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diff --git a/docs/htmldoc/mathcomp.fingroup.morphism.html b/docs/htmldoc/mathcomp.fingroup.morphism.html new file mode 100644 index 0000000..8c60de7 --- /dev/null +++ b/docs/htmldoc/mathcomp.fingroup.morphism.html @@ -0,0 +1,1422 @@ +<!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.morphism</title> +</head> + +<body> + +<div id="page"> + +<div id="header"> +</div> + +<div id="main"> + +<h1 class="libtitle">Library mathcomp.fingroup.morphism</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"> + This file contains the definitions of: + +<div class="paragraph"> </div> + + {morphism D >-> rT} == + the structure type of functions that are group morphisms mapping a + domain set D : {set aT} to a type rT; rT must have a finGroupType + structure, and D is usually a group (most of the theory expects this). + mfun == the coercion projecting {morphism D >-> rT} to aT -> rT + +<div class="paragraph"> </div> + + Basic examples: + idm D == the identity morphism with domain D, or more precisely + the identity function, but with a canonical + {morphism G -> gT} structure. + trivm D == the trivial morphism with domain D. + If f has a {morphism D >-> rT} structure + 'dom f == D, the domain of f. + f @* A == the image of A by f, where f is defined. + := f @: (D :&: A) + f @*^-1 R == the pre-image of R by f, where f is defined. + := D :&: f @^-1: R + 'ker f == the kernel of f. + := f @*^-1 1 + 'ker_G f == the kernel of f restricted to G. + := G :&: 'ker f (this is a pure notation) + 'injm f <=> f injective on D. + <-> ker f \subset 1 (this is a pure notation) + invm injf == the inverse morphism of f, with domain f @* D, when f + is injective (injf : 'injm f). + restrm f sDom == the restriction of f to a subset A of D, given + (sDom : A \subset D); restrm f sDom is transparently + identical to f; the restrmP and domP lemmas provide + opaque restrictions. + invm f infj == the inverse morphism for an injective f, with domain + f @* D, given (injf : 'injm f). + +<div class="paragraph"> </div> + + G \isog H <=> G and H are isomorphic as groups. + H \homg G <=> H is a homomorphic image of G. + isom G H f <=> f maps G isomorphically to H, provided D contains G. + := f @: G^# == H^# + +<div class="paragraph"> </div> + + If, moreover, g : {morphism G >-> gT} with G : {group aT}, + factm sKer sDom == the (natural) factor morphism mapping f @* G to g @* G + with sDom : G \subset D, sKer : 'ker f \subset 'ker g. + ifactm injf g == the (natural) factor morphism mapping f @* G to g @* G + when f is injective (injf : 'injm f); here g must + denote an actual morphism structure, not its function + projection. + +<div class="paragraph"> </div> + + If g has a {morphism G >-> aT} structure for any G : {group gT}, then + f \o g has a canonical {morphism g @*^-1 D >-> rT} structure. + +<div class="paragraph"> </div> + + Finally, for an arbitrary function f : aT -> rT + morphic D f <=> f preserves group multiplication in D, i.e., + f (x * y) = (f x) * (f y) for all x, y in D. + morphm fM == a function identical to f, but with a canonical + {morphism D >-> rT} structure, given fM : morphic D f. + misom D C f <=> f is a morphism that maps D isomorphically to C. + := morphic D f && isom D C f +</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">Reserved Notation</span> "x \isog y" (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70).<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="MorphismStructure"><span class="id" title="section">MorphismStructure</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="MorphismStructure.aT"><span class="id" title="variable">aT</span></a> <a name="MorphismStructure.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/> + +<br/> +<span class="id" title="keyword">Structure</span> <a name="morphism"><span class="id" title="record">morphism</span></a> (<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.morphism.html#MorphismStructure.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>) : <span class="id" title="keyword">Type</span> := <a name="Morphism"><span class="id" title="constructor">Morphism</span></a> {<br/> + <a name="mfun"><span class="id" title="projection">mfun</span></a> :> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure.aT"><span class="id" title="variable">aT</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.morphism.html#MorphismStructure.rT"><span class="id" title="variable">rT</span></a>;<br/> + <span class="id" title="var">_</span> : <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.morphism.html#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.morphism.html#mfun"><span class="id" title="method">mfun</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.morphism.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.morphism.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/> + +<br/> +</div> + +<div class="doc"> + We give the 'lightest' possible specification to define morphisms: local + congruence, in D, with the group law of aT. We then provide the properties + for the 'textbook' notion of morphism, when the required structures are + available (e.g. its domain is a group). +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="morphism_for"><span class="id" title="definition">morphism_for</span></a> <span class="id" title="var">D</span> <span class="id" title="keyword">of</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#phant"><span class="id" title="inductive">phant</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure.rT"><span class="id" title="variable">rT</span></a> := <a class="idref" href="mathcomp.fingroup.morphism.html#morphism"><span class="id" title="record">morphism</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#D"><span class="id" title="variable">D</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="clone_morphism"><span class="id" title="definition">clone_morphism</span></a> <span class="id" title="var">D</span> <span class="id" title="var">f</span> :=<br/> + <span class="id" title="keyword">let</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">fM</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</span></a><br/> + <span class="id" title="keyword">return</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><span class="id" title="notation">type</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><span class="id" title="notation">of</span></a> @<a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><span class="id" title="notation">for</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#d7f4c4dc11bb6b98f0e78c197b6ba346"><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.morphism.html#morphism_for"><span class="id" title="definition">morphism_for</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#D"><span class="id" title="variable">D</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#Phant"><span class="id" title="constructor">Phant</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure.rT"><span class="id" title="variable">rT</span></a>)<br/> + <span class="id" title="tactic">in</span> <span class="id" title="keyword">fun</span> <span class="id" title="var">k</span> ⇒ <a class="idref" href="mathcomp.fingroup.morphism.html#k"><span class="id" title="variable">k</span></a> <span class="id" title="var">fM</span>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="MorphismStructure.D"><span class="id" title="variable">D</span></a> <a name="MorphismStructure.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.morphism.html#MorphismStructure.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="MorphismStructure.R"><span class="id" title="variable">R</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.morphism.html#MorphismStructure.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>) (<a name="MorphismStructure.x"><span class="id" title="variable">x</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure.aT"><span class="id" title="variable">aT</span></a>) (<a name="MorphismStructure.y"><span class="id" title="variable">y</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure.rT"><span class="id" title="variable">rT</span></a>) (<a name="MorphismStructure.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure.aT"><span class="id" title="variable">aT</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#MorphismStructure.rT"><span class="id" title="variable">rT</span></a>).<br/> + +<br/> +<span class="id" title="keyword">CoInductive</span> <a name="morphim_spec"><span class="id" title="inductive">morphim_spec</span></a> : <span class="id" title="keyword">Prop</span> := <a name="MorphimSpec"><span class="id" title="constructor">MorphimSpec</span></a> <span class="id" title="var">z</span> & <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismStructure.D"><span class="id" title="variable">D</span></a> & <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismStructure.A"><span class="id" title="variable">A</span></a> & <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure.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.morphism.html#MorphismStructure.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#z"><span class="id" title="variable">z</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimP"><span class="id" title="lemma">morphimP</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.morphism.html#morphim_spec"><span class="id" title="inductive">morphim_spec</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure.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.morphism.html#MorphismStructure.f"><span class="id" title="variable">f</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#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure.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="mathcomp.fingroup.morphism.html#MorphismStructure.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>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreP"><span class="id" title="lemma">morphpreP</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.morphism.html#MorphismStructure.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.morphism.html#MorphismStructure.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#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure.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.morphism.html#MorphismStructure.R"><span class="id" title="variable">R</span></a>) (<a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure.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.morphism.html#MorphismStructure.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="mathcomp.fingroup.morphism.html#MorphismStructure.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#aa0c936d7feb8f26825e601735c0125f"><span class="id" title="notation">@^-1:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure.R"><span class="id" title="variable">R</span></a>).<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismStructure"><span class="id" title="section">MorphismStructure</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">"</span></a>{ 'morphism' D >-> T }" := (<a class="idref" href="mathcomp.fingroup.morphism.html#morphism_for"><span class="id" title="definition">morphism_for</span></a> <span class="id" title="var">D</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#Phant"><span class="id" title="constructor">Phant</span></a> <span class="id" title="var">T</span>))<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "{ 'morphism' D >-> T }") : <span class="id" title="var">group_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="fded068ead41ff834e75b45b29ee30bf"><span class="id" title="notation">"</span></a>[ 'morphism' D 'of' f ]" :=<br/> + (@<a class="idref" href="mathcomp.fingroup.morphism.html#clone_morphism"><span class="id" title="definition">clone_morphism</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">D</span> <span class="id" title="var">_</span> (<span class="id" title="keyword">fun</span> <span class="id" title="var">fM</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> <span class="id" title="var">D</span> <span class="id" title="var">f</span> <a class="idref" href="mathcomp.fingroup.morphism.html#fM"><span class="id" title="variable">fM</span></a>))<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'morphism' D 'of' f ]") : <span class="id" title="var">form_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="4638420a8c497f6fdfbc01376756a30a"><span class="id" title="notation">"</span></a>[ 'morphism' 'of' f ]" := (<a class="idref" href="mathcomp.fingroup.morphism.html#clone_morphism"><span class="id" title="definition">clone_morphism</span></a> (@<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> <span class="id" title="var">_</span> <span class="id" title="var">f</span>))<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 0, <span class="id" title="var">format</span> "[ 'morphism' 'of' f ]") : <span class="id" title="var">form_scope</span>.<br/> + +<br/> + +<br/> +</div> + +<div class="doc"> + Domain, image, preimage, kernel, using phantom types to infer the domain. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Section</span> <a name="MorphismOps1"><span class="id" title="section">MorphismOps1</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="MorphismOps1.aT"><span class="id" title="variable">aT</span></a> <a name="MorphismOps1.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="MorphismOps1.D"><span class="id" title="variable">D</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.morphism.html#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="MorphismOps1.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.morphism.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.morphism.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">Lemma</span> <a name="morphM"><span class="id" title="lemma">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.morphism.html#MorphismOps1.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.morphism.html#MorphismOps1.f"><span class="id" title="variable">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.morphism.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.morphism.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">Notation</span> <a name="morPhantom"><span class="id" title="abbreviation">morPhantom</span></a> := (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#phantom"><span class="id" title="inductive">phantom</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorphismOps1.aT"><span class="id" title="variable">aT</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#MorphismOps1.rT"><span class="id" title="variable">rT</span></a>)).<br/> +<span class="id" title="keyword">Definition</span> <a name="MorPhantom"><span class="id" title="definition">MorPhantom</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssreflect.html#Phantom"><span class="id" title="constructor">Phantom</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorphismOps1.aT"><span class="id" title="variable">aT</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#MorphismOps1.rT"><span class="id" title="variable">rT</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="dom"><span class="id" title="definition">dom</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.fingroup.morphism.html#morPhantom"><span class="id" title="abbreviation">morPhantom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismOps1.f"><span class="id" title="variable">f</span></a> := <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismOps1.D"><span class="id" title="variable">D</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="morphim"><span class="id" title="definition">morphim</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.fingroup.morphism.html#morPhantom"><span class="id" title="abbreviation">morPhantom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismOps1.f"><span class="id" title="variable">f</span></a> := <span class="id" title="keyword">fun</span> <span class="id" title="var">A</span> ⇒ <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismOps1.f"><span class="id" title="variable">f</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#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismOps1.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="mathcomp.fingroup.morphism.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>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="morphpre"><span class="id" title="definition">morphpre</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.fingroup.morphism.html#morPhantom"><span class="id" title="abbreviation">morPhantom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismOps1.f"><span class="id" title="variable">f</span></a> := <span class="id" title="keyword">fun</span> <span class="id" title="var">R</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.morphism.html#MorphismOps1.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> ⇒ <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismOps1.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="mathcomp.fingroup.morphism.html#MorphismOps1.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#aa0c936d7feb8f26825e601735c0125f"><span class="id" title="notation">@^-1:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="ker"><span class="id" title="definition">ker</span></a> <span class="id" title="var">mph</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#morphpre"><span class="id" title="definition">morphpre</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#mph"><span class="id" title="variable">mph</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismOps1"><span class="id" title="section">MorphismOps1</span></a>.<br/> + +<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="f47bac414ebbcaae273350b5c84ff313"><span class="id" title="notation">"</span></a>''dom' f" := (<a class="idref" href="mathcomp.fingroup.morphism.html#dom"><span class="id" title="definition">dom</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorPhantom"><span class="id" title="definition">MorPhantom</span></a> <span class="id" title="var">f</span>))<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10, <span class="id" title="var">f</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''dom' f") : <span class="id" title="var">group_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">"</span></a>''ker' f" := (<a class="idref" href="mathcomp.fingroup.morphism.html#ker"><span class="id" title="definition">ker</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorPhantom"><span class="id" title="definition">MorPhantom</span></a> <span class="id" title="var">f</span>))<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10, <span class="id" title="var">f</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''ker' f") : <span class="id" title="var">group_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="ad50b791c3cc7be9475b0d43c3f572dd"><span class="id" title="notation">"</span></a>''ker_' H f" := (<span class="id" title="var">H</span> <a class="idref" href="mathcomp.ssreflect.finset.html#cb41714a5a23482f7a48a98975fa8c59"><span class="id" title="notation">:&:</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> <span class="id" title="var">f</span>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10, <span class="id" title="var">H</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 2, <span class="id" title="var">f</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''ker_' H f")<br/> + : <span class="id" title="var">group_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="48cff845c81518398138031392d44c93"><span class="id" title="notation">"</span></a>f @* A" := (<a class="idref" href="mathcomp.fingroup.morphism.html#morphim"><span class="id" title="definition">morphim</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorPhantom"><span class="id" title="definition">MorPhantom</span></a> <span class="id" title="var">f</span>) <span class="id" title="var">A</span>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 24, <span class="id" title="var">format</span> "f @* A") : <span class="id" title="var">group_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">"</span></a>f @*^-1 R" := (<a class="idref" href="mathcomp.fingroup.morphism.html#morphpre"><span class="id" title="definition">morphpre</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorPhantom"><span class="id" title="definition">MorPhantom</span></a> <span class="id" title="var">f</span>) <span class="id" title="var">R</span>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 24, <span class="id" title="var">format</span> "f @*^-1 R") : <span class="id" title="var">group_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">"</span></a>''injm' f" := (<a class="idref" href="mathcomp.ssreflect.finset.html#pred_of_set"><span class="id" title="abbreviation">pred_of_set</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> <span class="id" title="var">f</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.ssreflect.finset.html#pred_of_set"><span class="id" title="abbreviation">pred_of_set</span></a> 1)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 10, <span class="id" title="var">f</span> <span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 8, <span class="id" title="var">format</span> "''injm' f") : <span class="id" title="var">group_scope</span>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="MorphismTheory"><span class="id" title="section">MorphismTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="MorphismTheory.aT"><span class="id" title="variable">aT</span></a> <a name="MorphismTheory.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.morphism.html#MorphismTheory.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>.<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.morphism.html#MorphismTheory.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>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">R</span> <span class="id" title="var">S</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.morphism.html#MorphismTheory.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">Types</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.morphism.html#MorphismTheory.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"> + Most properties of morphims hold only when the domain is a group. +</div> +<div class="code"> +<span class="id" title="keyword">Variables</span> (<a name="MorphismTheory.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.morphism.html#MorphismTheory.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="MorphismTheory.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.morphism.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.morphism.html#MorphismTheory.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">Lemma</span> <a name="morph1"><span class="id" title="lemma">morph1</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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> 1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morph_prod"><span class="id" title="lemma">morph_prod</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.morphism.html#I"><span class="id" title="variable">I</span></a>) <span class="id" title="var">F</span> :<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>, <a class="idref" href="mathcomp.fingroup.morphism.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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="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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.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.morphism.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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#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.morphism.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.morphism.html#P"><span class="id" title="variable">P</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#F"><span class="id" title="variable">F</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#i"><span class="id" title="variable">i</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphV"><span class="id" title="lemma">morphV</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.morphism.html#MorphismTheory.