_choiceMixin"><span class="id" title="definition">sdprod_choiceMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">choiceMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#6c8b2d90ff1fbb8e9926bbf12495cb70"><span class="id" title="notation"><:]</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sdprod_choiceType</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#Choice.Exports.ChoiceType"><span class="id" title="abbreviation">ChoiceType</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_choiceMixin"><span class="id" title="definition">sdprod_choiceMixin</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="sdprod_countMixin"><span class="id" title="definition">sdprod_countMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">countMixin</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#99c739c8f4212f142296b27d3077c65e"><span class="id" title="notation"><:]</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sdprod_countType</span> := <a class="idref" href="mathcomp.ssreflect.choice.html#Countable.Exports.CountType"><span class="id" title="abbreviation">CountType</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_countMixin"><span class="id" title="definition">sdprod_countMixin</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sdprod_subCountType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.choice.html#8a7192fa64a42310658fd5be07ae4fcc"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#8a7192fa64a42310658fd5be07ae4fcc"><span class="id" title="notation">subCountType</span></a> <a class="idref" href="mathcomp.ssreflect.choice.html#8a7192fa64a42310658fd5be07ae4fcc"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a><a class="idref" href="mathcomp.ssreflect.choice.html#8a7192fa64a42310658fd5be07ae4fcc"><span class="id" title="notation">]</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="sdprod_finMixin"><span class="id" title="definition">sdprod_finMixin</span></a> := <a class="idref" href="mathcomp.ssreflect.fintype.html#8c180561768185dd10396a5d3615104a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#8c180561768185dd10396a5d3615104a"><span class="id" title="notation">finMixin</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#8c180561768185dd10396a5d3615104a"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#8c180561768185dd10396a5d3615104a"><span class="id" title="notation">by</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#8c180561768185dd10396a5d3615104a"><span class="id" title="notation"><:]</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sdprod_finType</span> := <a class="idref" href="mathcomp.ssreflect.fintype.html#Finite.Exports.FinType"><span class="id" title="abbreviation">FinType</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_finMixin"><span class="id" title="definition">sdprod_finMixin</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sdprod_subFinType</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#a701c7b60b4a16f07950761d8bf90924"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#a701c7b60b4a16f07950761d8bf90924"><span class="id" title="notation">subFinType</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#a701c7b60b4a16f07950761d8bf90924"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#a701c7b60b4a16f07950761d8bf90924"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="sdprod_one"><span class="id" title="definition">sdprod_one</span></a> := <a class="idref" href="mathcomp.fingroup.gproduct.html#SdPair"><span class="id" title="constructor">SdPair</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#group1"><span class="id" title="lemma">group1</span></a> <span class="id" title="var">_</span>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_inv_proof"><span class="id" title="lemma">sdprod_inv_proof</span></a> (<span class="id" title="var">u</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="sdprod_inv"><span class="id" title="definition">sdprod_inv</span></a> <span class="id" title="var">u</span> := <a class="idref" href="mathcomp.fingroup.gproduct.html#SdPair"><span class="id" title="constructor">SdPair</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_inv_proof"><span class="id" title="lemma">sdprod_inv_proof</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_mul_proof"><span class="id" title="lemma">sdprod_mul_proof</span></a> (<span class="id" title="var">u</span> <span class="id" title="var">v</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a>) :<br/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#v"><span class="id" title="variable">v</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="sdprod_mul"><span class="id" title="definition">sdprod_mul</span></a> <span class="id" title="var">u</span> <span class="id" title="var">v</span> := <a class="idref" href="mathcomp.fingroup.gproduct.html#SdPair"><span class="id" title="constructor">SdPair</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.to"><span class="id" title="variable">to</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_mul_proof"><span class="id" title="lemma">sdprod_mul_proof</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#v"><span class="id" title="variable">v</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_mul1g"><span class="id" title="lemma">sdprod_mul1g</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_id"><span class="id" title="definition">left_id</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_one"><span class="id" title="definition">sdprod_one</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_mul"><span class="id" title="definition">sdprod_mul</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_mulVg"><span class="id" title="lemma">sdprod_mulVg</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#left_inverse"><span class="id" title="definition">left_inverse</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_one"><span class="id" title="definition">sdprod_one</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_inv"><span class="id" title="definition">sdprod_inv</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_mul"><span class="id" title="definition">sdprod_mul</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_mulgA"><span class="id" title="lemma">sdprod_mulgA</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#associative"><span class="id" title="definition">associative</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_mul"><span class="id" title="definition">sdprod_mul</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sdprod_groupMixin</span> :=<br/> + <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Mixin"><span