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
cosetpre_set1_coset xbar : coset H @*^-1 [set xbar] = xbar.	Proof. by case: (cosetP xbar) => x Nx ->; rewrite cosetpre_set1 ?val_coset. Qed.	Lemma	cosetpre_set1_coset	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "cosetP", "cosetpre_set1", "val_coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpreK C : coset H @*^-1 C / H = C.	Proof. by rewrite /quotient morphpreK ?sub_im_coset. Qed.	Lemma	cosetpreK	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "morphpreK", "quotient", "sub_im_coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivg_quotient : H / H = 1.	Proof. by rewrite -[X in X / _]ker_coset /quotient morphim_ker. Qed.	Lemma	trivg_quotient	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "ker_coset", "morphim_ker", "quotient" ]	Variant of morhphim_ker	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientS1 G : G \subset H -> G / H = 1.	Proof. by move=> sGH; apply/trivgP; rewrite -trivg_quotient quotientS. Qed.	Lemma	quotientS1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "quotientS", "sGH", "trivgP", "trivg_quotient" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_cosetpre M : H \subset coset H @*^-1 M.	Proof. by rewrite -{1}ker_coset; apply: ker_sub_pre. Qed.	Lemma	sub_cosetpre	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "ker_coset", "ker_sub_pre" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_proper G K : H <\| G -> H <\| K -> (G / H \proper K / H) = (G \proper K).	Proof. by move=> nHG nHK; rewrite -cosetpre_proper ?quotientGK. Qed.	Lemma	quotient_proper	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "cosetpre_proper", "nHG", "nHK", "proper", "quotientGK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
normal_cosetpre M : H <\| coset H @*^-1 M.	Proof. by rewrite -{1}ker_coset; apply: ker_normal_pre. Qed.	Lemma	normal_cosetpre	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "ker_coset", "ker_normal_pre" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpreSK C D : (coset H @^-1 C \subset coset H @^-1 D) = (C \subset D).	Proof. by rewrite morphpreSK ?sub_im_coset. Qed.	Lemma	cosetpreSK	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "morphpreSK", "sub_im_coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_quotient_pre A C : A \subset 'N(H) -> (A / H \subset C) = (A \subset coset H @*^-1 C).	Proof. exact: sub_morphim_pre. Qed.	Lemma	sub_quotient_pre	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "sub_morphim_pre" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_cosetpre_quo C G : H <\| G -> (coset H @*^-1 C \subset G) = (C \subset G / H).	Proof. by move=> nHG; rewrite -cosetpreSK quotientGK. Qed.	Lemma	sub_cosetpre_quo	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "cosetpreSK", "nHG", "quotientGK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_sub1 A : A \subset 'N(H) -> (A / H \subset [1]) = (A \subset H).	Proof. by move=> nHA /=; rewrite -[gval H in RHS]ker_coset ker_trivg_morphim nHA. Qed.	Lemma	quotient_sub1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "ker_coset", "ker_trivg_morphim" ]	Variant of ker_trivg_morphim.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientSK A B : A \subset 'N(H) -> (A / H \subset B / H) = (A \subset H * B).	Proof. by move=> nHA; rewrite morphimSK ?ker_coset. Qed.	Lemma	quotientSK	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "ker_coset", "morphimSK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientSGK A G : A \subset 'N(H) -> H \subset G -> (A / H \subset G / H) = (A \subset G).	Proof. by rewrite -{2}ker_coset; apply: morphimSGK. Qed.	Lemma	quotientSGK	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "ker_coset", "morphimSGK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_injG : {in [pred G : {group gT} \| H <\| G] &, injective (fun G => G / H)}.	Proof. by rewrite /normal -{1}ker_coset; apply: morphim_injG. Qed.	Lemma	quotient_injG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "gT", "group", "ker_coset", "morphim_injG", "normal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_inj G1 G2 : H <\| G1 -> H <\| G2 -> G1 / H = G2 / H -> G1 :=: G2.	Proof. by rewrite /normal -[in mem H]ker_coset; apply: morphim_inj. Qed.	