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphJ"><span class="id" title="lemma">morphJ</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.morphism.html#MorphismTheory.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">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.morphism.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.morphism.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="morphX"><span class="id" title="lemma">morphX</span></a> <span class="id" title="var">n</span> : <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.morphism.html#MorphismTheory.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">morph</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">:</span></a> <span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><span class="id" title="notation">/</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#86a04fb564fb97d388cad84a3a204260"><span class="id" title="notation">^+</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.ssrfun.html#c3c88e2b30b681cd767a54649faf5973"><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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphR"><span class="id" title="lemma">morphR</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.morphism.html#MorphismTheory.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">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.fingroup.html#d5d566f861753940ef0e9a18d348c2b8"><span class="id" title="notation">[~</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d5d566f861753940ef0e9a18d348c2b8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#y"><span class="id" title="variable">y</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d5d566f861753940ef0e9a18d348c2b8"><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.ssrbool.html#2bba53854f326a714d377124cccec593"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Morphic image, preimage properties w.r.t. set-theoretic operations. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimE"><span class="id" title="lemma">morphimE</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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="mathcomp.fingroup.morphism.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>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreE"><span class="id" title="lemma">morphpreE</span></a> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#aa0c936d7feb8f26825e601735c0125f"><span class="id" title="notation">@^-1:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="kerE"><span class="id" title="lemma">kerE</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.morphism.html#MorphismTheory.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> 1. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimEsub"><span class="id" title="lemma">morphimEsub</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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#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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimEdom"><span class="id" title="lemma">morphimEdom</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#MorphismTheory.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.D"><span class="id" title="variable">D</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimIdom"><span class="id" title="lemma">morphimIdom</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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.morphism.html#MorphismTheory.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="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</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.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreIdom"><span class="id" title="lemma">morphpreIdom</span></a> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreIim"><span class="id" title="lemma">morphpreIim</span></a> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#MorphismTheory.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="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimIim"><span class="id" title="lemma">morphimIim</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#MorphismTheory.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="mathcomp.fingroup.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="mem_morphim"><span class="id" title="lemma">mem_morphim</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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.morphism.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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="mem_morphpre"><span class="id" title="lemma">mem_morphpre</span></a> <span class="id" title="var">R</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#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.morphism.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimS"><span class="id" title="lemma">morphimS</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_sub"><span class="id" title="lemma">morphim_sub</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#MorphismTheory.D"><span class="id" title="variable">D</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="leq_morphim"><span class="id" title="lemma">leq_morphim</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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#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.morphism.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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreS"><span class="id" title="lemma">morphpreS</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</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#S"><span class="id" title="variable">S</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#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#S"><span class="id" title="variable">S</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_sub"><span class="id" title="lemma">morphpre_sub</span></a> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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#MorphismTheory.D"><span class="id" title="variable">D</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_setIpre"><span class="id" title="lemma">morphim_setIpre</span></a> <span class="id" title="var">A</span> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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.morphism.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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.morphism.html#MorphismTheory.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#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.morphism.html#R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim0"><span class="id" title="lemma">morphim0</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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.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#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="morphim_eq0"><span class="id" title="lemma">morphim_eq0</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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="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.morphism.html#MorphismTheory.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#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.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#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.morphism.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.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_set1"><span class="id" title="lemma">morphim_set1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><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#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim1"><span class="id" title="lemma">morphim1</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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> 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> 1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimV"><span class="id" title="lemma">morphimV</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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="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#a605acbeae7597f74f5a9b816ed8a717"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreV"><span class="id" title="lemma">morphpreV</span></a> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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">Lemma</span> <a name="morphimMl"><span class="id" title="lemma">morphimMl</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.morphism.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.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimMr"><span class="id" title="lemma">morphimMr</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.morphism.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.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreMl"><span class="id" title="lemma">morphpreMl</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</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#MorphismTheory.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#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#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.morphism.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#S"><span class="id" title="variable">S</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimJ"><span class="id" title="lemma">morphimJ</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.morphism.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.morphism.html#x"><span class="id" title="variable">x</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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreJ"><span class="id" title="lemma">morphpreJ</span></a> <span class="id" title="var">R</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#1deb3845cf16de446ae6619879e9d6db"><span class="id" title="notation">:^</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_class"><span class="id" title="lemma">morphim_class</span></a> <span class="id" title="var">x</span> <span class="id" title="var">A</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#MorphismTheory.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.morphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#30988ee242f08216f4b40cf90b42b816"><span class="id" title="notation">^:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#30988ee242f08216f4b40cf90b42b816"><span class="id" title="notation">^:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="classes_morphim"><span class="id" title="lemma">classes_morphim</span></a> <span class="id" title="var">A</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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.fingroup.html#classes"><span class="id" title="definition">classes</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#xA"><span class="id" title="variable">xA</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">|</span></a> <span class="id" title="var">xA</span> <a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#c4eb68ed64baca4028c54e8eaca3672a"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimT"><span class="id" title="lemma">morphimT</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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.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.morphism.html#MorphismTheory.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#MorphismTheory.D"><span class="id" title="variable">D</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimU"><span class="id" title="lemma">morphimU</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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.morphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#52f608a788da136ac97df132d7055463"><span class="id" title="notation">:|:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#52f608a788da136ac97df132d7055463"><span class="id" title="notation">:|:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimI"><span class="id" title="lemma">morphimI</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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.morphism.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.morphism.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="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#MorphismTheory.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#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.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre0"><span class="id" title="lemma">morphpre0</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <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#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="morphpreT"><span class="id" title="lemma">morphpreT</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</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.morphism.html#MorphismTheory.D"><span class="id" title="variable">D</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreU"><span class="id" title="lemma">morphpreU</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#52f608a788da136ac97df132d7055463"><span class="id" title="notation">:|:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#52f608a788da136ac97df132d7055463"><span class="id" title="notation">:|:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#S"><span class="id" title="variable">S</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreI"><span class="id" title="lemma">morphpreI</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#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.fingroup.morphism.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#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.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#S"><span class="id" title="variable">S</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreD"><span class="id" title="lemma">morphpreD</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#1db838ba797020f3b39c07ed7167bc93"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#1db838ba797020f3b39c07ed7167bc93"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#S"><span class="id" title="variable">S</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + kernel, domain properties +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="kerP"><span class="id" title="lemma">kerP</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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="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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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) (<a class="idref" href="mathcomp.fingroup.morphism.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.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="dom_ker"><span class="id" title="lemma">dom_ker</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">{</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">subset</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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">≤</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#53295d8f18390300c5a24f66e0bcda98"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="mker"><span class="id" title="lemma">mker</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.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.morphism.html#MorphismTheory.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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="mkerl"><span class="id" title="lemma">mkerl</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.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.morphism.html#MorphismTheory.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.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.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.morphism.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="mkerr"><span class="id" title="lemma">mkerr</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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.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.morphism.html#MorphismTheory.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.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.morphism.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="rcoset_kerP"><span class="id" title="lemma">rcoset_kerP</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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="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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#y"><span class="id" title="variable">y</span></a>) (<a class="idref" href="mathcomp.fingroup.morphism.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.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#291276ea06db0b00a2747a79d012bbe0"><span class="id" title="notation">:*</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#y"><span class="id" title="variable">y</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_rcoset"><span class="id" title="lemma">ker_rcoset</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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#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#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">exists2</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#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.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.morphism.html#MorphismTheory.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#28b18e493f7cb0bd8447607bdc385ff8"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#z"><span class="id" title="variable">z</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#y"><span class="id" title="variable">y</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_norm"><span class="id" title="lemma">ker_norm</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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="ker_normal"><span class="id" title="lemma">ker_normal</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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.D"><span class="id" title="variable">D</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimGI"><span class="id" title="lemma">morphimGI</span></a> <span class="id" title="var">G</span> <span class="id" title="var">A</span> : <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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#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.morphism.html#MorphismTheory.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.morphism.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.morphism.html#A"><span class="id" title="variable">A</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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimIG"><span class="id" title="lemma">morphimIG</span></a> <span class="id" title="var">A</span> <span class="id" title="var">G</span> : <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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#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.morphism.html#MorphismTheory.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.morphism.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.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimD"><span class="id" title="lemma">morphimD</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#1db838ba797020f3b39c07ed7167bc93"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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.morphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#1db838ba797020f3b39c07ed7167bc93"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimDG"><span class="id" title="lemma">morphimDG</span></a> <span class="id" title="var">A</span> <span class="id" title="var">G</span> : <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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#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.morphism.html#MorphismTheory.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.morphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#1db838ba797020f3b39c07ed7167bc93"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#1db838ba797020f3b39c07ed7167bc93"><span class="id" title="notation">:\:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimD1"><span class="id" title="lemma">morphimD1</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#adb8044960c962a921cca1bd48aae97d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#adb8044960c962a921cca1bd48aae97d"><span class="id" title="notation">)^#</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#MorphismTheory.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#adb8044960c962a921cca1bd48aae97d"><span class="id" title="notation">^#</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + group structure preservation +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_groupset"><span class="id" title="lemma">morphpre_groupset</span></a> <span class="id" title="var">M</span> : <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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#M"><span class="id" title="variable">M</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_groupset"><span class="id" title="lemma">morphim_groupset</span></a> <span class="id" title="var">G</span> : <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.morphism.html#MorphismTheory.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#G"><span class="id" title="variable">G</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">morphpre_group</span> <span class="id" title="var">fPh</span> <span class="id" title="var">M</span> :=<br/> + @<a class="idref" href="mathcomp.fingroup.fingroup.html#group"><span class="id" title="definition">group</span></a> <span class="id" title="var">_</span> (<a class="idref" href="mathcomp.fingroup.morphism.html#morphpre"><span class="id" title="definition">morphpre</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#fPh"><span class="id" title="variable">fPh</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#M"><span class="id" title="variable">M</span></a>) (<a class="idref" href="mathcomp.fingroup.morphism.html#morphpre_groupset"><span class="id" title="lemma">morphpre_groupset</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#M"><span class="id" title="variable">M</span></a>).<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">morphim_group</span> <span class="id" title="var">fPh</span> <span class="id" title="var">G</span> := @<a class="idref" href="mathcomp.fingroup.fingroup.html#group"><span class="id" title="definition">group</span></a> <span class="id" title="var">_</span> (<a class="idref" href="mathcomp.fingroup.morphism.html#morphim"><span class="id" title="definition">morphim</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#fPh"><span class="id" title="variable">fPh</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</span></a>) (<a class="idref" href="mathcomp.fingroup.morphism.html#morphim_groupset"><span class="id" title="lemma">morphim_groupset</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</span></a>).<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">ker_group</span> <span class="id" title="var">fPh</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.morphism.html#MorphismTheory.