phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
injm_normal A B : A \subset D -> B \subset D -> (f @* A <\| f @* B) = (A <\| B).	Proof. by move=> sAD sBD; rewrite /normal injmSK ?injm_norms. Qed.	Lemma	injm_normal	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injmSK", "injm_norms", "normal", "sAD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_subnorm A B : B \subset D -> f @* 'N_A(B) = 'N_(f @* A)(f @* B).	Proof. by move=> sBD; rewrite injmI injm_norm // setICA setIA morphimIim. Qed.	Lemma	injm_subnorm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injmI", "injm_norm", "morphimIim", "setIA", "setICA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_cent1 x : x \in D -> f @* 'C[x] = 'C_(f @* D)[f x].	Proof. by move=> Dx; rewrite injm_norm ?morphim_set1 ?sub1set. Qed.	Lemma	injm_cent1	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "injm_norm", "morphim_set1", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_subcent1 A x : x \in D -> f @* 'C_A[x] = 'C_(f @* A)[f x].	Proof. by move=> Dx; rewrite injm_subnorm ?morphim_set1 ?sub1set. Qed.	Lemma	injm_subcent1	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "injm_subnorm", "morphim_set1", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_cent A : A \subset D -> f @* 'C(A) = 'C_(f @* D)(f @* A).	Proof. move=> sAD; apply/eqP; rewrite -morphimIdom eqEsubset morphim_subcent. apply/subsetP=> fx; case/setIP; case/morphimP=> x Dx _ ->{fx} cAfx. rewrite mem_morphim // inE Dx -sub1set centsC cent_set1 -injmSK //. by rewrite injm_cent1 // subsetI morphimS // -cent_set1 centsC sub1set. Qed.	Lemma	injm_cent	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "cent_set1", "centsC", "eqEsubset", "inE", "injmSK", "injm_cent1", "mem_morphim", "morphimIdom", "morphimP", "morphimS", "morphim_subcent", "sAD", "setIP", "sub1set", "subsetI", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_cents A B : A \subset D -> B \subset D -> (f @* A \subset 'C(f @* B)) = (A \subset 'C(B)).	Proof. by move=> sAD sBD; rewrite -injmSK // injm_cent // subsetI morphimS. Qed.	Lemma	injm_cents	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injmSK", "injm_cent", "morphimS", "sAD", "subsetI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_subcent A B : B \subset D -> f @* 'C_A(B) = 'C_(f @* A)(f @* B).	Proof. by move=> sBD; rewrite injmI injm_cent // setICA setIA morphimIim. Qed.	Lemma	injm_subcent	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injmI", "injm_cent", "morphimIim", "setIA", "setICA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_abelian A : A \subset D -> abelian (f @* A) = abelian A.	Proof. by move=> sAD; rewrite /abelian -subsetIidl -injm_subcent // injmSK ?subsetIidl. Qed.	Lemma	injm_abelian	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "abelian", "injmSK", "injm_subcent", "sAD", "subsetIidl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_morphim (g : {morphism D >-> rT}): {in D, f =1 g} -> forall A, f @* A = g @* A.	Proof. by move=> efg A; apply: eq_in_imset; apply: sub_in1 efg => x /setIP[]. Qed.	Lemma	eq_morphim	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "eq_in_imset", "morphism", "setIP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_in_morphim B A (g : {morphism B >-> rT}) : D :&: A = B :&: A -> {in A, f =1 g} -> f @* A = g @* A.	Proof. move=> eqDBA eqAfg; rewrite /morphim /= eqDBA. by apply: eq_in_imset => x /setIP[_]/eqAfg. Qed.	Lemma	eq_in_morphim	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "eq_in_imset", "morphim", "morphism", "setIP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''ker' f"	:= (ker_group (MorPhantom f)) : Group_scope.	Notation	''ker' f	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "MorPhantom", "ker_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''ker_' G f"	:= (G :&: 'ker f)%G : Group_scope.	Notation	''ker_' G f	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"f @* G"	:= (morphim_group (MorPhantom f) G) : Group_scope.	Notation	f @* G	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "MorPhantom", "morphim_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"f @*^-1 M"	:= (morphpre_group (MorPhantom f) M) : Group_scope.	Notation	f @*^-1 M	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "MorPhantom", "morphpre_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"f @: D"	:= (morph_dom_group f D) : Group_scope.	Notation	f @: D	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morph_dom_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idm & {set gT}	:= fun x : gT => x : FinGroup.sort gT.	Definition	idm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "gT", "sort" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idm_morphM A : {in A & , {morph idm A : x y / x * y}}.	Proof. by []. Qed.	Lemma	idm_morphM	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "idm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idm_morphism A	:= Morphism (@idm_morphM A).	Canonical	idm_morphism	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "idm_morphM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_idm G : 'injm (idm G).	Proof. by apply/injmP=> x y _ _. Qed.	Lemma	injm_idm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "idm", "injmP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_idm G : 'ker (idm G) = 1.	Proof. by apply/trivgP; apply: injm_idm. Qed.	Lemma	ker_idm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "idm", "injm_idm", "ker", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_idm A B : B \subset A -> idm A @* B = B.	Proof. rewrite /morphim /= /idm => /setIidPr->. by apply/setP=> x; apply/imsetP/idP=> [[y By ->]\|Bx]; last exists x. Qed.	Lemma	morphim_idm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "idm", "imsetP", "last", "morphim", "setIidPr", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_idm A B : idm A @*^-1 B = A :&: B.	Proof. by apply/setP=> x; rewrite !inE. Qed.	Lemma	morphpre_idm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "idm", "inE", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_idm A : idm A @* A = A.	Proof. exact: morphim_idm. Qed.	Lemma	im_idm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "idm", "morphim_idm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restrm & A \subset D	:= @id (aT -> FinGroup.sort rT).	Definition	restrm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "id", "sort" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sAD : A \subset D.		Hypothesis	sAD	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fA	:= (restrm sAD (mfun f)).	Notation	fA	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "restrm", "sAD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restrm_morphism	:= @Morphism aT rT A fA (sub_in2 (subsetP sAD) (morphM f)).	