class="id" title="definition">FinGroup.Mixin</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_mulgA"><span class="id" title="lemma">sdprod_mulgA</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_mul1g"><span class="id" title="lemma">sdprod_mul1g</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_mulVg"><span class="id" title="lemma">sdprod_mulVg</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sdprod_baseFinGroupType</span> :=<br/> + <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.BaseFinGroupType"><span class="id" title="abbreviation">BaseFinGroupType</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_groupMixin"><span class="id" title="definition">sdprod_groupMixin</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sdprod_groupType</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.FinGroupType"><span class="id" title="abbreviation">FinGroupType</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_mulVg"><span class="id" title="lemma">sdprod_mulVg</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="sdpair1"><span class="id" title="definition">sdpair1</span></a> <span class="id" title="var">x</span> := <a class="idref" href="mathcomp.ssreflect.eqtype.html#insubd"><span class="id" title="definition">insubd</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_one"><span class="id" title="definition">sdprod_one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a>.<br/> +<span class="id" title="keyword">Definition</span> <a name="sdpair2"><span class="id" title="definition">sdpair2</span></a> <span class="id" title="var">a</span> := <a class="idref" href="mathcomp.ssreflect.eqtype.html#insubd"><span class="id" title="definition">insubd</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_one"><span class="id" title="definition">sdprod_one</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdpair1_morphM"><span class="id" title="lemma">sdpair1_morphM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdpair2_morphM"><span class="id" title="lemma">sdpair2_morphM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sdpair1_morphism</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1_morphM"><span class="id" title="lemma">sdpair1_morphM</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sdpair2_morphism</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2_morphM"><span class="id" title="lemma">sdpair2_morphM</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_sdpair1"><span class="id" title="lemma">injm_sdpair1</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_sdpair2"><span class="id" title="lemma">injm_sdpair2</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdpairE"><span class="id" title="lemma">sdpairE</span></a> (<span class="id" title="var">u</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#sdT"><span class="id" title="abbreviation">sdT</span></a>) : <a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c4877bbfe60d8f22b47ac99ace86216a"><span class="id" title="notation">.1</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#u"><span class="id" title="variable">u</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#f4827404159513e7fd691b60b7877737"><span class="id" title="notation">.2</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdpair_act"><span class="id" title="lemma">sdpair_act</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.D"><span class="id" title="variable">D</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a><br/> + <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">a</span>, <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#808c6b8e35e792f23899f360a21e4638"><span class="id" title="notation">^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdpair_setact"><span class="id" title="lemma">sdpair_setact</span></a> (<span class="id" title="var">G</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.R"><span class="id" title="variable">R</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.to"><span class="id" title="variable">to</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#8f28bbd804547edd8de802d63ef85617"><span class="id" title="notation">^~</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_sdpair_norm"><span class="id" title="lemma">im_sdpair_norm</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_sdpair_TI"><span class="id" title="lemma">im_sdpair_TI</span></a> : <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.D"><span class="id" title="variable">D</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_sdpair"><span class="id" title="lemma">im_sdpair</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.D"><span class="id" title="variable">D</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprod_sdpair"><span class="id" title="lemma">sdprod_sdpair</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="ExternalSDirProd.A"><span class="id" title="variable">A</span></a> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="ExternalSDirProd.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="gacentEsd"><span class="id" title="lemma">gacentEsd</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">(|</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#e2bbaaa2e35f8e6b75a00840566c57d7"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Hypotheses</span> (<a name="ExternalSDirProd.sAD"><span class="id" title="variable">sAD</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.D"><span class="id" title="variable">D</span></a>) (<a name="ExternalSDirProd.sGR"><span class="id" title="variable">sGR</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.R"><span class="id" title="variable">R</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="astabEsd"><span class="id" title="lemma">astabEsd</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#99d10685ba0de4584ba3a66908e81722"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="astabsEsd"><span class="id" title="lemma">astabsEsd</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#f49be8276ad4d3d8f37fd1509306940d"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="actsEsd"><span class="id" title="lemma">actsEsd</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">acts</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">on</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.to"><span class="id" title="variable">to</span></a><a class="idref" href="mathcomp.fingroup.action.html#4fabb88b896fd67dca370ff89a430b72"><span class="id" title="notation">]</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExternalSDirProd"><span class="id" title="section">ExternalSDirProd</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ProdMorph"><span class="id" title="section">ProdMorph</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="ProdMorph.gT"><span class="id" title="variable">gT</span></a> <a name="ProdMorph.