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> := <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#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">group</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">of</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ker"><span class="id" title="definition">ker</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#fPh"><span class="id" title="variable">fPh</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ccb763a84253e971fd106aeeb9cd3cb0"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morph_dom_groupset"><span class="id" title="lemma">morph_dom_groupset</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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.D"><span class="id" title="variable">D</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">morph_dom_group</span> := <a class="idref" href="mathcomp.fingroup.fingroup.html#group"><span class="id" title="definition">group</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#morph_dom_groupset"><span class="id" title="lemma">morph_dom_groupset</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreMr"><span class="id" title="lemma">morphpreMr</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#S"><span class="id" title="variable">S</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#MorphismTheory.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#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#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.morphism.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#S"><span class="id" title="variable">S</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimK"><span class="id" title="lemma">morphimK</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><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.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimGK"><span class="id" title="lemma">morphimGK</span></a> <span class="id" title="var">G</span> : <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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#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.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><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.morphism.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_set1"><span class="id" title="lemma">morphpre_set1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><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.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#291276ea06db0b00a2747a79d012bbe0"><span class="id" title="notation">:*</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreK"><span class="id" title="lemma">morphpreK</span></a> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</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#MorphismTheory.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#MorphismTheory.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#MorphismTheory.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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.morphism.html#R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_ker"><span class="id" title="lemma">morphim_ker</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_sub_pre"><span class="id" title="lemma">ker_sub_pre</span></a> <span class="id" title="var">M</span> : <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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#M"><span class="id" title="variable">M</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_normal_pre"><span class="id" title="lemma">ker_normal_pre</span></a> <span class="id" title="var">M</span> : <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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#M"><span class="id" title="variable">M</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpreSK"><span class="id" title="lemma">morphpreSK</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</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#MorphismTheory.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#MorphismTheory.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="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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#S"><span class="id" title="variable">S</span></a><a 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.morphism.html#R"><span class="id" title="variable">R</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#S"><span class="id" title="variable">S</span></a><a 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="sub_morphim_pre"><span class="id" title="lemma">sub_morphim_pre</span></a> <span class="id" title="var">A</span> <span class="id" title="var">R</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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="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.morphism.html#MorphismTheory.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#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.morphism.html#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#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.morphism.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_proper"><span class="id" title="lemma">morphpre_proper</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</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#MorphismTheory.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#MorphismTheory.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#S"><span class="id" title="variable">S</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#MorphismTheory.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#MorphismTheory.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="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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#S"><span class="id" title="variable">S</span></a><a 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.morphism.html#R"><span class="id" title="variable">R</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.morphism.html#S"><span class="id" title="variable">S</span></a><a 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="sub_morphpre_im"><span class="id" title="lemma">sub_morphpre_im</span></a> <span class="id" title="var">R</span> <span class="id" title="var">G</span> :<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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#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.morphism.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.morphism.html#MorphismTheory.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#R"><span class="id" title="variable">R</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#MorphismTheory.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#MorphismTheory.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="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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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#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="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.morphism.html#R"><span class="id" title="variable">R</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#MorphismTheory.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#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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_trivg_morphim"><span class="id" title="lemma">ker_trivg_morphim</span></a> <span class="id" title="var">A</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.morphism.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.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.morphism.html#MorphismTheory.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#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.morphism.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.morphism.html#MorphismTheory.D"><span class="id" title="variable">D</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="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.morphism.html#MorphismTheory.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#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#c33afa16525556de4ed568ad52c9389f"><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#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimSK"><span class="id" title="lemma">morphimSK</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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="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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#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.morphism.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.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#169fb610eeaa28cebf8ec36928167473"><span class="id" title="notation">×</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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="morphimSGK"><span class="id" title="lemma">morphimSGK</span></a> <span class="id" title="var">A</span> <span class="id" title="var">G</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#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="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.morphism.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.morphism.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ltn_morphim"><span class="id" title="lemma">ltn_morphim</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#c33afa16525556de4ed568ad52c9389f"><span class="id" title="notation">[1]</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.morphism.html#ad50b791c3cc7be9475b0d43c3f572dd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#ad50b791c3cc7be9475b0d43c3f572dd"><span class="id" title="notation">ker_A</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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#989c98e7ddd65d5bf37c334ff2076de8"><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.morphism.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>.<br/> + +<br/> +</div> + +<div class="doc"> + injectivity of image and preimage +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_inj"><span class="id" title="lemma">morphpre_inj</span></a> :<br/> + <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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ba4e2de877832f7c061554da0cb06dd3"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ba4e2de877832f7c061554da0cb06dd3"><span class="id" title="notation">pred</span></a> <span class="id" title="var">R</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ba4e2de877832f7c061554da0cb06dd3"><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.morphism.html#MorphismTheory.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> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#ba4e2de877832f7c061554da0cb06dd3"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</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#MorphismTheory.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#MorphismTheory.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#ba4e2de877832f7c061554da0cb06dd3"><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.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">R</span> ⇒ <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_injG"><span class="id" title="lemma">morphim_injG</span></a> :<br/> + <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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#6ea88f2b614e7a82f9b2bc6ef82e810a"><span class="id" title="notation">[</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#6ea88f2b614e7a82f9b2bc6ef82e810a"><span class="id" title="notation">pred</span></a> <span class="id" title="var">G</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#6ea88f2b614e7a82f9b2bc6ef82e810a"><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.morphism.html#MorphismTheory.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 class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#6ea88f2b614e7a82f9b2bc6ef82e810a"><span class="id" title="notation">|</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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#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#6ea88f2b614e7a82f9b2bc6ef82e810a"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#6ea88f2b614e7a82f9b2bc6ef82e810a"><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/> + <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#injective"><span class="id" title="definition">injective</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">G</span> ⇒ <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_inj"><span class="id" title="lemma">morphim_inj</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span> :<br/> + <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.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#G"><span class="id" title="variable">G</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="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.morphism.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.morphism.html#MorphismTheory.D"><span class="id" title="variable">D</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.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.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">(</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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#H"><span class="id" title="variable">H</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="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.morphism.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.morphism.html#MorphismTheory.D"><span class="id" title="variable">D</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.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#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.morphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><span class="id" title="notation">:=:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + commutation with generated groups and cycles +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_gen"><span class="id" title="lemma">morphim_gen</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</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.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</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">Lemma</span> <a name="morphim_cycle"><span class="id" title="lemma">morphim_cycle</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation"><[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><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#30152704c0ab4066186d0284456667e8"><span class="id" title="notation"><[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#30152704c0ab4066186d0284456667e8"><span class="id" title="notation">]></span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimY"><span class="id" title="lemma">morphimY</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#MorphismTheory.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.morphism.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.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_gen"><span class="id" title="lemma">morphpre_gen</span></a> <span class="id" title="var">R</span> :<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.morphism.html#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.morphism.html#R"><span class="id" title="variable">R</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#MorphismTheory.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#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</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#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</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.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation"><<</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#d2263119ac2870c795428c0a326d9d52"><span class="id" title="notation">>></span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + commutator, normaliser, normal, center properties +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphimR"><span class="id" title="lemma">morphimR</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#MorphismTheory.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.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><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#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">[~:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#44ce2df89b693f6f5ca2acfcd54d16b4"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_norm"><span class="id" title="lemma">morphim_norm</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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.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.morphism.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="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.morphism.html#MorphismTheory.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#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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_norms"><span class="id" title="lemma">morphim_norms</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.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="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#MorphismTheory.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#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.morphism.html#MorphismTheory.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#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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_subnorm"><span class="id" title="lemma">morphim_subnorm</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)</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#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_normal"><span class="id" title="lemma">morphim_normal</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_cent1"><span class="id" title="lemma">morphim_cent1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">]</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#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_cent1s"><span class="id" title="lemma">morphim_cent1s</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><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.morphism.html#MorphismTheory.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#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#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_subcent1"><span class="id" title="lemma">morphim_subcent1</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">C_A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">]</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#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">)[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_cent"><span class="id" title="lemma">morphim_cent</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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.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.morphism.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="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.morphism.html#MorphismTheory.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#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">Lemma</span> <a name="morphim_cents"><span class="id" title="lemma">morphim_cents</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.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#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.morphism.html#B"><span class="id" title="variable">B</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.morphism.html#MorphismTheory.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#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#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.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</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="morphim_subcent"><span class="id" title="lemma">morphim_subcent</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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.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_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.morphism.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">)</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#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.morphism.html#MorphismTheory.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#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><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#82eaaf0a77a7ca85d3ca7dcf4463ae79"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_abelian"><span class="id" title="lemma">morphim_abelian</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_norm"><span class="id" title="lemma">morphpre_norm</span></a> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#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> <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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#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="morphpre_norms"><span class="id" title="lemma">morphpre_norms</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</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.morphism.html#S"><span class="id" title="variable">S</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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#S"><span class="id" title="variable">S</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="morphpre_normal"><span class="id" title="lemma">morphpre_normal</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</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#MorphismTheory.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#MorphismTheory.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#S"><span class="id" title="variable">S</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#MorphismTheory.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#MorphismTheory.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="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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#S"><span class="id" title="variable">S</span></a><a 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.morphism.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#c27c638e534bbb5b7de2d4b4aa0a3e82"><span class="id" title="notation"><|</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#S"><span class="id" title="variable">S</span></a><a 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="morphpre_subnorm"><span class="id" title="lemma">morphpre_subnorm</span></a> <span class="id" title="var">R</span> <span class="id" title="var">S</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_R</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)</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#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#S"><span class="id" title="variable">S</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_normG"><span class="id" title="lemma">morphim_normG</span></a> <span class="id" title="var">G</span> :<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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#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.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.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.morphism.html#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> <a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#MorphismTheory.D"><span class="id" title="variable">D</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_subnormG"><span class="id" title="lemma">morphim_subnormG</span></a> <span class="id" title="var">A</span> <span class="id" title="var">G</span> :<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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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#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.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><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#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_cent1"><span class="id" title="lemma">morphpre_cent1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">C_D</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">]</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#MorphismTheory.f"><span class="id" title="variable">f</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#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_cent1s"><span class="id" title="lemma">morphpre_cent1s</span></a> <span class="id" title="var">R</span> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#R"><span class="id" title="variable">R</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#MorphismTheory.