Canonical	restrm_morphism	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "fA", "morphM", "sAD", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_restrm B : fA @* B = f @* (A :&: B).	Proof. by rewrite {2}/morphim setIA (setIidPr sAD). Qed.	Lemma	morphim_restrm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "fA", "morphim", "sAD", "setIA", "setIidPr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restrmEsub B : B \subset A -> fA @* B = f @* B.	Proof. by rewrite morphim_restrm => /setIidPr->. Qed.	Lemma	restrmEsub	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "fA", "morphim_restrm", "setIidPr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_restrm : fA @* A = f @* A.	Proof. exact: restrmEsub. Qed.	Lemma	im_restrm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "fA", "restrmEsub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_restrm R : fA @^-1 R = A :&: f @^-1 R.	Proof. by rewrite setIA (setIidPl sAD). Qed.	Lemma	morphpre_restrm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "fA", "sAD", "setIA", "setIidPl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_restrm : 'ker fA = 'ker_A f.	Proof. exact: morphpre_restrm. Qed.	Lemma	ker_restrm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "fA", "ker", "morphpre_restrm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_restrm : 'injm f -> 'injm fA.	Proof. by apply: subset_trans; rewrite ker_restrm subsetIr. Qed.	Lemma	injm_restrm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "fA", "ker_restrm", "subsetIr", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restrmP (f : {morphism D >-> rT}) : A \subset 'dom f -> {g : {morphism A >-> rT} \| [/\ g = f :> (aT -> rT), 'ker g = 'ker_A f, forall R, g @^-1 R = A :&: f @^-1 R & forall B, B \subset A -> g @* B = f @* B]}.	Proof. move=> sAD; exists (restrm_morphism sAD f). split=> // [\|R\|B sBA]; first 1 [exact: ker_restrm \| exact: morphpre_restrm]. by rewrite morphim_restrm (setIidPr sBA). Qed.	Lemma	restrmP	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "dom", "ker", "ker_restrm", "morphim_restrm", "morphism", "morphpre_restrm", "restrm_morphism", "sAD", "setIidPr", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
domP (f : {morphism D >-> rT}) : 'dom f = A -> {g : {morphism A >-> rT} \| [/\ g = f :> (aT -> rT), 'ker g = 'ker f, forall R, g @^-1 R = f @^-1 R & forall B, g @* B = f @* B]}.	Proof. by move <-; exists f. Qed.	Lemma	domP	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "dom", "ker", "morphism" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivm & {set aT} & aT	:= 1 : FinGroup.sort rT.	Definition	trivm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "sort" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivm_morphM (A : {set aT}) : {in A &, {morph trivm A : x y / x * y}}.	Proof. by move=> x y /=; rewrite mulg1. Qed.	Lemma	trivm_morphM	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "mulg1", "trivm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
triv_morph A	:= Morphism (@trivm_morphM A).	Canonical	triv_morph	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "trivm_morphM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_trivm (G H : {group aT}) : trivm G @* H = 1.	Proof. apply/setP=> /= y; rewrite inE; apply/idP/eqP=> [\|->]; first by case/morphimP. by apply/morphimP; exists (1 : aT); rewrite /= ?group1. Qed.	Lemma	morphim_trivm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "group", "group1", "inE", "morphimP", "setP", "trivm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_trivm (G : {group aT}) : 'ker (trivm G) = G.	Proof. by apply/setIidPl/subsetP=> x _; rewrite !inE /=. Qed.	Lemma	ker_trivm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "group", "inE", "ker", "setIidPl", "subsetP", "trivm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gof	:= (mfun g \o mfun f).	Notation	gof	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_morphM : {in f @^-1 H &, {morph gof: x y / x y}}.	Proof. by move=> x y; rewrite /= !inE => /andP[? ?] /andP[? ?]; rewrite !morphM. Qed.	Lemma	comp_morphM	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "gof", "inE", "morphM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_morphism	:= Morphism comp_morphM.	Canonical	comp_morphism	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "comp_morphM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_comp : 'ker gof = f @*^-1 'ker g.	Proof. by apply/setP=> x; rewrite !inE andbA. Qed.	Lemma	ker_comp	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "gof", "inE", "ker", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_comp : 'injm f -> 'injm g -> 'injm gof.	Proof. by move=> injf; rewrite ker_comp; move/trivgP=> ->. Qed.	Lemma	injm_comp	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "gof", "injf", "ker_comp", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_comp (A : {set gT}) : gof @* A = g @* (f @* A).	Proof. apply/setP=> z; apply/morphimP/morphimP=> [[x]\|[y Hy fAy ->{z}]]. rewrite !inE => /andP[Gx Hfx]; exists (f x) => //. by apply/morphimP; exists x. by case/morphimP: fAy Hy => x Gx Ax ->{y} Hfx; exists x; rewrite ?inE ?Gx. Qed.	Lemma	morphim_comp	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "gT", "gof", "inE", "morphimP", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_comp (C : {set rT}) : gof @^-1 C = f @^-1 (g @*^-1 C).	Proof. by apply/setP=> z; rewrite !inE andbA. Qed.	Lemma	morphpre_comp	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "gof", "inE", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
factm & 'ker q \subset 'ker f & G \subset H	:= fun x => f (repr (q @*^-1 [set x])).	Definition	factm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sKqKf : 'ker q \subset 'ker f.		Hypothesis	sKqKf	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sGH : G \subset H.		Hypothesis	sGH	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ff	:= (factm sKqKf sGH).	Notation	ff	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "factm", "sGH", "sKqKf" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
factmE x : x \in G -> ff (q x) = f x.	Proof. rewrite /ff => Gx; have Hx := subsetP sGH x Gx. have /mem_repr: x \in q @*^-1 [set q x] by rewrite !inE Hx /=. case/morphpreP; move: (repr _) => y Hy /set1P. by case/ker_rcoset=> // z Kz ->; rewrite mkerl ?(subsetP sKqKf). Qed.	Lemma	factmE	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ff", "inE", "ker_rcoset", "mem_repr", "mkerl", "morphpreP", "repr", "sGH", "sKqKf", "set1P", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