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> <span class="id" title="var">K</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">C</span> <span class="id" title="var">D</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">L</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ProdMorph.defs"><span class="id" title="section">defs</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="ProdMorph.defs.A"><span class="id" title="variable">A</span></a> <a name="ProdMorph.defs.B"><span class="id" title="variable">B</span></a> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) (<a name="ProdMorph.defs.fA"><span class="id" title="variable">fA</span></a> <a name="ProdMorph.defs.fB"><span class="id" title="variable">fB</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.sort"><span class="id" title="projection">FinGroup.sort</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.rT"><span class="id" title="variable">rT</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="pprodm"><span class="id" title="definition">pprodm</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a> & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#morph_act"><span class="id" title="definition">morph_act</span></a> <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.fA"><span class="id" title="variable">fA</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.fB"><span class="id" title="variable">fB</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a><br/> + & <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.B"><span class="id" title="variable">B</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.fA"><span class="id" title="variable">fA</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.fB"><span class="id" title="variable">fB</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a> :=<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.fA"><span class="id" title="variable">fA</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#divgr"><span class="id" title="definition">divgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.fB"><span class="id" title="variable">fB</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#remgr"><span class="id" title="definition">remgr</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs.B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a>).<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.defs"><span class="id" title="section">defs</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ProdMorph.Props"><span class="id" title="section">Props</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="ProdMorph.Props.H"><span class="id" title="variable">H</span></a> <a name="ProdMorph.Props.K"><span class="id" title="variable">K</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> +<span class="id" title="keyword">Variables</span> (<a name="ProdMorph.Props.fH"><span class="id" title="variable">fH</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>) (<a name="ProdMorph.Props.fK"><span class="id" title="variable">fK</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="ProdMorph.Props.nHK"><span class="id" title="variable">nHK</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="ProdMorph.Props.actf"><span class="id" title="variable">actf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#morph_act"><span class="id" title="definition">morph_act</span></a> <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fK"><span class="id" title="variable">fK</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="ProdMorph.Props.eqfHK"><span class="id" title="variable">eqfHK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fK"><span class="id" title="variable">fK</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pprodmE"><span class="id" title="lemma">pprodmE</span></a> <span class="id" title="var">x</span> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#f"><span class="id" title="abbreviation">f</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pprodmEl"><span class="id" title="lemma">pprodmEl</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.H"><span class="id" title="variable">H</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fH"><span class="id" title="variable">fH</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pprodmEr"><span class="id" title="lemma">pprodmEr</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fK"><span class="id" title="variable">fK</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="pprodmM"><span class="id" title="lemma">pprodmM</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">&,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#f"><span class="id" title="abbreviation">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#3014e73af2a90fd800d8681479d76336"><span class="id" title="notation">}</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">pprodm_morphism</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#pprodmM"><span class="id" title="lemma">pprodmM</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_pprodm"><span class="id" title="lemma">morphim_pprodm</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_pprodml"><span class="id" title="lemma">morphim_pprodml</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_pprodmr"><span class="id" title="lemma">morphim_pprodmr</span></a> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_pprodm"><span class="id" title="lemma">ker_pprodm</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">ker</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">|</span></a> <span class="id" title="var">x</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">,</span></a> <span class="id" title="var">a</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_pprodm"><span class="id" title="lemma">injm_pprodm</span></a> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#f"><span class="id" title="abbreviation">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fH"><span class="id" title="variable">fH</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props.K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Props"><span class="id" title="section">Props</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ProdMorph.Sdprodm"><span class="id" title="section">Sdprodm</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a> <a name="ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a> <a name="ProdMorph.Sdprodm.