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#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><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.morphism.html#R"><span class="id" title="variable">R</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#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_subcent1"><span class="id" title="lemma">morphpre_subcent1</span></a> <span class="id" title="var">R</span> <span class="id" title="var">x</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">)[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">]</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#MorphismTheory.f"><span class="id" title="variable">f</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#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">C_R</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_cent"><span class="id" title="lemma">morphpre_cent</span></a> <span class="id" title="var">A</span> : <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_D</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.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.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#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#MorphismTheory.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#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">Lemma</span> <a name="morphpre_cents"><span class="id" title="lemma">morphpre_cents</span></a> <span class="id" title="var">A</span> <span class="id" title="var">R</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</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#MorphismTheory.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#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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.morphism.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.morphism.html#R"><span class="id" title="variable">R</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.morphism.html#MorphismTheory.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#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">Lemma</span> <a name="morphpre_subcent"><span class="id" title="lemma">morphpre_subcent</span></a> <span class="id" title="var">R</span> <span class="id" title="var">A</span> : <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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#82eaaf0a77a7ca85d3ca7dcf4463ae79"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.morphism.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> <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#MorphismTheory.f"><span class="id" title="variable">f</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#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">C_R</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#507fd39a15bb9cb7e52e1aaa9e2285b5"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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/> +</div> + +<div class="doc"> + local injectivity properties +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injmP"><span class="id" title="lemma">injmP</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="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.morphism.html#MorphismTheory.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#injective"><span class="id" title="definition">injective</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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="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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="card_im_injm"><span class="id" title="lemma">card_im_injm</span></a> : <a 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.morphism.html#MorphismTheory.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#MorphismTheory.D"><span class="id" title="variable">D</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><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.morphism.html#MorphismTheory.D"><span class="id" title="variable">D</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="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.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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="MorphismTheory.Injective"><span class="id" title="section">Injective</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="MorphismTheory.Injective.injf"><span class="id" title="variable">injf</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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_injm"><span class="id" title="lemma">ker_injm</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.morphism.html#MorphismTheory.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> 1.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injmK"><span class="id" title="lemma">injmK</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><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.morphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_morphim_inj"><span class="id" title="lemma">injm_morphim_inj</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#MorphismTheory.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#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.morphism.html#MorphismTheory.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#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.morphism.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.morphism.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="card_injm"><span class="id" title="lemma">card_injm</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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="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.morphism.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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="order_injm"><span class="id" title="lemma">order_injm</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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.fingroup.html#89402f0d9375903caa99ad84144160d5"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#89402f0d9375903caa99ad84144160d5"><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#89402f0d9375903caa99ad84144160d5"><span class="id" title="notation">#[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#89402f0d9375903caa99ad84144160d5"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm1"><span class="id" title="lemma">injm1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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 <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#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="morph_injm_eq1"><span class="id" title="lemma">morph_injm_eq1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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="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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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> 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.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.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> 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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injmSK"><span class="id" title="lemma">injmSK</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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="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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#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.morphism.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.morphism.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="sub_morphpre_injm"><span class="id" title="lemma">sub_morphpre_injm</span></a> <span class="id" title="var">R</span> <span class="id" title="var">A</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#R"><span class="id" title="variable">R</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#MorphismTheory.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#MorphismTheory.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="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.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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#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="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.morphism.html#R"><span class="id" title="variable">R</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#MorphismTheory.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#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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_eq"><span class="id" title="lemma">injm_eq</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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="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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#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.morphism.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.morphism.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="morphim_injm_eq1"><span class="id" title="lemma">morphim_injm_eq1</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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="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.morphism.html#MorphismTheory.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#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.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.morphism.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.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injmI"><span class="id" title="lemma">injmI</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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.morphism.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.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injmD1"><span class="id" title="lemma">injmD1</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#adb8044960c962a921cca1bd48aae97d"><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#adb8044960c962a921cca1bd48aae97d"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</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="nclasses_injm"><span class="id" title="lemma">nclasses_injm</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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="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.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_norm"><span class="id" title="lemma">injm_norm</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.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.morphism.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.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#MorphismTheory.D"><span class="id" title="variable">D</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_norms"><span class="id" title="lemma">injm_norms</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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="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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#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="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.morphism.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.morphism.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="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="injm_normal"><span class="id" title="lemma">injm_normal</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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="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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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#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.morphism.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.morphism.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="injm_subnorm"><span class="id" title="lemma">injm_subnorm</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><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#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">N_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#7193b23d12b4f3c2146b0e77ee974b2b"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_cent1"><span class="id" title="lemma">injm_cent1</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">C</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#b59c327df8172426dc1fbd94c21cf201"><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#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#MorphismTheory.D"><span class="id" title="variable">D</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">)[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_subcent1"><span class="id" title="lemma">injm_subcent1</span></a> <span class="id" title="var">A</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">C_A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#addacbae2e0ffbfd03aaa03c308b39d7"><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#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">C_</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">)[</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#8ad9bf5af7bbbf04d2034213ec4c29f6"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_cent"><span class="id" title="lemma">injm_cent</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.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.morphism.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#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.morphism.html#MorphismTheory.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#MorphismTheory.D"><span class="id" title="variable">D</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#82eaaf0a77a7ca85d3ca7dcf4463ae79"><span class="id" title="notation">)(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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/> +<span class="id" title="keyword">Lemma</span> <a name="injm_cents"><span class="id" title="lemma">injm_cents</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#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.morphism.html#MorphismTheory.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="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.morphism.html#MorphismTheory.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#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#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.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</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#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.morphism.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#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.morphism.html#B"><span class="id" title="variable">B</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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_subcent"><span class="id" title="lemma">injm_subcent</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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#MorphismTheory.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.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_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.morphism.html#B"><span class="id" title="variable">B</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.morphism.html#MorphismTheory.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#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><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#B"><span class="id" title="variable">B</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#82eaaf0a77a7ca85d3ca7dcf4463ae79"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_abelian"><span class="id" title="lemma">injm_abelian</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#MorphismTheory.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.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.Injective"><span class="id" title="section">Injective</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="eq_morphim"><span class="id" title="lemma">eq_morphim</span></a> (<span class="id" title="var">g</span> : <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.morphism.html#MorphismTheory.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.morphism.html#MorphismTheory.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/> + <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.morphism.html#MorphismTheory.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">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.morphism.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#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> <span class="id" title="keyword">∀</span> <span class="id" title="var">A</span>, <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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.morphism.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="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="eq_in_morphim"><span class="id" title="lemma">eq_in_morphim</span></a> <span class="id" title="var">B</span> <span class="id" title="var">A</span> (<span class="id" title="var">g</span> : <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.morphism.html#B"><span class="id" title="variable">B</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#MorphismTheory.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/> + <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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="mathcomp.fingroup.morphism.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.morphism.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.morphism.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#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.morphism.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.f"><span class="id" title="variable">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.morphism.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#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> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory.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#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.morphism.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="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismTheory"><span class="id" title="section">MorphismTheory</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="9b5e56aac3b1ca31e22775c2f0cc87c3"><span class="id" title="notation">"</span></a>''ker' f" := (<a class="idref" href="mathcomp.fingroup.morphism.html#ker_group"><span class="id" title="definition">ker_group</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorPhantom"><span class="id" title="definition">MorPhantom</span></a> <span class="id" title="var">f</span>)) : <span class="id" title="var">Group_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="0d4b089495183b6790bbcf78bc015b5f"><span class="id" title="notation">"</span></a>''ker_' G f" := (<span class="id" title="var">G</span> <a class="idref" href="mathcomp.fingroup.fingroup.html#afecdac1581fada66d09c1ab40cfc23e"><span class="id" title="notation">:&:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#9b5e56aac3b1ca31e22775c2f0cc87c3"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#9b5e56aac3b1ca31e22775c2f0cc87c3"><span class="id" title="notation">ker</span></a> <span class="id" title="var">f</span>)%<span class="id" title="var">G</span> : <span class="id" title="var">Group_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="3b106b15610dbcdb2f3d7e59c9b527c5"><span class="id" title="notation">"</span></a>f @* G" := (<a class="idref" href="mathcomp.fingroup.morphism.html#morphim_group"><span class="id" title="definition">morphim_group</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorPhantom"><span class="id" title="definition">MorPhantom</span></a> <span class="id" title="var">f</span>) <span class="id" title="var">G</span>) : <span class="id" title="var">Group_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="ba46f4001d675d19d4eefacace1dabbe"><span class="id" title="notation">"</span></a>f @*^-1 M" := (<a class="idref" href="mathcomp.fingroup.morphism.html#morphpre_group"><span class="id" title="definition">morphpre_group</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#MorPhantom"><span class="id" title="definition">MorPhantom</span></a> <span class="id" title="var">f</span>) <span class="id" title="var">M</span>) : <span class="id" title="var">Group_scope</span>.<br/> +<span class="id" title="keyword">Notation</span> <a name="03d290dc6fac626a44a8b31b17b086c8"><span class="id" title="notation">"</span></a>f @: D" := (<a class="idref" href="mathcomp.fingroup.morphism.html#morph_dom_group"><span class="id" title="definition">morph_dom_group</span></a> <span class="id" title="var">f</span> <span class="id" title="var">D</span>) : <span class="id" title="var">Group_scope</span>.<br/> + +<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="IdentityMorphism"><span class="id" title="section">IdentityMorphism</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="IdentityMorphism.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> : <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.morphism.html#IdentityMorphism.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">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.morphism.html#IdentityMorphism.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="idm"><span class="id" title="definition">idm</span></a> <span class="id" title="keyword">of</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.morphism.html#IdentityMorphism.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="keyword">fun</span> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#IdentityMorphism.gT"><span class="id" title="variable">gT</span></a> ⇒ <a class="idref" href="mathcomp.fingroup.morphism.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.ssreflect.html#4509b22bf26e3d6d771897e22bd8bc8f"><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.morphism.html#IdentityMorphism.gT"><span class="id" title="variable">gT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="idm_morphM"><span class="id" title="lemma">idm_morphM</span></a> <span class="id" title="var">A</span> : <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.morphism.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#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">,</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.morphism.html#idm"><span class="id" title="definition">idm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.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.morphism.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.morphism.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">idm_morphism</span> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> (@<a class="idref" href="mathcomp.fingroup.morphism.html#idm_morphM"><span class="id" title="lemma">idm_morphM</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_idm"><span class="id" title="lemma">injm_idm</span></a> <span class="id" title="var">G</span> : <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.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#idm"><span class="id" title="definition">idm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_idm"><span class="id" title="lemma">ker_idm</span></a> <span class="id" title="var">G</span> : <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.