factm_morphM : {in q @* G &, {morph ff : x y / x * y}}.	Proof. move=> _ _ /morphimP[x Hx Gx ->] /morphimP[y Hy Gy ->]. by rewrite -morphM ?factmE ?groupM // morphM. Qed.	Lemma	factm_morphM	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "factmE", "ff", "groupM", "morphM", "morphimP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
factm_morphism	:= Morphism factm_morphM.	Canonical	factm_morphism	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "factm_morphM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_factm (A : {set aT}) : ff @* (q @* A) = f @* A.	Proof. rewrite -morphim_comp /= {1}/morphim /= morphimGK //. by rewrite (subset_trans sKqKf) ?subsetIl. apply/setP=> y; apply/morphimP/morphimP; by case=> x Gx Ax ->{y}; exists x; rewrite //= factmE. Qed.	Lemma	morphim_factm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "factmE", "ff", "morphim", "morphimGK", "morphimP", "morphim_comp", "sKqKf", "setP", "subsetIl", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_factm (C : {set rT}) : ff @^-1 C = q @ (f @*^-1 C).	Proof. apply/setP=> y /[!inE]/=; apply/andP/morphimP=> [[]\|[x Hx]]; last first. by case/morphpreP=> Gx Cfx ->; rewrite factmE ?imset_f ?inE ?Hx. case/morphimP=> x Hx Gx ->; rewrite factmE //. by exists x; rewrite // !inE Gx. Qed.	Lemma	morphpre_factm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "factmE", "ff", "imset_f", "inE", "last", "morphimP", "morphpreP", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_factm : 'ker ff = q @* 'ker f.	Proof. exact: morphpre_factm. Qed.	Lemma	ker_factm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ff", "ker", "morphpre_factm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_factm : 'injm f -> 'injm ff.	Proof. by rewrite ker_factm => /trivgP->; rewrite morphim1. Qed.	Lemma	injm_factm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ff", "ker_factm", "morphim1", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_factmP : reflect ('ker f = 'ker q) ('injm ff).	Proof. rewrite ker_factm -morphimIdom sub_morphim_pre ?subsetIl //. rewrite setIA (setIidPr sGH) (sameP setIidPr eqP) (setIidPl _) // eq_sym. exact: eqP. Qed.	Lemma	injm_factmP	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "eq_sym", "ff", "ker", "ker_factm", "morphimIdom", "sGH", "setIA", "setIidPl", "setIidPr", "sub_morphim_pre", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_factm_loc (K : {group aT}) : 'ker_(q @* K) ff = q @* 'ker_K f.	Proof. by rewrite ker_factm -morphimIG. Qed.	Lemma	ker_factm_loc	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "ff", "group", "ker_factm", "morphimIG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invm_subker : 'ker f \subset 'ker (idm G).	Proof. by rewrite ker_idm. Qed.	Lemma	invm_subker	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "idm", "ker", "ker_idm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invm	:= factm invm_subker (subxx _).	Definition	invm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "factm", "invm_subker", "subxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invm_morphism	:= Eval hnf in [morphism of invm].	Canonical	invm_morphism	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "invm", "morphism" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invmE : {in G, cancel f invm}.	Proof. exact: factmE. Qed.	Lemma	invmE	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "factmE", "invm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invmK : {in f @* G, cancel invm f}.	Proof. by move=> fx; case/morphimP=> x _ Gx ->; rewrite invmE. Qed.	Lemma	invmK	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "invm", "invmE", "morphimP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_invm A : invm @^-1 A = f @ A.	Proof. by rewrite morphpre_factm morphpre_idm morphimIdom. Qed.	Lemma	morphpre_invm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "invm", "morphimIdom", "morphpre_factm", "morphpre_idm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_invm A : A \subset G -> invm @* (f @* A) = A.	Proof. by move=> sAG; rewrite morphim_factm morphim_idm. Qed.	Lemma	morphim_invm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "invm", "morphim_factm", "morphim_idm", "sAG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_invmE C : invm @* C = f @*^-1 C.	Proof. rewrite -morphpreIdom -(morphim_invm (subsetIl _ _)). by rewrite morphimIdom -morphpreIim morphpreK (subsetIl, morphimIdom). Qed.	Lemma	morphim_invmE	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "invm", "morphimIdom", "morphim_invm", "morphpreIdom", "morphpreIim", "morphpreK", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_proper A B : A \subset G -> B \subset G -> (f @* A \proper f @* B) = (A \proper B).	Proof. move=> dA dB; rewrite -morphpre_invm -(morphpre_invm B). by rewrite morphpre_proper ?morphim_invm. Qed.	Lemma	injm_proper	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphim_invm", "morphpre_invm", "morphpre_proper", "proper" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_invm : 'injm invm.	Proof. by move/can_in_inj/injmP: invmK. Qed.	Lemma	injm_invm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injmP", "invm", "invmK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_invm : 'ker invm = 1.	Proof. by move/trivgP: injm_invm. Qed.	Lemma	ker_invm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injm_invm", "invm", "ker", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_invm : invm @* (f @* G) = G.	Proof. exact: morphim_invm. Qed.	Lemma	im_invm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "invm", "morphim_invm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ifactm	:= tag (domP [morphism of g \o invm injf] (morphpre_invm injf G)).	Definition	ifactm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "domP", "injf", "invm", "morphism", "morphpre_invm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ifactmE : {in D, forall x, ifactm (f x) = g x}.	Proof. rewrite /ifactm => x Dx; case: domP => f' /= [def_f' _ _ _]. by rewrite {f'}def_f' //= invmE. Qed.	Lemma	ifactmE	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "domP", "ifactm", "invmE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_ifactm (A : {set gT}) : A \subset D -> ifactm @* (f @* A) = g @* A.	Proof. rewrite /ifactm => sAD; case: domP => _ /= [_ _ _ ->]. by rewrite morphim_comp morphim_invm. Qed.	Lemma	morphim_ifactm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "domP", "gT", "ifactm", "morphim_comp", "morphim_invm", "sAD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_ifactm : G \subset D -> ifactm @* (f @* G) = g @* G.	Proof. exact: morphim_ifactm. Qed.	Lemma	im_ifactm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ifactm", "morphim_ifactm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_ifactm C : ifactm @^-1 C = f @ (g @*^-1 C).	