Lemma	quotient_inj	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "G1", "apply", "ker_coset", "morphim_inj", "normal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_neq1 A : H <\| A -> (A / H != 1) = (H \proper A).	Proof. case/andP=> sHA nHA; rewrite /proper sHA -trivg_quotient eqEsubset andbC. by rewrite quotientS //= quotientSGK. Qed.	Lemma	quotient_neq1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "eqEsubset", "proper", "quotientS", "quotientSGK", "trivg_quotient" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_gen A : A \subset 'N(H) -> <<A>> / H = <<A / H>>.	Proof. exact: morphim_gen. Qed.	Lemma	quotient_gen	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphim_gen" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpre_gen C : 1 \in C -> coset H @^-1 <<C>> = <<coset H @^-1 C>>.	Proof. by move=> C1; rewrite morphpre_gen ?sub_im_coset. Qed.	Lemma	cosetpre_gen	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "morphpre_gen", "sub_im_coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientR A B : A \subset 'N(H) -> B \subset 'N(H) -> [~: A, B] / H = [~: A / H, B / H].	Proof. exact: morphimR. Qed.	Lemma	quotientR	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimR" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_norm A : 'N(A) / H \subset 'N(A / H).	Proof. exact: morphim_norm. Qed.	Lemma	quotient_norm	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphim_norm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_norms A B : A \subset 'N(B) -> A / H \subset 'N(B / H).	Proof. exact: morphim_norms. Qed.	Lemma	quotient_norms	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphim_norms" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_subnorm A B : 'N_A(B) / H \subset 'N_(A / H)(B / H).	Proof. exact: morphim_subnorm. Qed.	Lemma	quotient_subnorm	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphim_subnorm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_normal A B : A <\| B -> A / H <\| B / H.	Proof. exact: morphim_normal. Qed.	Lemma	quotient_normal	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphim_normal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_cent1 x : 'C[x] / H \subset 'C[coset H x].	Proof. case Nx: (x \in 'N(H)); first exact: morphim_cent1. by rewrite coset_default // cent11T subsetT. Qed.	Lemma	quotient_cent1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "cent11T", "coset", "coset_default", "morphim_cent1", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_cent1s A x : A \subset 'C[x] -> A / H \subset 'C[coset H x].	Proof. by move=> sAC; apply: subset_trans (quotientS sAC) (quotient_cent1 x). Qed.	Lemma	quotient_cent1s	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "quotientS", "quotient_cent1", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_subcent1 A x : 'C_A[x] / H \subset 'C_(A / H)[coset H x].	Proof. exact: subset_trans (quotientI _ _) (setIS _ (quotient_cent1 x)). Qed.	Lemma	quotient_subcent1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "quotientI", "quotient_cent1", "setIS", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_cent A : 'C(A) / H \subset 'C(A / H).	Proof. exact: morphim_cent. Qed.	Lemma	quotient_cent	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphim_cent" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_cents A B : A \subset 'C(B) -> A / H \subset 'C(B / H).	Proof. exact: morphim_cents. Qed.	Lemma	quotient_cents	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphim_cents" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_abelian A : abelian A -> abelian (A / H).	Proof. exact: morphim_abelian. Qed.	Lemma	quotient_abelian	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "abelian", "morphim_abelian" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_subcent A B : 'C_A(B) / H \subset 'C_(A / H)(B / H).	Proof. exact: morphim_subcent. Qed.	Lemma	quotient_subcent	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphim_subcent" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