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#idm"><span class="id" title="definition">idm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><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="morphim_idm"><span class="id" title="lemma">morphim_idm</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#idm"><span class="id" title="definition">idm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</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#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.morphism.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_idm"><span class="id" title="lemma">morphpre_idm</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#idm"><span class="id" title="definition">idm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</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.morphism.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.morphism.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.morphism.html#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_idm"><span class="id" title="lemma">im_idm</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#idm"><span class="id" title="definition">idm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</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#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.morphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#IdentityMorphism"><span class="id" title="section">IdentityMorphism</span></a>.<br/> + +<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="RestrictedMorphism"><span class="id" title="section">RestrictedMorphism</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="RestrictedMorphism.aT"><span class="id" title="variable">aT</span></a> <a name="RestrictedMorphism.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="RestrictedMorphism.A"><span class="id" title="variable">A</span></a> <a name="RestrictedMorphism.D"><span class="id" title="variable">D</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.morphism.html#RestrictedMorphism.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>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</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.morphism.html#RestrictedMorphism.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>.<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Type</span> <span class="id" title="var">R</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.morphism.html#RestrictedMorphism.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">Definition</span> <a name="restrm"><span class="id" title="definition">restrm</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.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.morphism.html#RestrictedMorphism.D"><span class="id" title="variable">D</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#18cd0ece1c60c28b29a57eea4cb098ca"><span class="id" title="notation">@</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#18cd0ece1c60c28b29a57eea4cb098ca"><span class="id" title="notation">id</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#18cd0ece1c60c28b29a57eea4cb098ca"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.aT"><span class="id" title="variable">aT</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.morphism.html#RestrictedMorphism.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#18cd0ece1c60c28b29a57eea4cb098ca"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="RestrictedMorphism.Props"><span class="id" title="section">Props</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="RestrictedMorphism.Props.sAD"><span class="id" title="variable">sAD</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.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.morphism.html#RestrictedMorphism.D"><span class="id" title="variable">D</span></a>.<br/> +<span class="id" title="keyword">Variable</span> <a name="RestrictedMorphism.Props.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.morphism.html#RestrictedMorphism.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.morphism.html#RestrictedMorphism.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">Canonical</span> <span class="id" title="var">restrm_morphism</span> :=<br/> + @<a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.aT"><span class="id" title="variable">aT</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.rT"><span class="id" title="variable">rT</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#fA"><span class="id" title="abbreviation">fA</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrbool.html#sub_in2"><span class="id" title="lemma">sub_in2</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#subsetP"><span class="id" title="lemma">subsetP</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.Props.sAD"><span class="id" title="variable">sAD</span></a>) (<a class="idref" href="mathcomp.fingroup.morphism.html#morphM"><span class="id" title="lemma">morphM</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.Props.f"><span class="id" title="variable">f</span></a>)).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_restrm"><span class="id" title="lemma">morphim_restrm</span></a> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#fA"><span class="id" title="abbreviation">fA</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#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.morphism.html#RestrictedMorphism.Props.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.morphism.html#RestrictedMorphism.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.morphism.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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="restrmEsub"><span class="id" title="lemma">restrmEsub</span></a> <span class="id" title="var">B</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#RestrictedMorphism.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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#fA"><span class="id" title="abbreviation">fA</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#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.morphism.html#RestrictedMorphism.Props.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#B"><span class="id" title="variable">B</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_restrm"><span class="id" title="lemma">im_restrm</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#fA"><span class="id" title="abbreviation">fA</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#RestrictedMorphism.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.morphism.html#RestrictedMorphism.Props.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#RestrictedMorphism.A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_restrm"><span class="id" title="lemma">morphpre_restrm</span></a> <span class="id" title="var">R</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#fA"><span class="id" title="abbreviation">fA</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.morphism.html#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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.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.morphism.html#RestrictedMorphism.Props.f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_restrm"><span class="id" title="lemma">ker_restrm</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.morphism.html#fA"><span class="id" title="abbreviation">fA</span></a> <a 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.morphism.html#ad50b791c3cc7be9475b0d43c3f572dd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#ad50b791c3cc7be9475b0d43c3f572dd"><span class="id" title="notation">ker_A</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.Props.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_restrm"><span class="id" title="lemma">injm_restrm</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.morphism.html#RestrictedMorphism.Props.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.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.morphism.html#fA"><span class="id" title="abbreviation">fA</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.Props"><span class="id" title="section">Props</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="restrmP"><span class="id" title="lemma">restrmP</span></a> (<span class="id" title="var">f</span> : <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.morphism.html#RestrictedMorphism.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.morphism.html#RestrictedMorphism.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 class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.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.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.morphism.html#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><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">g</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.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.morphism.html#RestrictedMorphism.A"><span class="id" title="variable">A</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#RestrictedMorphism.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 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="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.morphism.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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.aT"><span class="id" title="variable">aT</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#RestrictedMorphism.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><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.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.morphism.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ad50b791c3cc7be9475b0d43c3f572dd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#ad50b791c3cc7be9475b0d43c3f572dd"><span class="id" title="notation">ker_A</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</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/> + <span class="id" title="keyword">∀</span> <span class="id" title="var">R</span>, <a class="idref" href="mathcomp.fingroup.morphism.html#g"><span class="id" title="variable">g</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.morphism.html#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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.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.morphism.html#f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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> <span class="id" title="keyword">∀</span> <span class="id" title="var">B</span>, <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#RestrictedMorphism.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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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="mathcomp.fingroup.morphism.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.morphism.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#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="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="domP"><span class="id" title="lemma">domP</span></a> (<span class="id" title="var">f</span> : <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.morphism.html#RestrictedMorphism.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.morphism.html#RestrictedMorphism.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 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.morphism.html#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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.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#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">g</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.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.morphism.html#RestrictedMorphism.A"><span class="id" title="variable">A</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#RestrictedMorphism.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 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="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.morphism.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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#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#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">:></span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism.aT"><span class="id" title="variable">aT</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#RestrictedMorphism.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#8f9364556521ebb498093f28eea2240f"><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.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.morphism.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</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.morphism.html#f"><span class="id" title="variable">f</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/> + <span class="id" title="keyword">∀</span> <span class="id" title="var">R</span>, <a class="idref" href="mathcomp.fingroup.morphism.html#g"><span class="id" title="variable">g</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.morphism.html#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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</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.morphism.html#R"><span class="id" title="variable">R</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> <span class="id" title="keyword">∀</span> <span class="id" title="var">B</span>, <a class="idref" href="mathcomp.fingroup.morphism.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="mathcomp.fingroup.morphism.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.morphism.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#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="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">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#RestrictedMorphism"><span class="id" title="section">RestrictedMorphism</span></a>.<br/> + +<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="TrivMorphism"><span class="id" title="section">TrivMorphism</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="TrivMorphism.aT"><span class="id" title="variable">aT</span></a> <a name="TrivMorphism.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/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="trivm"><span class="id" title="definition">trivm</span></a> <span class="id" title="keyword">of</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.morphism.html#TrivMorphism.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 class="idref" href="mathcomp.fingroup.morphism.html#TrivMorphism.aT"><span class="id" title="variable">aT</span></a> := 1 <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.fingroup.html#FinGroup.sort"><span class="id" title="projection">FinGroup.sort</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#TrivMorphism.rT"><span class="id" title="variable">rT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="trivm_morphM"><span class="id" title="lemma">trivm_morphM</span></a> (<span class="id" title="var">A</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.morphism.html#TrivMorphism.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 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.morphism.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#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.morphism.html#trivm"><span class="id" title="definition">trivm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.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.morphism.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.morphism.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">triv_morph</span> <span class="id" title="var">A</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#Morphism"><span class="id" title="constructor">Morphism</span></a> (@<a class="idref" href="mathcomp.fingroup.morphism.html#trivm_morphM"><span class="id" title="lemma">trivm_morphM</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_trivm"><span class="id" title="lemma">morphim_trivm</span></a> (<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.morphism.html#TrivMorphism.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 class="idref" href="mathcomp.fingroup.morphism.html#trivm"><span class="id" title="definition">trivm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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="mathcomp.fingroup.morphism.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.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_trivm"><span class="id" title="lemma">ker_trivm</span></a> (<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.morphism.html#TrivMorphism.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 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.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#trivm"><span class="id" title="definition">trivm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><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.morphism.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.morphism.html#TrivMorphism"><span class="id" title="section">TrivMorphism</span></a>.<br/> + +<br/> + +<br/> +</div> + +<div class="doc"> + The composition of two morphisms is a Canonical morphism instance. +</div> +<div class="code"> +<span class="id" title="keyword">Section</span> <a name="MorphismComposition"><span class="id" title="section">MorphismComposition</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="MorphismComposition.gT"><span class="id" title="variable">gT</span></a> <a name="MorphismComposition.hT"><span class="id" title="variable">hT</span></a> <a name="MorphismComposition.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="MorphismComposition.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.morphism.html#MorphismComposition.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="MorphismComposition.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.morphism.html#MorphismComposition.hT"><span class="id" title="variable">hT</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">Variable</span> <a name="MorphismComposition.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.morphism.html#MorphismComposition.G"><span class="id" title="variable">G</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#MorphismComposition.hT"><span class="id" title="variable">hT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>.<br/> +<span class="id" title="keyword">Variable</span> <a name="MorphismComposition.g"><span class="id" title="variable">g</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.morphism.html#MorphismComposition.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.morphism.html#MorphismComposition.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/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="comp_morphM"><span class="id" title="lemma">comp_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.morphism.html#MorphismComposition.f"><span class="id" title="variable">f</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.morphism.html#MorphismComposition.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#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.morphism.html#gof"><span class="id" title="abbreviation">gof</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.morphism.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.morphism.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">comp_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.morphism.html#comp_morphM"><span class="id" title="lemma">comp_morphM</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_comp"><span class="id" title="lemma">ker_comp</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.morphism.html#gof"><span class="id" title="abbreviation">gof</span></a> <a 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.morphism.html#MorphismComposition.f"><span class="id" title="variable">f</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.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.morphism.html#MorphismComposition.g"><span class="id" title="variable">g</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_comp"><span class="id" title="lemma">injm_comp</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.morphism.html#MorphismComposition.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.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.morphism.html#MorphismComposition.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.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.morphism.html#gof"><span class="id" title="abbreviation">gof</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_comp"><span class="id" title="lemma">morphim_comp</span></a> (<span class="id" title="var">A</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.morphism.html#MorphismComposition.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.morphism.html#gof"><span class="id" title="abbreviation">gof</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#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.morphism.html#MorphismComposition.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="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismComposition.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#A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_comp"><span class="id" title="lemma">morphpre_comp</span></a> (<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.morphism.html#MorphismComposition.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>) : <a class="idref" href="mathcomp.fingroup.morphism.html#gof"><span class="id" title="abbreviation">gof</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.morphism.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.morphism.html#MorphismComposition.f"><span class="id" title="variable">f</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.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#MorphismComposition.g"><span class="id" title="variable">g</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.morphism.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#MorphismComposition"><span class="id" title="section">MorphismComposition</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + The factor morphism +</div> +<div class="code"> +<span class="id" title="keyword">Section</span> <a name="FactorMorphism"><span class="id" title="section">FactorMorphism</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="FactorMorphism.aT"><span class="id" title="variable">aT</span></a> <a name="FactorMorphism.qT"><span class="id" title="variable">qT</span></a> <a name="FactorMorphism.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/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="FactorMorphism.G"><span class="id" title="variable">G</span></a> <a name="FactorMorphism.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.morphism.html#FactorMorphism.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>.<br/> +<span class="id" title="keyword">Variable</span> <a name="FactorMorphism.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.morphism.html#FactorMorphism.G"><span class="id" title="variable">G</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#FactorMorphism.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">Variable</span> <a name="FactorMorphism.q"><span class="id" title="variable">q</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.morphism.html#FactorMorphism.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.morphism.html#FactorMorphism.qT"><span class="id" title="variable">qT</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">Definition</span> <a name="factm"><span class="id" title="definition">factm</span></a> <span class="id" title="keyword">of</span> <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.morphism.html#FactorMorphism.q"><span class="id" title="variable">q</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#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.morphism.html#FactorMorphism.f"><span class="id" title="variable">f</span></a> & <a class="idref" href="mathcomp.fingroup.morphism.html#FactorMorphism.