Proof. rewrite /ifactm; case: domP => _ /= [_ _ -> _]. by rewrite morphpre_comp morphpre_invm. Qed.	Lemma	morphpre_ifactm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "domP", "ifactm", "morphpre_comp", "morphpre_invm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_ifactm : 'ker ifactm = f @* 'ker g.	Proof. exact: morphpre_ifactm. Qed.	Lemma	ker_ifactm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ifactm", "ker", "morphpre_ifactm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_ifactm : 'injm g -> 'injm ifactm.	Proof. by rewrite ker_ifactm => /trivgP->; rewrite morphim1. Qed.	Lemma	injm_ifactm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ifactm", "ker_ifactm", "morphim1", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphic (f : aT -> rT)	:= [forall u in [predX A & A], f (u.1 * u.2) == f u.1 * f u.2].	Definition	morphic	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "predX" ]	morphic is the morphM property of morphisms seen through morphicP.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isom f	:= f @: A^# == B^#.	Definition	isom	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
misom f	:= morphic f && isom f.	Definition	misom	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "isom", "morphic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog	:= [exists f : {ffun aT -> rT}, misom f].	Definition	isog	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "misom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphicP : reflect {in A &, {morph f : x y / x * y}} (morphic f).	Proof. apply: (iffP forallP) => [fM x y Ax Ay \| fM [x y] /=]. by apply/eqP; have:= fM (x, y); rewrite inE /= Ax Ay. by apply/implyP=> /andP[Ax Ay]; rewrite fM. Qed.	Lemma	morphicP	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "fM", "forallP", "inE", "morphic" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphm & morphic f	:= f : aT -> FinGroup.sort rT.	Definition	morphm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "morphic", "sort" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphmE fM : morphm fM = f.	Proof. by []. Qed.	Lemma	morphmE	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "fM", "morphm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphm_morphism fM	:= @Morphism _ _ A (morphm fM) (morphicP fM).	Canonical	morphm_morphism	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "fM", "morphicP", "morphm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
misomP f : reflect {fM : morphic f & isom (morphm fM)} (misom f).	Proof. by apply: (iffP andP) => [] [fM fiso] //; exists fM. Qed.	Lemma	misomP	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "fM", "isom", "misom", "morphic", "morphm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
misom_isog f : misom f -> isog.	Proof. case/andP=> fM iso_f; apply/existsP; exists (finfun f). apply/andP; split; last by rewrite /misom /isom !(eq_imset _ (ffunE f)). by apply/forallP=> u; rewrite !ffunE; apply: forallP fM u. Qed.	Lemma	misom_isog	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "eq_imset", "existsP", "fM", "ffunE", "forallP", "isog", "isom", "last", "misom", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isom_isog (D : {group aT}) (f : {morphism D >-> rT}) : A \subset D -> isom f -> isog.	Proof. move=> sAD isof; apply: (@misom_isog f); rewrite /misom isof andbT. by apply/morphicP; apply: (sub_in2 (subsetP sAD) (morphM f)). Qed.	Lemma	isom_isog	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "group", "isog", "isom", "misom", "misom_isog", "morphM", "morphicP", "morphism", "sAD", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_isom : isog -> {f : {morphism A >-> rT} \| isom f}.	Proof. by case/existsP/sigW=> f /misomP[fM isom_f]; exists (morphm_morphism fM). Qed.	Lemma	isog_isom	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "existsP", "fM", "isog", "isom", "misomP", "morphism", "morphm_morphism", "sigW" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isomP (f : {morphism G >-> rT}) : reflect ('injm f /\ f @* G = H) (isom G H f).	Proof. apply: (iffP eqP) => [eqfGH \| [injf <-]]; last first. by rewrite -injmD1 // morphimEsub ?subsetDl. split. apply/subsetP=> x /morphpreP[Gx fx1]; have: f x \notin H^# by rewrite inE fx1. by apply: contraR => ntx; rewrite -eqfGH imset_f // inE ntx. rewrite morphimEdom -{2}(setD1K (group1 G)) imsetU eqfGH. by ...	Lemma	isomP	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "group1", "imsetU", "imset_f", "imset_set1", "inE", "injf", "injmD1", "isom", "last", "morph1", "morphimEdom", "morphimEsub", "morphism", "morphpreP", "setD1K", "split", "subsetDl", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isogP : reflect (exists2 f : {morphism G >-> rT}, 'injm f & f @* G = H) (G \isog H).	Proof. apply: (iffP idP) => [/isog_isom[f /isomP[]] \| [f injf fG]]; first by exists f. by apply: (isom_isog f) => //; apply/isomP. Qed.	Lemma	isogP	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "injf", "isog", "isog_isom", "isomP", "isom_isog", "morphism" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isoGH : isom G H f.		Hypothesis	isoGH	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isom_inj : 'injm f.	Proof. by have /isomP[] := isoGH. Qed.	Lemma	isom_inj	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "isoGH", "isomP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isom_im : f @* G = H.	Proof. by have /isomP[] := isoGH. Qed.	Lemma	isom_im	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "isoGH", "isomP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isom_card : #\|G\| = #\|H\|.	Proof. by rewrite -isom_im card_injm ?isom_inj. Qed.	Lemma	isom_card	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "card_injm", "isom_im", "isom_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isom_sub_im : H \subset f @* G.	Proof. by rewrite isom_im. Qed.	Lemma	isom_sub_im	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "isom_im" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isom_inv	:= restrm isom_sub_im (invm isom_inj).	Definition	isom_inv	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "invm", "isom_inj", "isom_sub_im", "restrm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_isom (H : {group aT}) (K : {group rT}) : H \subset G -> isom H K f -> f @* H = K.	Proof. by case/(restrmP f)=> g [gf _ _ <- //]; rewrite -gf; case/isomP. Qed.	Lemma	morphim_isom	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "group", "isom", "isomP", "restrmP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_normal A B : A \subset D -> B \subset D -> (f @* A <| f @* B) = (A <| B).