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> +<span class="id" title="keyword">Variables</span> (<a name="ProdMorph.Sdprodm.fH"><span class="id" title="variable">fH</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>) (<a name="ProdMorph.Sdprodm.fK"><span class="id" title="variable">fK</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="ProdMorph.Sdprodm.eqHK_G"><span class="id" title="variable">eqHK_G</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ff5a974c523b8d4c8927273818a26a02"><span class="id" title="notation">><|</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.G"><span class="id" title="variable">G</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="ProdMorph.Sdprodm.actf"><span class="id" title="variable">actf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#morph_act"><span class="id" title="definition">morph_act</span></a> <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fK"><span class="id" title="variable">fK</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodm_norm"><span class="id" title="lemma">sdprodm_norm</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodm_sub"><span class="id" title="lemma">sdprodm_sub</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodm_eqf"><span class="id" title="lemma">sdprodm_eqf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fK"><span class="id" title="variable">fK</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="sdprodm"><span class="id" title="definition">sdprodm</span></a> :=<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#restrm"><span class="id" title="definition">restrm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm_sub"><span class="id" title="lemma">sdprodm_sub</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#pprodm"><span class="id" title="definition">pprodm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm_norm"><span class="id" title="lemma">sdprodm_norm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.actf"><span class="id" title="variable">actf</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm_eqf"><span class="id" title="lemma">sdprodm_eqf</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">sdprodm_morphism</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm"><span class="id" title="definition">sdprodm</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodmE"><span class="id" title="lemma">sdprodmE</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm"><span class="id" title="definition">sdprodm</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodmEl"><span class="id" title="lemma">sdprodmEl</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm"><span class="id" title="definition">sdprodm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sdprodmEr"><span class="id" title="lemma">sdprodmEr</span></a> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm"><span class="id" title="definition">sdprodm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_sdprodm"><span class="id" title="lemma">morphim_sdprodm</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm"><span class="id" title="definition">sdprodm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_sdprodm"><span class="id" title="lemma">im_sdprodm</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm"><span class="id" title="definition">sdprodm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_sdprodml"><span class="id" title="lemma">morphim_sdprodml</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm"><span class="id" title="definition">sdprodm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_sdprodmr"><span class="id" title="lemma">morphim_sdprodmr</span></a> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm"><span class="id" title="definition">sdprodm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_sdprodm"><span class="id" title="lemma">ker_sdprodm</span></a> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">ker</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm"><span class="id" title="definition">sdprodm</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">|</span></a> <span class="id" title="var">a</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_sdprodm"><span class="id" title="lemma">injm_sdprodm</span></a> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm"><span class="id" title="definition">sdprodm</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fH"><span class="id" title="variable">fH</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Sdprodm"><span class="id" title="section">Sdprodm</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ProdMorph.Cprodm"><span class="id" title="section">Cprodm</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a> <a name="ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a> <a name="ProdMorph.Cprodm.G"><span class="id" title="variable">G</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> +<span class="id" title="keyword">Variables</span> (<a name="ProdMorph.Cprodm.fH"><span class="id" title="variable">fH</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>) (<a name="ProdMorph.Cprodm.fK"><span class="id" title="variable">fK</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="ProdMorph.Cprodm.eqHK_G"><span class="id" title="variable">eqHK_G</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.G"><span class="id" title="variable">G</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="ProdMorph.Cprodm.cfHK"><span class="id" title="variable">cfHK</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="ProdMorph.Cprodm.eqfHK"><span class="id" title="variable">eqfHK</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fK"><span class="id" title="variable">fK</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodm_norm"><span class="id" title="lemma">cprodm_norm</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">N</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#3cae19671031307d430e5b14ccbd1058"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodm_sub"><span class="id" title="lemma">cprodm_sub</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#80208730563aa86aa7861f6fe1b846da"><span class="id" title="notation"><*></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodm_actf"><span class="id" title="lemma">cprodm_actf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#morph_act"><span class="id" title="definition">morph_act</span></a> <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fK"><span class="id" title="variable">fK</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="cprodm"><span