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.morphism.html#FactorMorphism.H"><span class="id" title="variable">H</span></a> :=<br/> + <span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="mathcomp.fingroup.morphism.html#FactorMorphism.f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#repr"><span class="id" title="definition">repr</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#FactorMorphism.q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#619a2190d60a66179f3396458e2a09ae"><span class="id" title="notation">@*^-1</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">set</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="mathcomp.ssreflect.finset.html#b08e42f5c9c65aa9346e7b6dc26e3b5a"><span class="id" title="notation">]</span></a>)).<br/> + +<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="FactorMorphism.sKqKf"><span class="id" title="variable">sKqKf</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.morphism.html#FactorMorphism.q"><span class="id" title="variable">q</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#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.morphism.html#FactorMorphism.f"><span class="id" title="variable">f</span></a>.<br/> +<span class="id" title="keyword">Hypothesis</span> <a name="FactorMorphism.sGH"><span class="id" title="variable">sGH</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#FactorMorphism.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.morphism.html#FactorMorphism.H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="ff"><span class="id" title="abbreviation">ff</span></a> := (<a class="idref" href="mathcomp.fingroup.morphism.html#factm"><span class="id" title="definition">factm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#FactorMorphism.sKqKf"><span class="id" title="variable">sKqKf</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#FactorMorphism.sGH"><span class="id" title="variable">sGH</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="factmE"><span class="id" title="lemma">factmE</span></a> <span class="id" title="var">x</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#FactorMorphism.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.morphism.html#ff"><span class="id" title="abbreviation">ff</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#FactorMorphism.q"><span class="id" title="variable">q</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#FactorMorphism.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#x"><span class="id" title="variable">x</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="factm_morphM"><span class="id" title="lemma">factm_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.morphism.html#FactorMorphism.q"><span class="id" title="variable">q</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#FactorMorphism.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.morphism.html#ff"><span class="id" title="abbreviation">ff</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.morphism.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.morphism.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">factm_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.morphism.html#factm_morphM"><span class="id" title="lemma">factm_morphM</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_factm"><span class="id" title="lemma">morphim_factm</span></a> (<span class="id" title="var">A</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.morphism.html#FactorMorphism.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 class="idref" href="mathcomp.fingroup.morphism.html#ff"><span class="id" title="abbreviation">ff</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.morphism.html#FactorMorphism.q"><span class="id" title="variable">q</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#A"><span class="id" title="variable">A</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.morphism.html#FactorMorphism.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#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_factm"><span class="id" title="lemma">morphpre_factm</span></a> (<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.morphism.html#FactorMorphism.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>) : <a class="idref" href="mathcomp.fingroup.morphism.html#ff"><span class="id" title="abbreviation">ff</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.morphism.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.morphism.html#FactorMorphism.q"><span class="id" title="variable">q</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.morphism.html#FactorMorphism.f"><span class="id" title="variable">f</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.morphism.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_factm"><span class="id" title="lemma">ker_factm</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.morphism.html#ff"><span class="id" title="abbreviation">ff</span></a> <a 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.morphism.html#FactorMorphism.q"><span class="id" title="variable">q</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#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.morphism.html#FactorMorphism.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_factm"><span class="id" title="lemma">injm_factm</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.morphism.html#FactorMorphism.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.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.morphism.html#ff"><span class="id" title="abbreviation">ff</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_factmP"><span class="id" title="lemma">injm_factmP</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.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.morphism.html#FactorMorphism.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</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.morphism.html#FactorMorphism.q"><span class="id" title="variable">q</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.morphism.html#ff"><span class="id" title="abbreviation">ff</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_factm_loc"><span class="id" title="lemma">ker_factm_loc</span></a> (<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.morphism.html#FactorMorphism.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 class="idref" href="mathcomp.fingroup.morphism.html#ad50b791c3cc7be9475b0d43c3f572dd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#ad50b791c3cc7be9475b0d43c3f572dd"><span class="id" title="notation">ker_</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#ad50b791c3cc7be9475b0d43c3f572dd"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#FactorMorphism.q"><span class="id" title="variable">q</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#K"><span class="id" title="variable">K</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#ad50b791c3cc7be9475b0d43c3f572dd"><span class="id" title="notation">)</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ff"><span class="id" title="abbreviation">ff</span></a> <a 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.morphism.html#FactorMorphism.q"><span class="id" title="variable">q</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#ad50b791c3cc7be9475b0d43c3f572dd"><span class="id" title="notation">'</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#ad50b791c3cc7be9475b0d43c3f572dd"><span class="id" title="notation">ker_K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#FactorMorphism.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#FactorMorphism"><span class="id" title="section">FactorMorphism</span></a>.<br/> + +<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="InverseMorphism"><span class="id" title="section">InverseMorphism</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="InverseMorphism.aT"><span class="id" title="variable">aT</span></a> <a name="InverseMorphism.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.morphism.html#InverseMorphism.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>.<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.morphism.html#InverseMorphism.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">Variables</span> (<a name="InverseMorphism.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.morphism.html#InverseMorphism.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="InverseMorphism.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.morphism.html#G"><span class="id" title="variable">G</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#InverseMorphism.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="InverseMorphism.injf"><span class="id" title="variable">injf</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.morphism.html#InverseMorphism.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="invm_subker"><span class="id" title="lemma">invm_subker</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.morphism.html#InverseMorphism.f"><span class="id" title="variable">f</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#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.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#idm"><span class="id" title="definition">idm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#InverseMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="invm"><span class="id" title="definition">invm</span></a> := <a class="idref" href="mathcomp.fingroup.morphism.html#factm"><span class="id" title="definition">factm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#invm_subker"><span class="id" title="lemma">invm_subker</span></a> (<a class="idref" href="mathcomp.ssreflect.fintype.html#subxx"><span class="id" title="lemma">subxx</span></a> <span class="id" title="var">_</span>).<br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">invm_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.morphism.html#invm"><span class="id" title="definition">invm</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="invmE"><span class="id" title="lemma">invmE</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.morphism.html#InverseMorphism.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#InverseMorphism.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#invm"><span class="id" title="definition">invm</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="invmK"><span class="id" title="lemma">invmK</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.morphism.html#InverseMorphism.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#InverseMorphism.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#cancel"><span class="id" title="definition">cancel</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#invm"><span class="id" title="definition">invm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#InverseMorphism.f"><span class="id" title="variable">f</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="morphpre_invm"><span class="id" title="lemma">morphpre_invm</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#invm"><span class="id" title="definition">invm</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.morphism.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.morphism.html#InverseMorphism.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#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_invm"><span class="id" title="lemma">morphim_invm</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#InverseMorphism.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.morphism.html#invm"><span class="id" title="definition">invm</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.morphism.html#InverseMorphism.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#A"><span class="id" title="variable">A</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.morphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_invmE"><span class="id" title="lemma">morphim_invmE</span></a> <span class="id" title="var">C</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#invm"><span class="id" title="definition">invm</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#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.morphism.html#InverseMorphism.f"><span class="id" title="variable">f</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.morphism.html#C"><span class="id" title="variable">C</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_proper"><span class="id" title="lemma">injm_proper</span></a> <span class="id" title="var">A</span> <span class="id" title="var">B</span> :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#InverseMorphism.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.morphism.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.morphism.html#InverseMorphism.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#InverseMorphism.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#A"><span class="id" title="variable">A</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.morphism.html#InverseMorphism.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#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.morphism.html#A"><span class="id" title="variable">A</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.morphism.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="injm_invm"><span class="id" title="lemma">injm_invm</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.morphism.html#invm"><span class="id" title="definition">invm</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_invm"><span class="id" title="lemma">ker_invm</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.morphism.html#invm"><span class="id" title="definition">invm</span></a> <a 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_invm"><span class="id" title="lemma">im_invm</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#invm"><span class="id" title="definition">invm</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.morphism.html#InverseMorphism.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#InverseMorphism.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.morphism.html#InverseMorphism.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#InverseMorphism"><span class="id" title="section">InverseMorphism</span></a>.<br/> + +<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="InjFactm"><span class="id" title="section">InjFactm</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="InjFactm.gT"><span class="id" title="variable">gT</span></a> <a name="InjFactm.aT"><span class="id" title="variable">aT</span></a> <a name="InjFactm.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="InjFactm.D"><span class="id" title="variable">D</span></a> <a name="InjFactm.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.morphism.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/> +<span class="id" title="keyword">Variables</span> (<a name="InjFactm.g"><span class="id" title="variable">g</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.morphism.html#InjFactm.G"><span class="id" title="variable">G</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#InjFactm.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="InjFactm.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.morphism.html#InjFactm.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.morphism.html#InjFactm.aT"><span class="id" title="variable">aT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>) (<a name="InjFactm.injf"><span class="id" title="variable">injf</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.morphism.html#f"><span class="id" title="variable">f</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="ifactm"><span class="id" title="definition">ifactm</span></a> :=<br/> + <a class="idref" href="mathcomp.ssreflect.eqtype.html#tag"><span class="id" title="definition">tag</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#domP"><span class="id" title="lemma">domP</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#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.morphism.html#InjFactm.g"><span class="id" title="variable">g</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">\</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.ssr.ssrfun.html#1b4394c5c1740ef3dc9e4224084970bb"><span class="id" title="notation">o</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#invm"><span class="id" title="definition">invm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#InjFactm.injf"><span class="id" title="variable">injf</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#morphpre_invm"><span class="id" title="lemma">morphpre_invm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#InjFactm.injf"><span class="id" title="variable">injf</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#InjFactm.G"><span class="id" title="variable">G</span></a>)).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ifactmE"><span class="id" title="lemma">ifactmE</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.morphism.html#InjFactm.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">,</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="mathcomp.fingroup.morphism.html#ifactm"><span class="id" title="definition">ifactm</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#InjFactm.f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#InjFactm.g"><span class="id" title="variable">g</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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#5c59b35a0b51db520cf1fba473ecf127"><span class="id" title="notation">}</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_ifactm"><span class="id" title="lemma">morphim_ifactm</span></a> (<span class="id" title="var">A</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.morphism.html#InjFactm.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/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#InjFactm.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#ifactm"><span class="id" title="definition">ifactm</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.morphism.html#InjFactm.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#A"><span class="id" title="variable">A</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.morphism.html#InjFactm.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="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_ifactm"><span class="id" title="lemma">im_ifactm</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#InjFactm.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.morphism.html#InjFactm.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#ifactm"><span class="id" title="definition">ifactm</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.morphism.html#InjFactm.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#InjFactm.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.morphism.html#InjFactm.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="mathcomp.fingroup.morphism.html#InjFactm.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphpre_ifactm"><span class="id" title="lemma">morphpre_ifactm</span></a> <span class="id" title="var">C</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#ifactm"><span class="id" title="definition">ifactm</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.morphism.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.morphism.html#InjFactm.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.morphism.html#InjFactm.g"><span class="id" title="variable">g</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.morphism.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_ifactm"><span class="id" title="lemma">ker_ifactm</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.morphism.html#ifactm"><span class="id" title="definition">ifactm</span></a> <a 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.morphism.html#InjFactm.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#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.morphism.html#InjFactm.g"><span class="id" title="variable">g</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_ifactm"><span class="id" title="lemma">injm_ifactm</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.morphism.html#InjFactm.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.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.morphism.html#ifactm"><span class="id" title="definition">ifactm</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#InjFactm"><span class="id" title="section">InjFactm</span></a>.<br/> + +<br/> +</div> + +<div class="doc"> + Reflected (boolean) form of morphism and isomorphism properties. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ReflectProp"><span class="id" title="section">ReflectProp</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="ReflectProp.aT"><span class="id" title="variable">aT</span></a> <a name="ReflectProp.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/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ReflectProp.Defs"><span class="id" title="section">Defs</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="ReflectProp.Defs.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.morphism.html#ReflectProp.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="ReflectProp.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.morphism.html#ReflectProp.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/> +</div> + +<div class="doc"> + morphic is the morphM property of morphisms seen through morphicP. +</div> +<div class="code"> +<span class="id" title="keyword">Definition</span> <a name="morphic"><span class="id" title="definition">morphic</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.aT"><span class="id" title="variable">aT</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#ReflectProp.rT"><span class="id" title="variable">rT</span></a>) :=<br/> + <a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">∀</span></a> <span class="id" title="var">u</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">in</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#a4c99a0dbc2a758b24afbd951fc3a580"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#a4c99a0dbc2a758b24afbd951fc3a580"><span class="id" title="notation">predX</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Defs.A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#a4c99a0dbc2a758b24afbd951fc3a580"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Defs.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.ssreflect.eqtype.html#a4c99a0dbc2a758b24afbd951fc3a580"><span class="id" title="notation">]</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#46e5a4123d46e6b126f7788a77176785"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.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.morphism.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.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.ssreflect.fintype.html#0a2353937835d965c09d6cd592199019"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="isom"><span class="id" title="definition">isom</span></a> <span class="id" title="var">f</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Defs.A"><span class="id" title="variable">A</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#adb8044960c962a921cca1bd48aae97d"><span class="id" title="notation">^#</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Defs.B"><span class="id" title="variable">B</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">Definition</span> <a name="misom"><span class="id" title="definition">misom</span></a> <span class="id" title="var">f</span> := <a class="idref" href="mathcomp.fingroup.morphism.html#morphic"><span class="id" title="definition">morphic</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</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.