Proof. by move=> sAD sBD; rewrite /normal injmSK ?injm_norms. Qed.

Lemma

injm_normal

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "injmSK", "injm_norms", "normal", "sAD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_subnorm A B : B \subset D -> f @* 'N_A(B) = 'N_(f @* A)(f @* B).

Proof. by move=> sBD; rewrite injmI injm_norm // setICA setIA morphimIim. Qed.

Lemma

injm_subnorm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "injmI", "injm_norm", "morphimIim", "setIA", "setICA" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_cent1 x : x \in D -> f @* 'C[x] = 'C_(f @* D)[f x].

Proof. by move=> Dx; rewrite injm_norm ?morphim_set1 ?sub1set. Qed.

Lemma

injm_cent1

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "Dx", "injm_norm", "morphim_set1", "sub1set" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_subcent1 A x : x \in D -> f @* 'C_A[x] = 'C_(f @* A)[f x].

Proof. by move=> Dx; rewrite injm_subnorm ?morphim_set1 ?sub1set. Qed.

Lemma

injm_subcent1

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "Dx", "injm_subnorm", "morphim_set1", "sub1set" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_cent A : A \subset D -> f @* 'C(A) = 'C_(f @* D)(f @* A).

Proof. move=> sAD; apply/eqP; rewrite -morphimIdom eqEsubset morphim_subcent. apply/subsetP=> fx; case/setIP; case/morphimP=> x Dx _ ->{fx} cAfx. rewrite mem_morphim // inE Dx -sub1set centsC cent_set1 -injmSK //. by rewrite injm_cent1 // subsetI morphimS // -cent_set1 centsC sub1set. Qed.

Lemma

injm_cent

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "Dx", "apply", "cent_set1", "centsC", "eqEsubset", "inE", "injmSK", "injm_cent1", "mem_morphim", "morphimIdom", "morphimP", "morphimS", "morphim_subcent", "sAD", "setIP", "sub1set", "subsetI", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_cents A B : A \subset D -> B \subset D -> (f @* A \subset 'C(f @* B)) = (A \subset 'C(B)).

Proof. by move=> sAD sBD; rewrite -injmSK // injm_cent // subsetI morphimS. Qed.

Lemma

injm_cents

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "injmSK", "injm_cent", "morphimS", "sAD", "subsetI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_subcent A B : B \subset D -> f @* 'C_A(B) = 'C_(f @* A)(f @* B).

Proof. by move=> sBD; rewrite injmI injm_cent // setICA setIA morphimIim. Qed.

Lemma

injm_subcent

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "injmI", "injm_cent", "morphimIim", "setIA", "setICA" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_abelian A : A \subset D -> abelian (f @* A) = abelian A.

Proof. by move=> sAD; rewrite /abelian -subsetIidl -injm_subcent // injmSK ?subsetIidl. Qed.

Lemma

injm_abelian

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "abelian", "injmSK", "injm_subcent", "sAD", "subsetIidl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_morphim (g : {morphism D >-> rT}): {in D, f =1 g} -> forall A, f @* A = g @* A.

Proof. by move=> efg A; apply: eq_in_imset; apply: sub_in1 efg => x /setIP[]. Qed.

Lemma

eq_morphim

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "eq_in_imset", "morphism", "setIP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_in_morphim B A (g : {morphism B >-> rT}) : D :&: A = B :&: A -> {in A, f =1 g} -> f @* A = g @* A.

Proof. move=> eqDBA eqAfg; rewrite /morphim /= eqDBA. by apply: eq_in_imset => x /setIP[_]/eqAfg. Qed.

Lemma

eq_in_morphim

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "eq_in_imset", "morphim", "morphism", "setIP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''ker' f"

:= (ker_group (MorPhantom f)) : Group_scope.

Notation

''ker' f

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "MorPhantom", "ker_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''ker_' G f"

:= (G :&: 'ker f)%G : Group_scope.

Notation

''ker_' G f

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "ker" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"f @* G"

:= (morphim_group (MorPhantom f) G) : Group_scope.

Notation

f @* G

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "MorPhantom", "morphim_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"f @*^-1 M"

:= (morphpre_group (MorPhantom f) M) : Group_scope.

Notation

f @*^-1 M

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "MorPhantom", "morphpre_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"f @: D"

:= (morph_dom_group f D) : Group_scope.

Notation

f @: D

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "morph_dom_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idm & {set gT}

:= fun x : gT => x : FinGroup.sort gT.

Definition

idm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "gT", "sort" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idm_morphM A : {in A & , {morph idm A : x y / x * y}}.

Proof. by []. Qed.

Lemma

idm_morphM

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "idm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idm_morphism A

:= Morphism (@idm_morphM A).

Canonical

idm_morphism

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "idm_morphM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_idm G : 'injm (idm G).

Proof. by apply/injmP=> x y _ _. Qed.

Lemma

injm_idm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "idm", "injmP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_idm G : 'ker (idm G) = 1.

Proof. by apply/trivgP; apply: injm_idm. Qed.

Lemma

ker_idm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "idm", "injm_idm", "ker", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_idm A B : B \subset A -> idm A @* B = B.

Proof. rewrite /morphim /= /idm => /setIidPr->. by apply/setP=> x; apply/imsetP/idP=> [[y By ->]|Bx]; last exists x. Qed.

Lemma

morphim_idm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "idm", "imsetP", "last", "morphim", "setIidPr", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_idm A B : idm A @*^-1 B = A :&: B.

Proof. by apply/setP=> x; rewrite !inE. Qed.

Lemma

morphpre_idm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "idm", "inE", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_idm A : idm A @* A = A.

Proof. exact: morphim_idm. Qed.

Lemma

im_idm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "idm", "morphim_idm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restrm & A \subset D

:= @id (aT -> FinGroup.sort rT).

Definition

restrm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "id", "sort" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sAD : A \subset D.

Hypothesis

sAD

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fA

:= (restrm sAD (mfun f)).

Notation

fA

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "restrm", "sAD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restrm_morphism

:= @Morphism aT rT A fA (sub_in2 (subsetP sAD) (morphM f)).

Canonical

restrm_morphism

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "fA", "morphM", "sAD", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_restrm B : fA @* B = f @* (A :&: B).