class="id" title="definition">cprodm</span></a> := <a class="idref" href="mathcomp.fingroup.morphism.html#restrm"><span class="id" title="definition">restrm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm_sub"><span class="id" title="lemma">cprodm_sub</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#pprodm"><span class="id" title="definition">pprodm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm_norm"><span class="id" title="lemma">cprodm_norm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm_actf"><span class="id" title="lemma">cprodm_actf</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.eqfHK"><span class="id" title="variable">eqfHK</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">cprodm_morphism</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm"><span class="id" title="definition">cprodm</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodmE"><span class="id" title="lemma">cprodmE</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm"><span class="id" title="definition">cprodm</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodmEl"><span class="id" title="lemma">cprodmEl</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm"><span class="id" title="definition">cprodm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="cprodmEr"><span class="id" title="lemma">cprodmEr</span></a> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm"><span class="id" title="definition">cprodm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_cprodm"><span class="id" title="lemma">morphim_cprodm</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm"><span class="id" title="definition">cprodm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_cprodm"><span class="id" title="lemma">im_cprodm</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm"><span class="id" title="definition">cprodm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_cprodml"><span class="id" title="lemma">morphim_cprodml</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm"><span class="id" title="definition">cprodm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_cprodmr"><span class="id" title="lemma">morphim_cprodmr</span></a> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm"><span class="id" title="definition">cprodm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_cprodm"><span class="id" title="lemma">ker_cprodm</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">ker</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm"><span class="id" title="definition">cprodm</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">|</span></a> <span class="id" title="var">a</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_cprodm"><span class="id" title="lemma">injm_cprodm</span></a> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm"><span class="id" title="definition">cprodm</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fH"><span class="id" title="variable">fH</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm.K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Cprodm"><span class="id" title="section">Cprodm</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ProdMorph.Dprodm"><span class="id" title="section">Dprodm</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="ProdMorph.Dprodm.G"><span class="id" title="variable">G</span></a> <a name="ProdMorph.Dprodm.H"><span class="id" title="variable">H</span></a> <a name="ProdMorph.Dprodm.K"><span class="id" title="variable">K</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> +<span class="id" title="keyword">Variables</span> (<a name="ProdMorph.Dprodm.fH"><span class="id" title="variable">fH</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>) (<a name="ProdMorph.Dprodm.fK"><span class="id" title="variable">fK</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="ProdMorph.Dprodm.eqHK_G"><span class="id" title="variable">eqHK_G</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.G"><span class="id" title="variable">G</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="ProdMorph.Dprodm.cfHK"><span class="id" title="variable">cfHK</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#67c26168baa7671aab03da2a0fb7dafa"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodm_cprod"><span class="id" title="lemma">dprodm_cprod</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#9607c0b7b0a7e59f4327b220d5a93330"><span class="id" title="notation">\*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodm_eqf"><span class="id" title="lemma">dprodm_eqf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#2500d48ed8e862ccfda98a44dff88963"><span class="id" title="notation">=1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fK"><span class="id" title="variable">fK</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="dprodm"><span class="id" title="definition">dprodm</span></a> := <a class="idref" href="mathcomp.fingroup.gproduct.html#cprodm"><span class="id" title="definition">cprodm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodm_cprod"><span class="id" title="lemma">dprodm_cprod</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.cfHK"><span class="id" title="variable">cfHK</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodm_eqf"><span class="id" title="lemma">dprodm_eqf</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">dprodm_morphism</span> := <span class="id" title="keyword">Eval</span> <span class="id" title="tactic">hnf</span> <span class="id" title="tactic">in</span> <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodm"><span class="id" title="definition">dprodm</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodmE"><span class="id" title="lemma">dprodmE</span></a> <span class="id" title="var">a</span> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodm"><span class="id" title="definition">dprodm</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodmEl"><span class="id" title="lemma">dprodmEl</span></a> <span class="id" title="var">a</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodm"><span class="id" title="definition">dprodm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dprodmEr"><span class="id" title="lemma">dprodmEr</span></a> <span class="id" title="var">b</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#46c9e8232fa09401e24f1934bb65029f"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodm"><span class="id" title="definition">dprodm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_dprodm"><span class="id" title="lemma">morphim_dprodm</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodm"><span