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="isog"><span class="id" title="definition">isog</span></a> := <a class="idref" href="mathcomp.ssreflect.fintype.html#a843dcbb9dc2e69b147054d3e1465e78"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#a843dcbb9dc2e69b147054d3e1465e78"><span class="id" title="notation">∃</span></a> <span class="id" title="var">f</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#a843dcbb9dc2e69b147054d3e1465e78"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#9f24a6f16bf73832c2d9aa4e2c16f692"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#9f24a6f16bf73832c2d9aa4e2c16f692"><span class="id" title="notation">ffun</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.aT"><span class="id" title="variable">aT</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#ReflectProp.rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#9f24a6f16bf73832c2d9aa4e2c16f692"><span class="id" title="notation">}</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#46e5a4123d46e6b126f7788a77176785"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#misom"><span class="id" title="definition">misom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#a843dcbb9dc2e69b147054d3e1465e78"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ReflectProp.Defs.MorphicProps"><span class="id" title="section">MorphicProps</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="ReflectProp.Defs.MorphicProps.f"><span class="id" title="variable">f</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.aT"><span class="id" title="variable">aT</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#ReflectProp.rT"><span class="id" title="variable">rT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphicP"><span class="id" title="lemma">morphicP</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="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.morphism.html#ReflectProp.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#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.morphism.html#ReflectProp.Defs.MorphicProps.f"><span class="id" title="variable">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.morphism.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.morphism.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> (<a class="idref" href="mathcomp.fingroup.morphism.html#morphic"><span class="id" title="definition">morphic</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Defs.MorphicProps.f"><span class="id" title="variable">f</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Definition</span> <a name="morphm"><span class="id" title="definition">morphm</span></a> <span class="id" title="keyword">of</span> <a class="idref" href="mathcomp.fingroup.morphism.html#morphic"><span class="id" title="definition">morphic</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Defs.MorphicProps.f"><span class="id" title="variable">f</span></a> := <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Defs.MorphicProps.f"><span class="id" title="variable">f</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.morphism.html#ReflectProp.aT"><span class="id" title="variable">aT</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.morphism.html#ReflectProp.rT"><span class="id" title="variable">rT</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphmE"><span class="id" title="lemma">morphmE</span></a> <span class="id" title="var">fM</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#morphm"><span class="id" title="definition">morphm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#fM"><span class="id" title="variable">fM</span></a> <a 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.morphism.html#ReflectProp.Defs.MorphicProps.f"><span class="id" title="variable">f</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">morphm_morphism</span> <span class="id" title="var">fM</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.fingroup.morphism.html#ReflectProp.Defs.A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#morphm"><span class="id" title="definition">morphm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#fM"><span class="id" title="variable">fM</span></a>) (<a class="idref" href="mathcomp.fingroup.morphism.html#morphicP"><span class="id" title="lemma">morphicP</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#fM"><span class="id" title="variable">fM</span></a>).<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Defs.MorphicProps"><span class="id" title="section">MorphicProps</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="misomP"><span class="id" title="lemma">misomP</span></a> <span class="id" title="var">f</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="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">{</span></a><span class="id" title="var">fM</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#morphic"><span class="id" title="definition">morphic</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><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.morphism.html#morphm"><span class="id" title="definition">morphm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#fM"><span class="id" title="variable">fM</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Specif.html#50b5d8dd6be4fba768e35617e518ad76"><span class="id" title="notation">}</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#misom"><span class="id" title="definition">misom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="misom_isog"><span class="id" title="lemma">misom_isog</span></a> <span class="id" title="var">f</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#misom"><span class="id" title="definition">misom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#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.morphism.html#isog"><span class="id" title="definition">isog</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isom_isog"><span class="id" title="lemma">isom_isog</span></a> (<span class="id" title="var">D</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.morphism.html#ReflectProp.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>) (<span class="id" title="var">f</span> : <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.morphism.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.morphism.html#ReflectProp.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/> + <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Defs.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.morphism.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#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#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.morphism.html#isog"><span class="id" title="definition">isog</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isog_isom"><span class="id" title="lemma">isog_isom</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#isog"><span class="id" title="definition">isog</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.Specif.html#72ca3fac4636a1b19c963b12162882cf"><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#72ca3fac4636a1b19c963b12162882cf"><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.morphism.html#ReflectProp.Defs.A"><span class="id" title="variable">A</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#ReflectProp.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 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.morphism.html#f"><span class="id" title="variable">f</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">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Defs"><span class="id" title="section">Defs</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Infix</span> <a name="6202b790dcb0ea1946ff79ad4eb4448e"><span class="id" title="notation">"</span></a>\isog" := <a class="idref" href="mathcomp.fingroup.morphism.html#isog"><span class="id" title="definition">isog</span></a>.<br/> + +<br/> + +<br/> +</div> + +<div class="doc"> + The real reflection properties only hold for true groups and morphisms. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Section</span> <a name="ReflectProp.Main"><span class="id" title="section">Main</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="ReflectProp.Main.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.morphism.html#ReflectProp.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="ReflectProp.Main.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.morphism.html#ReflectProp.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">Lemma</span> <a name="isomP"><span class="id" title="lemma">isomP</span></a> (<span class="id" title="var">f</span> : <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.morphism.html#ReflectProp.Main.G"><span class="id" title="variable">G</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#ReflectProp.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/> + <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.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.morphism.html#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#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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#ReflectProp.Main.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.morphism.html#ReflectProp.Main.H"><span class="id" title="variable">H</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.morphism.html#ReflectProp.Main.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Main.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isogP"><span class="id" title="lemma">isogP</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> (<a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">exists2</span></a> <span class="id" title="var">f</span> <a class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><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.morphism.html#ReflectProp.Main.G"><span class="id" title="variable">G</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#ReflectProp.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 class="idref" href="http://coq.inria.fr/distrib/8.8.0/stdlib//Coq.Init.Logic.html#fe60c20831f772c0c3c288abf68cc42a"><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.morphism.html#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#fe60c20831f772c0c3c288abf68cc42a"><span class="id" title="notation">&</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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#ReflectProp.Main.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.morphism.html#ReflectProp.Main.H"><span class="id" title="variable">H</span></a>) (<a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Main.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#6202b790dcb0ea1946ff79ad4eb4448e"><span class="id" title="notation">\</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#6202b790dcb0ea1946ff79ad4eb4448e"><span class="id" title="notation">isog</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Main.H"><span class="id" title="variable">H</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Variable</span> <a name="ReflectProp.Main.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.morphism.html#ReflectProp.Main.G"><span class="id" title="variable">G</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#ReflectProp.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="ReflectProp.Main.isoGH"><span class="id" title="variable">isoGH</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.morphism.html#ReflectProp.Main.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Main.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Main.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isom_inj"><span class="id" title="lemma">isom_inj</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.morphism.html#ReflectProp.Main.f"><span class="id" title="variable">f</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="isom_im"><span class="id" title="lemma">isom_im</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Main.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#ReflectProp.Main.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.morphism.html#ReflectProp.Main.H"><span class="id" title="variable">H</span></a>. <br/> +<span class="id" title="keyword">Lemma</span> <a name="isom_card"><span class="id" title="lemma">isom_card</span></a> : <a class="idref" href="mathcomp.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Main.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#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.morphism.html#ReflectProp.Main.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>.<br/> + <span class="id" title="keyword">Lemma</span> <a name="isom_sub_im"><span class="id" title="lemma">isom_sub_im</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Main.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.morphism.html#ReflectProp.Main.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#ReflectProp.Main.G"><span class="id" title="variable">G</span></a>. <br/> +<span class="id" title="keyword">Definition</span> <a name="isom_inv"><span class="id" title="definition">isom_inv</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.morphism.html#isom_sub_im"><span class="id" title="lemma">isom_sub_im</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#invm"><span class="id" title="definition">invm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#isom_inj"><span class="id" title="lemma">isom_inj</span></a>).<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.Main"><span class="id" title="section">Main</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="ReflectProp.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.morphism.html#ReflectProp.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="ReflectProp.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.morphism.html#G"><span class="id" title="variable">G</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#ReflectProp.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">Lemma</span> <a name="morphim_isom"><span class="id" title="lemma">morphim_isom</span></a> (<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.morphism.html#ReflectProp.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>) (<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.morphism.html#ReflectProp.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/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#ReflectProp.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.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.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.morphism.html#ReflectProp.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#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.morphism.html#K"><span class="id" title="variable">K</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sub_isom"><span class="id" title="lemma">sub_isom</span></a> (<span class="id" title="var">A</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.morphism.html#ReflectProp.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>) (<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.morphism.html#ReflectProp.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/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#ReflectProp.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.morphism.html#ReflectProp.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#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.morphism.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#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.morphism.html#ReflectProp.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.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.f"><span class="id" title="variable">f</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sub_isog"><span class="id" title="lemma">sub_isog</span></a> (<span class="id" title="var">A</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.morphism.html#ReflectProp.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 class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#ReflectProp.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.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.morphism.html#ReflectProp.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.morphism.html#isog"><span class="id" title="definition">isog</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.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#A"><span class="id" title="variable">A</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="restr_isom_to"><span class="id" title="lemma">restr_isom_to</span></a> (<span class="id" title="var">A</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.morphism.html#ReflectProp.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>) (<span class="id" title="var">C</span> <span class="id" title="var">R</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.morphism.html#ReflectProp.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>) (<span class="id" title="var">sAG</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#ReflectProp.G"><span class="id" title="variable">G</span></a>) :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.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#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.morphism.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#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.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.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#C"><span class="id" title="variable">C</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.morphism.html#sAG"><span class="id" title="variable">sAG</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.f"><span class="id" title="variable">f</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="restr_isom"><span class="id" title="lemma">restr_isom</span></a> (<span class="id" title="var">A</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.morphism.html#ReflectProp.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>) (<span class="id" title="var">R</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.morphism.html#ReflectProp.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>) (<span class="id" title="var">sAG</span> : <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#ReflectProp.G"><span class="id" title="variable">G</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.morphism.html#ReflectProp.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#R"><span class="id" title="variable">R</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.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.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.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#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.morphism.html#sAG"><span class="id" title="variable">sAG</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp.f"><span class="id" title="variable">f</span></a>).<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#ReflectProp"><span class="id" title="section">ReflectProp</span></a>.<br/> + +<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="cec6c3028572f2d4d267ecf02dc64058"><span class="id" title="notation">"</span></a>x \isog y":= (<a class="idref" href="mathcomp.fingroup.morphism.html#isog"><span class="id" title="definition">isog</span></a> <span class="id" title="var">x</span> <span class="id" title="var">y</span>).<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Isomorphisms"><span class="id" title="section">Isomorphisms</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="Isomorphisms.gT"><span class="id" title="variable">gT</span></a> <a name="Isomorphisms.hT"><span class="id" title="variable">hT</span></a> <a name="Isomorphisms.kT"><span class="id" title="variable">kT</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="Isomorphisms.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.morphism.html#Isomorphisms.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="Isomorphisms.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.morphism.html#Isomorphisms.hT"><span class="id" title="variable">hT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="Isomorphisms.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.morphism.html#Isomorphisms.kT"><span class="id" title="variable">kT</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="idm_isom"><span class="id" title="lemma">idm_isom</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.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#idm"><span class="id" title="definition">idm</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isog_refl"><span class="id" title="lemma">isog_refl</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</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.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</span></a>. <br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="card_isog"><span class="id" title="lemma">card_isog</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</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.morphism.html#Isomorphisms.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.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.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#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.morphism.html#Isomorphisms.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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isog_abelian"><span class="id" title="lemma">isog_abelian</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</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.morphism.html#Isomorphisms.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.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.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.fingroup.html#abelian"><span class="id" title="definition">abelian</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="trivial_isog"><span class="id" title="lemma">trivial_isog</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><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.morphism.html#Isomorphisms.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#67291ec55239f54fa5aa0b0bb974446c"><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.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</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.morphism.html#Isomorphisms.H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isog_eq1"><span class="id" title="lemma">isog_eq1</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</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.morphism.html#Isomorphisms.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="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.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#24f47bb7b1a372904563d2bdb8a213a4"><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.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#24f47bb7b1a372904563d2bdb8a213a4"><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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isom_sym"><span class="id" title="lemma">isom_sym</span></a> (<span class="id" title="var">f</span> : <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.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</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#Isomorphisms.hT"><span class="id" title="variable">hT</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#c5b2825fcd994c4c5cc69df8802f5376"><span class="id" title="notation">}</span></a>) (<span class="id" title="var">isoGH</span> : <a class="idref" href="mathcomp.fingroup.morphism.html#isom"><span class="id" title="definition">isom</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</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.morphism.html#Isomorphisms.H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.html#isom_inv"><span class="id" title="definition">isom_inv</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#isoGH"><span class="id" title="variable">isoGH</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isog_symr"><span class="id" title="lemma">isog_symr</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</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.morphism.html#Isomorphisms.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.morphism.html#Isomorphisms.H"><span class="id" title="variable">H</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.