Proof. by rewrite {2}/morphim setIA (setIidPr sAD). Qed.

Lemma

morphim_restrm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "fA", "morphim", "sAD", "setIA", "setIidPr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restrmEsub B : B \subset A -> fA @* B = f @* B.

Proof. by rewrite morphim_restrm => /setIidPr->. Qed.

Lemma

restrmEsub

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "fA", "morphim_restrm", "setIidPr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_restrm : fA @* A = f @* A.

Proof. exact: restrmEsub. Qed.

Lemma

im_restrm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "fA", "restrmEsub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_restrm R : fA @*^-1 R = A :&: f @*^-1 R.

Proof. by rewrite setIA (setIidPl sAD). Qed.

Lemma

morphpre_restrm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "fA", "sAD", "setIA", "setIidPl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_restrm : 'ker fA = 'ker_A f.

Proof. exact: morphpre_restrm. Qed.

Lemma

ker_restrm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "fA", "ker", "morphpre_restrm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_restrm : 'injm f -> 'injm fA.

Proof. by apply: subset_trans; rewrite ker_restrm subsetIr. Qed.

Lemma

injm_restrm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "fA", "ker_restrm", "subsetIr", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restrmP (f : {morphism D >-> rT}) : A \subset 'dom f -> {g : {morphism A >-> rT} | [/\ g = f :> (aT -> rT), 'ker g = 'ker_A f, forall R, g @*^-1 R = A :&: f @*^-1 R & forall B, B \subset A -> g @* B = f @* B]}.

Proof. move=> sAD; exists (restrm_morphism sAD f). split=> // [|R|B sBA]; first 1 [exact: ker_restrm | exact: morphpre_restrm]. by rewrite morphim_restrm (setIidPr sBA). Qed.

Lemma

restrmP

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "dom", "ker", "ker_restrm", "morphim_restrm", "morphism", "morphpre_restrm", "restrm_morphism", "sAD", "setIidPr", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

domP (f : {morphism D >-> rT}) : 'dom f = A -> {g : {morphism A >-> rT} | [/\ g = f :> (aT -> rT), 'ker g = 'ker f, forall R, g @*^-1 R = f @*^-1 R & forall B, g @* B = f @* B]}.

Proof. by move <-; exists f. Qed.

Lemma

domP

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "dom", "ker", "morphism" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivm & {set aT} & aT

:= 1 : FinGroup.sort rT.

Definition

trivm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "sort" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivm_morphM (A : {set aT}) : {in A &, {morph trivm A : x y / x * y}}.

Proof. by move=> x y /=; rewrite mulg1. Qed.

Lemma

trivm_morphM

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "mulg1", "trivm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

triv_morph A

:= Morphism (@trivm_morphM A).

Canonical

triv_morph

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "trivm_morphM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_trivm (G H : {group aT}) : trivm G @* H = 1.

Proof. apply/setP=> /= y; rewrite inE; apply/idP/eqP=> [|->]; first by case/morphimP. by apply/morphimP; exists (1 : aT); rewrite /= ?group1. Qed.

Lemma

morphim_trivm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "apply", "group", "group1", "inE", "morphimP", "setP", "trivm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_trivm (G : {group aT}) : 'ker (trivm G) = G.

Proof. by apply/setIidPl/subsetP=> x _; rewrite !inE /=. Qed.

Lemma

ker_trivm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "apply", "group", "inE", "ker", "setIidPl", "subsetP", "trivm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gof

:= (mfun g \o mfun f).

Notation

gof

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_morphM : {in f @*^-1 H &, {morph gof: x y / x * y}}.

Proof. by move=> x y; rewrite /= !inE => /andP[? ?] /andP[? ?]; rewrite !morphM. Qed.

Lemma

comp_morphM

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "gof", "inE", "morphM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comp_morphism

:= Morphism comp_morphM.

Canonical

comp_morphism

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "comp_morphM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_comp : 'ker gof = f @*^-1 'ker g.

Proof. by apply/setP=> x; rewrite !inE andbA. Qed.

Lemma

ker_comp

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "gof", "inE", "ker", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_comp : 'injm f -> 'injm g -> 'injm gof.

Proof. by move=> injf; rewrite ker_comp; move/trivgP=> ->. Qed.

Lemma

injm_comp

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "gof", "injf", "ker_comp", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_comp (A : {set gT}) : gof @* A = g @* (f @* A).

Proof. apply/setP=> z; apply/morphimP/morphimP=> [[x]|[y Hy fAy ->{z}]]. rewrite !inE => /andP[Gx Hfx]; exists (f x) => //. by apply/morphimP; exists x. by case/morphimP: fAy Hy => x Gx Ax ->{y} Hfx; exists x; rewrite ?inE ?Gx. Qed.

Lemma

morphim_comp

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "gT", "gof", "inE", "morphimP", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_comp (C : {set rT}) : gof @*^-1 C = f @*^-1 (g @*^-1 C).

Proof. by apply/setP=> z; rewrite !inE andbA. Qed.

Lemma

morphpre_comp

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "gof", "inE", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

factm & 'ker q \subset 'ker f & G \subset H

:= fun x => f (repr (q @*^-1 [set x])).

Definition

factm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "ker", "repr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sKqKf : 'ker q \subset 'ker f.

Hypothesis

sKqKf

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "ker" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sGH : G \subset H.

Hypothesis

sGH

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ff

:= (factm sKqKf sGH).

Notation

ff

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "factm", "sGH", "sKqKf" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

factmE x : x \in G -> ff (q x) = f x.

Proof. rewrite /ff => Gx; have Hx := subsetP sGH x Gx. have /mem_repr: x \in q @*^-1 [set q x] by rewrite !inE Hx /=. case/morphpreP; move: (repr _) => y Hy /set1P. by case/ker_rcoset=> // z Kz ->; rewrite mkerl ?(subsetP sKqKf). Qed.

Lemma

factmE

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "ff", "inE", "ker_rcoset", "mem_repr", "mkerl", "morphpreP", "repr", "sGH", "sKqKf", "set1P", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

factm_morphM : {in q @* G &, {morph ff : x y / x * y}}.