class="id" title="definition">dprodm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_dprodm"><span class="id" title="lemma">im_dprodm</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodm"><span class="id" title="definition">dprodm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.K"><span class="id" title="variable">K</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_dprodml"><span class="id" title="lemma">morphim_dprodml</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodm"><span class="id" title="definition">dprodm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_dprodmr"><span class="id" title="lemma">morphim_dprodmr</span></a> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodm"><span class="id" title="definition">dprodm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_dprodm"><span class="id" title="lemma">ker_dprodm</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">ker</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodm"><span class="id" title="definition">dprodm</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">^-1</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">|</span></a> <span class="id" title="var">a</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.H"><span class="id" title="variable">H</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">,</span></a> <span class="id" title="var">b</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#cbd91f75d319d6d9a2d587d4cdfd1a36"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_dprodm"><span class="id" title="lemma">injm_dprodm</span></a> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#dprodm"><span class="id" title="definition">dprodm</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fH"><span class="id" title="variable">fH</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph.Dprodm"><span class="id" title="section">Dprodm</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isog_dprod"><span class="id" title="lemma">isog_dprod</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> <span class="id" title="var">G</span> <span class="id" title="var">C</span> <span class="id" title="var">D</span> <span class="id" title="var">L</span> :<br/> + <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#L"><span class="id" title="variable">L</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#isog"><span class="id" title="definition">isog</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#isog"><span class="id" title="definition">isog</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#isog"><span class="id" title="definition">isog</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#L"><span class="id" title="variable">L</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#ProdMorph"><span class="id" title="section">ProdMorph</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ExtSdprodm"><span class="id" title="section">ExtSdprodm</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="ExtSdprodm.gT"><span class="id" title="variable">gT</span></a> <a name="ExtSdprodm.aT"><span class="id" title="variable">aT</span></a> <a name="ExtSdprodm.rT"><span class="id" title="variable">rT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> +<span class="id" title="keyword">Variables</span> (<a name="ExtSdprodm.H"><span class="id" title="variable">H</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="ExtSdprodm.K"><span class="id" title="variable">K</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="ExtSdprodm.to"><span class="id" title="variable">to</span></a> : <a class="idref" href="mathcomp.fingroup.action.html#groupAction"><span class="id" title="record">groupAction</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H"><span class="id" title="variable">H</span></a>).<br/> +<span class="id" title="keyword">Variables</span> (<a name="ExtSdprodm.fH"><span class="id" title="variable">fH</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>) (<a name="ExtSdprodm.fK"><span class="id" title="variable">fK</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">>-></span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="ExtSdprodm.actf"><span class="id" title="variable">actf</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.K"><span class="id" title="variable">K</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#morph_act"><span class="id" title="definition">morph_act</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.fK"><span class="id" title="variable">fK</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Let</span> <a name="ExtSdprodm.DgH"><span class="id" title="variable">DgH</span></a> := <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair1"><span class="id" title="definition">sdpair1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.H"><span class="id" title="variable">H</span></a>.<br/> +<span class="id" title="keyword">Let</span> <a name="ExtSdprodm.DgK"><span class="id" title="variable">DgK</span></a> := <a class="idref" href="mathcomp.fingroup.gproduct.html#sdpair2"><span class="id" title="definition">sdpair2</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.to"><span class="id" title="variable">to</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.K"><span class="id" title="variable">K</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="xsdprodm_dom1"><span class="id" title="lemma">xsdprodm_dom1</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.DgH"><span class="id" title="variable">DgH</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f47bac414ebbcaae273350b5c84ff313"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#f47bac414ebbcaae273350b5c84ff313"><span class="id" title="notation">dom</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#fsH"><span class="id" title="abbreviation">fsH</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="xsdprodm_dom2"><span class="id" title="lemma">xsdprodm_dom2</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.DgK"><span class="id" title="variable">DgK</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#826eae8d7598a787ea56f4249e6e210e"><span class="id" title="notation">subset</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f47bac414ebbcaae273350b5c84ff313"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#f47bac414ebbcaae273350b5c84ff313"><span class="id" title="notation">dom</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#fsK"><span class="id" title="abbreviation">fsK</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_sdprodm1"><span class="id" title="lemma">im_sdprodm1</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#gH"><span