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isog_trans"><span class="id" title="lemma">isog_trans</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</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.morphism.html#Isomorphisms.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.morphism.html#Isomorphisms.H"><span class="id" title="variable">H</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.morphism.html#Isomorphisms.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.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</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.morphism.html#Isomorphisms.K"><span class="id" title="variable">K</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="nclasses_isog"><span class="id" title="lemma">nclasses_isog</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.G"><span class="id" title="variable">G</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.morphism.html#Isomorphisms.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.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.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#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.fingroup.html#classes"><span class="id" title="definition">classes</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms.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>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#Isomorphisms"><span class="id" title="section">Isomorphisms</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="IsoBoolEquiv"><span class="id" title="section">IsoBoolEquiv</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> <a name="IsoBoolEquiv.gT"><span class="id" title="variable">gT</span></a> <a name="IsoBoolEquiv.hT"><span class="id" title="variable">hT</span></a> <a name="IsoBoolEquiv.kT"><span class="id" title="variable">kT</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="IsoBoolEquiv.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.morphism.html#IsoBoolEquiv.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="IsoBoolEquiv.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.morphism.html#IsoBoolEquiv.hT"><span class="id" title="variable">hT</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#ab072eb546972c7e5cdaf33b8a35ce6b"><span class="id" title="notation">}</span></a>) (<a name="IsoBoolEquiv.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.morphism.html#IsoBoolEquiv.kT"><span class="id" title="variable">kT</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="isog_sym"><span class="id" title="lemma">isog_sym</span></a> : <a 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.morphism.html#IsoBoolEquiv.G"><span class="id" title="variable">G</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.morphism.html#IsoBoolEquiv.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="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.morphism.html#IsoBoolEquiv.H"><span class="id" title="variable">H</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.morphism.html#IsoBoolEquiv.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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isog_transl"><span class="id" title="lemma">isog_transl</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#IsoBoolEquiv.G"><span class="id" title="variable">G</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.morphism.html#IsoBoolEquiv.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="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.morphism.html#IsoBoolEquiv.G"><span class="id" title="variable">G</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.morphism.html#IsoBoolEquiv.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="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.morphism.html#IsoBoolEquiv.H"><span class="id" title="variable">H</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.morphism.html#IsoBoolEquiv.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>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isog_transr"><span class="id" title="lemma">isog_transr</span></a> : <a class="idref" href="mathcomp.fingroup.morphism.html#IsoBoolEquiv.G"><span class="id" title="variable">G</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.morphism.html#IsoBoolEquiv.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="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.morphism.html#IsoBoolEquiv.K"><span class="id" title="variable">K</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.morphism.html#IsoBoolEquiv.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="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.morphism.html#IsoBoolEquiv.K"><span class="id" title="variable">K</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.morphism.html#IsoBoolEquiv.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>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#IsoBoolEquiv"><span class="id" title="section">IsoBoolEquiv</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Section</span> <a name="Homg"><span class="id" title="section">Homg</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Implicit</span> <span class="id" title="keyword">Types</span> <span class="id" title="var">rT</span> <span class="id" title="var">gT</span> <span class="id" title="var">aT</span> : <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="homg"><span class="id" title="definition">homg</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">aT</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.morphism.html#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">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.morphism.html#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>) :=<br/> + <a class="idref" href="mathcomp.ssreflect.fintype.html#158832d8c1ec7121db752c45c87cc3aa"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#158832d8c1ec7121db752c45c87cc3aa"><span class="id" title="notation">∃</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#158832d8c1ec7121db752c45c87cc3aa"><span class="id" title="notation">(</span></a><span class="id" title="var">f</span> <a class="idref" href="mathcomp.ssreflect.fintype.html#158832d8c1ec7121db752c45c87cc3aa"><span class="id" title="notation">:</span></a> <a class="idref" href="mathcomp.ssreflect.finfun.html#9f24a6f16bf73832c2d9aa4e2c16f692"><span class="id" title="notation">{</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#9f24a6f16bf73832c2d9aa4e2c16f692"><span class="id" title="notation">ffun</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#aT"><span class="id" title="variable">aT</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#rT"><span class="id" title="variable">rT</span></a><a class="idref" href="mathcomp.ssreflect.finfun.html#9f24a6f16bf73832c2d9aa4e2c16f692"><span class="id" title="notation">}</span></a> <a class="idref" href="mathcomp.ssreflect.fintype.html#158832d8c1ec7121db752c45c87cc3aa"><span class="id" title="notation">|</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#morphic"><span class="id" title="definition">morphic</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#158832d8c1ec7121db752c45c87cc3aa"><span class="id" title="notation">)</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#46e5a4123d46e6b126f7788a77176785"><span class="id" title="notation">,</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="mathcomp.ssreflect.finset.html#f2bdcb40cf423bf8d54f091f6cec6964"><span class="id" title="notation">@:</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#D"><span class="id" title="variable">D</span></a> <a class="idref" href="mathcomp.ssreflect.eqtype.html#17d28d004d0863cb022d4ce832ddaaae"><span class="id" title="notation">==</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#C"><span class="id" title="variable">C</span></a><a class="idref" href="mathcomp.ssreflect.fintype.html#158832d8c1ec7121db752c45c87cc3aa"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="homgP"><span class="id" title="lemma">homgP</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">aT</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.morphism.html#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">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.morphism.html#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>) : <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">f</span> : <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.morphism.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.morphism.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><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.morphism.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#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.morphism.html#C"><span class="id" title="variable">C</span></a>) (<a class="idref" href="mathcomp.fingroup.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#D"><span class="id" title="variable">D</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="morphim_homg"><span class="id" title="lemma">morphim_homg</span></a> <span class="id" title="var">aT</span> <span class="id" title="var">rT</span> (<span class="id" title="var">A</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.morphism.html#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>) (<span class="id" title="var">f</span> : <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.morphism.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.morphism.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/> + <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.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#homg"><span class="id" title="definition">homg</span></a> (<a class="idref" href="mathcomp.fingroup.morphism.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#A"><span class="id" title="variable">A</span></a>) <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="leq_homg"><span class="id" title="lemma">leq_homg</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">aT</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.morphism.html#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">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.morphism.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>) :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#C"><span class="id" title="variable">C</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.morphism.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="homg_refl"><span class="id" title="lemma">homg_refl</span></a> <span class="id" title="var">aT</span> (<span class="id" title="var">A</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.morphism.html#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 class="idref" href="mathcomp.fingroup.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="homg_trans"><span class="id" title="lemma">homg_trans</span></a> <span class="id" title="var">aT</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.morphism.html#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>) <span class="id" title="var">rT</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.morphism.html#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">gT</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.morphism.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/> + <a class="idref" href="mathcomp.fingroup.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#B"><span class="id" title="variable">B</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#C"><span class="id" title="variable">C</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isogEcard"><span class="id" title="lemma">isogEcard</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">aT</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.morphism.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>) (<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.morphism.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>) :<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.morphism.html#G"><span class="id" title="variable">G</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.morphism.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="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.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.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.ssreflect.fintype.html#f01714bb99e6c7abc6cfb2e43eff7f6e"><span class="id" title="notation">#|</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#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="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.morphism.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.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">)</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isog_hom"><span class="id" title="lemma">isog_hom</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">aT</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.morphism.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>) (<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.morphism.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 class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</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.morphism.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.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isogEhom"><span class="id" title="lemma">isogEhom</span></a> <span class="id" title="var">rT</span> <span class="id" title="var">aT</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.morphism.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>) (<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.morphism.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>) :<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.morphism.html#G"><span class="id" title="variable">G</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.morphism.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="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.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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.Datatypes.html#49ac24efa716d8b0ee8943bc1d1769a9"><span class="id" title="notation">&&</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="eq_homgl"><span class="id" title="lemma">eq_homgl</span></a> <span class="id" title="var">gT</span> <span class="id" title="var">aT</span> <span class="id" title="var">rT</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.morphism.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>) (<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.morphism.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>) (<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.morphism.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/> + <a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</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.morphism.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.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#H"><span class="id" title="variable">H</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#K"><span class="id" title="variable">K</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="eq_homgr"><span class="id" title="lemma">eq_homgr</span></a> <span class="id" title="var">gT</span> <span class="id" title="var">rT</span> <span class="id" title="var">aT</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.morphism.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>) (<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.morphism.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>) (<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.morphism.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>) :<br/> + <a class="idref" href="mathcomp.fingroup.morphism.html#G"><span class="id" title="variable">G</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.morphism.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.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.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#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#homg"><span class="id" title="definition">homg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#K"><span class="id" title="variable">K</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#H"><span class="id" title="variable">H</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#Homg"><span class="id" title="section">Homg</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Notation</span> <a name="660be9f1a393f4f575133c9df42aec65"><span class="id" title="notation">"</span></a>G \homg H" := (<a class="idref" href="mathcomp.fingroup.morphism.html#homg"><span class="id" title="definition">homg</span></a> <span class="id" title="var">G</span> <span class="id" title="var">H</span>)<br/> + (<span class="id" title="tactic">at</span> <span class="id" title="keyword">level</span> 70, <span class="id" title="keyword">no</span> <span class="id" title="keyword">associativity</span>) : <span class="id" title="var">group_scope</span>.<br/> + +<br/> + +<br/> +</div> + +<div class="doc"> + Isomorphism between a group and its subtype. +</div> +<div class="code"> + +<br/> +<span class="id" title="keyword">Section</span> <a name="SubMorphism"><span class="id" title="section">SubMorphism</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Variables</span> (<a name="SubMorphism.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="SubMorphism.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.morphism.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">Canonical</span> <span class="id" title="var">sgval_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.fingroup.html#sgvalM"><span class="id" title="lemma">sgvalM</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a>).<br/> +<span class="id" title="keyword">Canonical</span> <span class="id" title="var">subg_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.fingroup.html#subgM"><span class="id" title="lemma">subgM</span></a> <span class="id" title="var">_</span> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_sgval"><span class="id" title="lemma">injm_sgval</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.fingroup.html#sgval"><span class="id" title="definition">sgval</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="injm_subg"><span class="id" title="lemma">injm_subg</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.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#subg"><span class="id" title="definition">subg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#14bfb149f00fa839cfb11397f4fe629f"><span class="id" title="notation">)</span></a>.<br/> + <span class="id" title="keyword">Hint Resolve</span> <span class="id" title="var">injm_sgval</span> <span class="id" title="var">injm_subg</span>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="ker_sgval"><span class="id" title="lemma">ker_sgval</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.fingroup.html#sgval"><span class="id" title="definition">sgval</span></a> <a 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/> +<span class="id" title="keyword">Lemma</span> <a name="ker_subg"><span class="id" title="lemma">ker_subg</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.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#subg"><span class="id" title="definition">subg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.morphism.html#034cc0eb573e9a86d9574eaed7b27a13"><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_subg"><span class="id" title="lemma">im_subg</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#subg"><span class="id" title="definition">subg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.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="mathcomp.fingroup.morphism.html#SubMorphism.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.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">subg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sgval_sub"><span class="id" title="lemma">sgval_sub</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#sgval"><span class="id" title="definition">sgval</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#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.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="sgvalmK"><span class="id" title="lemma">sgvalmK</span></a> <span class="id" title="var">A</span> : <a class="idref" href="mathcomp.fingroup.fingroup.html#subg"><span class="id" title="definition">subg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.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="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">(</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#sgval"><span class="id" title="definition">sgval</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#A"><span class="id" title="variable">A</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.morphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="subgmK"><span class="id" title="lemma">subgmK</span></a> (<span class="id" title="var">A</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.morphism.html#SubMorphism.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.morphism.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.morphism.html#SubMorphism.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#sgval"><span class="id" title="definition">sgval</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.fingroup.html#subg"><span class="id" title="definition">subg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.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="mathcomp.fingroup.morphism.html#A"><span class="id" title="variable">A</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.morphism.html#A"><span class="id" title="variable">A</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="im_sgval"><span class="id" title="lemma">im_sgval</span></a> : <a class="idref" href="mathcomp.fingroup.fingroup.html#sgval"><span class="id" title="definition">sgval</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#48cff845c81518398138031392d44c93"><span class="id" title="notation">@*</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">subg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><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.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isom_subg"><span class="id" title="lemma">isom_subg</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.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">subg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">]</span></a> (<a class="idref" href="mathcomp.fingroup.fingroup.html#subg"><span class="id" title="definition">subg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a>).<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isom_sgval"><span class="id" title="lemma">isom_sgval</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.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">subg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">]</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#sgval"><span class="id" title="definition">sgval</span></a>.<br/> + +<br/> +<span class="id" title="keyword">Lemma</span> <a name="isog_subg"><span class="id" title="lemma">isog_subg</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.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a> <a class="idref" href="mathcomp.fingroup.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">[</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">subg</span></a> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism.G"><span class="id" title="variable">G</span></a><a class="idref" href="mathcomp.fingroup.fingroup.html#062f7a5a3435bce4208812f8529aae0f"><span class="id" title="notation">]</span></a>.<br/> + +<br/> +<span class="id" title="keyword">End</span> <a class="idref" href="mathcomp.fingroup.morphism.html#SubMorphism"><span class="id" title="section">SubMorphism</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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