Proof. move=> _ _ /morphimP[x Hx Gx ->] /morphimP[y Hy Gy ->]. by rewrite -morphM ?factmE ?groupM // morphM. Qed.

Lemma

factm_morphM

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "factmE", "ff", "groupM", "morphM", "morphimP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

factm_morphism

:= Morphism factm_morphM.

Canonical

factm_morphism

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "factm_morphM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_factm (A : {set aT}) : ff @* (q @* A) = f @* A.

Proof. rewrite -morphim_comp /= {1}/morphim /= morphimGK //. by rewrite (subset_trans sKqKf) ?subsetIl. apply/setP=> y; apply/morphimP/morphimP; by case=> x Gx Ax ->{y}; exists x; rewrite //= factmE. Qed.

Lemma

morphim_factm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "apply", "factmE", "ff", "morphim", "morphimGK", "morphimP", "morphim_comp", "sKqKf", "setP", "subsetIl", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_factm (C : {set rT}) : ff @*^-1 C = q @* (f @*^-1 C).

Proof. apply/setP=> y /[!inE]/=; apply/andP/morphimP=> [[]|[x Hx]]; last first. by case/morphpreP=> Gx Cfx ->; rewrite factmE ?imset_f ?inE ?Hx. case/morphimP=> x Hx Gx ->; rewrite factmE //. by exists x; rewrite // !inE Gx. Qed.

Lemma

morphpre_factm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "factmE", "ff", "imset_f", "inE", "last", "morphimP", "morphpreP", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_factm : 'ker ff = q @* 'ker f.

Proof. exact: morphpre_factm. Qed.

Lemma

ker_factm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "ff", "ker", "morphpre_factm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_factm : 'injm f -> 'injm ff.

Proof. by rewrite ker_factm => /trivgP->; rewrite morphim1. Qed.

Lemma

injm_factm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "ff", "ker_factm", "morphim1", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_factmP : reflect ('ker f = 'ker q) ('injm ff).

Proof. rewrite ker_factm -morphimIdom sub_morphim_pre ?subsetIl //. rewrite setIA (setIidPr sGH) (sameP setIidPr eqP) (setIidPl _) // eq_sym. exact: eqP. Qed.

Lemma

injm_factmP

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "eq_sym", "ff", "ker", "ker_factm", "morphimIdom", "sGH", "setIA", "setIidPl", "setIidPr", "sub_morphim_pre", "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_factm_loc (K : {group aT}) : 'ker_(q @* K) ff = q @* 'ker_K f.

Proof. by rewrite ker_factm -morphimIG. Qed.

Lemma

ker_factm_loc

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "ff", "group", "ker_factm", "morphimIG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invm_subker : 'ker f \subset 'ker (idm G).

Proof. by rewrite ker_idm. Qed.

Lemma

invm_subker

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "idm", "ker", "ker_idm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invm

:= factm invm_subker (subxx _).

Definition

invm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "factm", "invm_subker", "subxx" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invm_morphism

:= Eval hnf in [morphism of invm].

Canonical

invm_morphism

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "invm", "morphism" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invmE : {in G, cancel f invm}.

Proof. exact: factmE. Qed.

Lemma

invmE

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "factmE", "invm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invmK : {in f @* G, cancel invm f}.

Proof. by move=> fx; case/morphimP=> x _ Gx ->; rewrite invmE. Qed.

Lemma

invmK

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "invm", "invmE", "morphimP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_invm A : invm @*^-1 A = f @* A.

Proof. by rewrite morphpre_factm morphpre_idm morphimIdom. Qed.

Lemma

morphpre_invm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "invm", "morphimIdom", "morphpre_factm", "morphpre_idm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_invm A : A \subset G -> invm @* (f @* A) = A.

Proof. by move=> sAG; rewrite morphim_factm morphim_idm. Qed.

Lemma

morphim_invm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "invm", "morphim_factm", "morphim_idm", "sAG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_invmE C : invm @* C = f @*^-1 C.

Proof. rewrite -morphpreIdom -(morphim_invm (subsetIl _ _)). by rewrite morphimIdom -morphpreIim morphpreK (subsetIl, morphimIdom). Qed.

Lemma

morphim_invmE

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "invm", "morphimIdom", "morphim_invm", "morphpreIdom", "morphpreIim", "morphpreK", "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_proper A B : A \subset G -> B \subset G -> (f @* A \proper f @* B) = (A \proper B).

Proof. move=> dA dB; rewrite -morphpre_invm -(morphpre_invm B). by rewrite morphpre_proper ?morphim_invm. Qed.

Lemma

injm_proper

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "morphim_invm", "morphpre_invm", "morphpre_proper", "proper" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_invm : 'injm invm.

Proof. by move/can_in_inj/injmP: invmK. Qed.

Lemma

injm_invm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "injmP", "invm", "invmK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_invm : 'ker invm = 1.

Proof. by move/trivgP: injm_invm. Qed.

Lemma

ker_invm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "injm_invm", "invm", "ker", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_invm : invm @* (f @* G) = G.

Proof. exact: morphim_invm. Qed.

Lemma

im_invm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "invm", "morphim_invm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ifactm

:= tag (domP [morphism of g \o invm injf] (morphpre_invm injf G)).

Definition

ifactm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "domP", "injf", "invm", "morphism", "morphpre_invm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ifactmE : {in D, forall x, ifactm (f x) = g x}.

Proof. rewrite /ifactm => x Dx; case: domP => f' /= [def_f' _ _ _]. by rewrite {f'}def_f' //= invmE. Qed.

Lemma

ifactmE

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "Dx", "domP", "ifactm", "invmE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_ifactm (A : {set gT}) : A \subset D -> ifactm @* (f @* A) = g @* A.

Proof. rewrite /ifactm => sAD; case: domP => _ /= [_ _ _ ->]. by rewrite morphim_comp morphim_invm. Qed.

Lemma

morphim_ifactm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "domP", "gT", "ifactm", "morphim_comp", "morphim_invm", "sAD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_ifactm : G \subset D -> ifactm @* (f @* G) = g @* G.

Proof. exact: morphim_ifactm. Qed.