class="id" title="abbreviation">gH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.DgH"><span class="id" title="variable">DgH</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_sdprodm2"><span class="id" title="lemma">im_sdprodm2</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#gK"><span class="id" title="abbreviation">gK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.DgK"><span class="id" title="variable">DgK</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.K"><span class="id" title="variable">K</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="xsdprodm_act"><span class="id" title="lemma">xsdprodm_act</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.DgH"><span class="id" title="variable">DgH</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.DgK"><span class="id" title="variable">DgK</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.action.html#morph_act"><span class="id" title="definition">morph_act</span></a> <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.action.html#3d89e389df5b2c87554b2b824e5fe7d1"><span class="id" title="notation">J</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#gH"><span class="id" title="abbreviation">gH</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#gK"><span class="id" title="abbreviation">gK</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#208bc995000a6307bdbc043c43919d97"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="xsdprodm"><span class="id" title="definition">xsdprodm</span></a> := <a class="idref" href="mathcomp.fingroup.gproduct.html#sdprodm"><span class="id" title="definition">sdprodm</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#sdprod_sdpair"><span class="id" title="lemma">sdprod_sdpair</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.to"><span class="id" title="variable">to</span></a>) <a class="idref" href="mathcomp.fingroup.gproduct.html#xsdprodm_act"><span class="id" title="lemma">xsdprodm_act</span></a>.<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">xsdprod_morphism</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">morphism</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#xsdprodm"><span class="id" title="definition">xsdprodm</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_xsdprodm"><span class="id" title="lemma">im_xsdprodm</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#xsdprodm"><span class="id" title="definition">xsdprodm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setT"><span class="id" title="abbreviation">setT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.K"><span class="id" title="variable">K</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_xsdprodm"><span class="id" title="lemma">injm_xsdprodm</span></a> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#xsdprodm"><span class="id" title="definition">xsdprodm</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">[&&</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.fH"><span class="id" title="variable">fH</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">injm</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.fH"><span class="id" title="variable">fH</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.fK"><span class="id" title="variable">fK</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm.K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#2e9317c5f71a1305fb695cdc49716482"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#ExtSdprodm"><span class="id" title="section">ExtSdprodm</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="DirprodIsom"><span class="id" title="section">DirprodIsom</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="DirprodIsom.gT"><span class="id" title="variable">gT</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#FinGroup.Exports.finGroupType"><span class="id" title="abbreviation">finGroupType</span></a>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">G</span> <span class="id" title="var">H</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#DirprodIsom.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="mulgm"><span class="id" title="definition">mulgm</span></a> : <a class="idref" href="mathcomp.fingroup.gproduct.html#DirprodIsom.gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#DirprodIsom.gT"><span class="id" title="variable">gT</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <span class="id" title="var">_</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Datatypes.html#prod_curry"><span class="id" title="definition">prod_curry</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#mulg"><span class="id" title="definition">mulg</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="imset_mulgm"><span class="id" title="lemma">imset_mulgm</span></a> (<span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#DirprodIsom.gT"><span class="id" title="variable">gT</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#0fec877de6d09ef39abb9b599a84eb0e"><span class="id" title="notation">}</span></a>) : <a class="idref" href="mathcomp.fingroup.gproduct.html#mulgm"><span class="id" title="definition">mulgm</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="mulgmP"><span class="id" title="lemma">mulgmP</span></a> <span class="id" title="var">H1</span> <span class="id" title="var">H2</span> <span class="id" title="var">G</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#reflect"><span class="id" title="abbreviation">reflect</span></a> (<a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.gproduct.html#3733c0e43956ad2062ab5f1e57ceb9a8"><span class="id" title="notation">x</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.fingroup.morphism.html#misom"><span class="id" title="definition">misom</span></a> (<a class="idref" href="mathcomp.ssreflect.finset.html#setX"><span class="id" title="definition">setX</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H1"><span class="id" title="variable">H1</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#H2"><span class="id" title="variable">H2</span></a>) <a class="idref" href="mathcomp.fingroup.gproduct.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.gproduct.html#mulgm"><span class="id" title="definition">mulgm</span></a>).<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.gproduct.html#DirprodIsom"><span class="id" title="section">DirprodIsom</span></a>.<br/> + +<br/> +</div> +</div> + +<div id="footer"> +<hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a> +</div> + +</div> + +</body> +</html>
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