Lemma

im_ifactm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "ifactm", "morphim_ifactm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_ifactm C : ifactm @*^-1 C = f @* (g @*^-1 C).

Proof. rewrite /ifactm; case: domP => _ /= [_ _ -> _]. by rewrite morphpre_comp morphpre_invm. Qed.

Lemma

morphpre_ifactm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "domP", "ifactm", "morphpre_comp", "morphpre_invm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_ifactm : 'ker ifactm = f @* 'ker g.

Proof. exact: morphpre_ifactm. Qed.

Lemma

ker_ifactm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "ifactm", "ker", "morphpre_ifactm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_ifactm : 'injm g -> 'injm ifactm.

Proof. by rewrite ker_ifactm => /trivgP->; rewrite morphim1. Qed.

Lemma

injm_ifactm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "ifactm", "ker_ifactm", "morphim1", "trivgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphic (f : aT -> rT)

:= [forall u in [predX A & A], f (u.1 * u.2) == f u.1 * f u.2].

Definition

morphic

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "predX" ]

morphic is the morphM property of morphisms seen through morphicP.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isom f

:= f @: A^# == B^#.

Definition

isom

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

misom f

:= morphic f && isom f.

Definition

misom

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "isom", "morphic" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog

:= [exists f : {ffun aT -> rT}, misom f].

Definition

isog

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "misom" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphicP : reflect {in A &, {morph f : x y / x * y}} (morphic f).

Proof. apply: (iffP forallP) => [fM x y Ax Ay | fM [x y] /=]. by apply/eqP; have:= fM (x, y); rewrite inE /= Ax Ay. by apply/implyP=> /andP[Ax Ay]; rewrite fM. Qed.

Lemma

morphicP

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "fM", "forallP", "inE", "morphic" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphm & morphic f

:= f : aT -> FinGroup.sort rT.

Definition

morphm

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "morphic", "sort" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphmE fM : morphm fM = f.

Proof. by []. Qed.

Lemma

morphmE

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "fM", "morphm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphm_morphism fM

:= @Morphism _ _ A (morphm fM) (morphicP fM).

Canonical

morphm_morphism

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "fM", "morphicP", "morphm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

misomP f : reflect {fM : morphic f & isom (morphm fM)} (misom f).

Proof. by apply: (iffP andP) => [] [fM fiso] //; exists fM. Qed.

Lemma

misomP

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "fM", "isom", "misom", "morphic", "morphm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

misom_isog f : misom f -> isog.

Proof. case/andP=> fM iso_f; apply/existsP; exists (finfun f). apply/andP; split; last by rewrite /misom /isom !(eq_imset _ (ffunE f)). by apply/forallP=> u; rewrite !ffunE; apply: forallP fM u. Qed.

Lemma

misom_isog

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "eq_imset", "existsP", "fM", "ffunE", "forallP", "isog", "isom", "last", "misom", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isom_isog (D : {group aT}) (f : {morphism D >-> rT}) : A \subset D -> isom f -> isog.

Proof. move=> sAD isof; apply: (@misom_isog f); rewrite /misom isof andbT. by apply/morphicP; apply: (sub_in2 (subsetP sAD) (morphM f)). Qed.

Lemma

isom_isog

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "apply", "group", "isog", "isom", "misom", "misom_isog", "morphM", "morphicP", "morphism", "sAD", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_isom : isog -> {f : {morphism A >-> rT} | isom f}.

Proof. by case/existsP/sigW=> f /misomP[fM isom_f]; exists (morphm_morphism fM). Qed.

Lemma

isog_isom

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "existsP", "fM", "isog", "isom", "misomP", "morphism", "morphm_morphism", "sigW" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isomP (f : {morphism G >-> rT}) : reflect ('injm f /\ f @* G = H) (isom G H f).

Proof. apply: (iffP eqP) => [eqfGH | [injf <-]]; last first. by rewrite -injmD1 // morphimEsub ?subsetDl. split. apply/subsetP=> x /morphpreP[Gx fx1]; have: f x \notin H^# by rewrite inE fx1. by apply: contraR => ntx; rewrite -eqfGH imset_f // inE ntx. rewrite morphimEdom -{2}(setD1K (group1 G)) imsetU eqfGH. by ...

Lemma

isomP

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "group1", "imsetU", "imset_f", "imset_set1", "inE", "injf", "injmD1", "isom", "last", "morph1", "morphimEdom", "morphimEsub", "morphism", "morphpreP", "setD1K", "split", "subsetDl", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isogP : reflect (exists2 f : {morphism G >-> rT}, 'injm f & f @* G = H) (G \isog H).

Proof. apply: (iffP idP) => [/isog_isom[f /isomP[]] | [f injf fG]]; first by exists f. by apply: (isom_isog f) => //; apply/isomP. Qed.

Lemma

isogP

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "apply", "injf", "isog", "isog_isom", "isomP", "isom_isog", "morphism" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isoGH : isom G H f.

Hypothesis

isoGH

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "isom" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isom_inj : 'injm f.

Proof. by have /isomP[] := isoGH. Qed.

Lemma

isom_inj

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "isoGH", "isomP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isom_im : f @* G = H.

Proof. by have /isomP[] := isoGH. Qed.

Lemma

isom_im

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "isoGH", "isomP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isom_card : #|G| = #|H|.

Proof. by rewrite -isom_im card_injm ?isom_inj. Qed.

Lemma

isom_card

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "card_injm", "isom_im", "isom_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isom_sub_im : H \subset f @* G.

Proof. by rewrite isom_im. Qed.

Lemma

isom_sub_im

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "isom_im" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isom_inv

:= restrm isom_sub_im (invm isom_inj).

Definition

isom_inv

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "invm", "isom_inj", "isom_sub_im", "restrm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_isom (H : {group aT}) (K : {group rT}) : H \subset G -> isom H K f -> f @* H = K.

Proof. by case/(restrmP f)=> g [gf _ _ <- //]; rewrite -gf; case/isomP. Qed.

Lemma

morphim_isom

finite_group

finite_group/morphism.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]

[ "aT", "group", "isom", "isomP", "restrmP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d