norm_quotient_pre A C : A \subset 'N(H) -> A / H \subset 'N(C) -> A \subset 'N(coset H @*^-1 C).	Proof. by move/sub_quotient_pre=> -> /subset_trans-> //; apply: morphpre_norm. Qed.	Lemma	norm_quotient_pre	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "morphpre_norm", "sub_quotient_pre", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpre_normal C D : (coset H @^-1 C <\| coset H @^-1 D) = (C <\| D).	Proof. by rewrite morphpre_normal ?sub_im_coset. Qed.	Lemma	cosetpre_normal	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "morphpre_normal", "sub_im_coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_normG G : H <\| G -> 'N(G) / H = 'N(G / H).	Proof. case/andP=> sHG nHG. by rewrite [_ / _]morphim_normG ?ker_coset // im_coset setTI. Qed.	Lemma	quotient_normG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "im_coset", "ker_coset", "morphim_normG", "nHG", "sHG", "setTI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_subnormG A G : H <\| G -> 'N_A(G) / H = 'N_(A / H)(G / H).	Proof. by case/andP=> sHG nHG; rewrite -morphim_subnormG ?ker_coset. Qed.	Lemma	quotient_subnormG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "ker_coset", "morphim_subnormG", "nHG", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpre_cent1 x : 'C_('N(H))[x] \subset coset H @*^-1 'C[coset H x].	Proof. case Nx: (x \in 'N(H)); first by rewrite morphpre_cent1. by rewrite coset_default // cent11T morphpreT subsetIl. Qed.	Lemma	cosetpre_cent1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "cent11T", "coset", "coset_default", "morphpreT", "morphpre_cent1", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpre_cent1s C x : coset H @*^-1 C \subset 'C[x] -> C \subset 'C[coset H x].	Proof. move=> sC; rewrite -cosetpreSK; apply: subset_trans (cosetpre_cent1 x). by rewrite subsetI subsetIl. Qed.	Lemma	cosetpre_cent1s	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "cosetpreSK", "cosetpre_cent1", "subsetI", "subsetIl", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpre_subcent1 C x : 'C_(coset H @^-1 C)[x] \subset coset H @^-1 'C_C[coset H x].	Proof. by rewrite -morphpreIdom -setIA setICA morphpreI setIS // cosetpre_cent1. Qed.	Lemma	cosetpre_subcent1	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "cosetpre_cent1", "morphpreI", "morphpreIdom", "setIA", "setICA", "setIS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpre_cent A : 'C_('N(H))(A) \subset coset H @*^-1 'C(A / H).	Proof. exact: morphpre_cent. Qed.	Lemma	cosetpre_cent	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "morphpre_cent" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpre_cents A C : coset H @*^-1 C \subset 'C(A) -> C \subset 'C(A / H).	Proof. by apply: morphpre_cents; rewrite ?sub_im_coset. Qed.	Lemma	cosetpre_cents	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "morphpre_cents", "sub_im_coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cosetpre_subcent C A : 'C_(coset H @^-1 C)(A) \subset coset H @^-1 'C_C(A / H).	Proof. exact: morphpre_subcent. Qed.	Lemma	cosetpre_subcent	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "morphpre_subcent" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
restrm_quotientE G A (nHG : G \subset 'N(H)) : A \subset G -> restrm nHG (coset H) @* A = A / H.	Proof. exact: restrmEsub. Qed.	Lemma	restrm_quotientE	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "nHG", "restrm", "restrmEsub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nHG : H <\| G.		Hypothesis	nHG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inv_quotient_spec (P : pred {group gT}) : Prop	:= InvQuotientSpec K of Kbar :=: K / H & H \subset K & P K.	Variant	inv_quotient_spec	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inv_quotientS : Kbar \subset G / H -> inv_quotient_spec (fun K => K \subset G).	Proof. move=> sKH; exists (coset H @*^-1 Kbar); first by rewrite cosetpreK. by rewrite sub_cosetpre. by rewrite sub_cosetpre_quo. Qed.	Lemma	inv_quotientS	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "cosetpreK", "inv_quotient_spec", "sub_cosetpre", "sub_cosetpre_quo" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inv_quotientN : Kbar <\| G / H -> inv_quotient_spec (fun K => K <\| G).	Proof. move=> nKbar; case/inv_quotientS: (normal_sub nKbar) => K defKbar sHK sKG. exists K => //; rewrite defKbar -cosetpre_normal !quotientGK // in nKbar. exact: normalS nHG. Qed.	Lemma	inv_quotientN	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "cosetpre_normal", "inv_quotientS", "inv_quotient_spec", "nHG", "normalS", "normal_sub", "quotientGK", "sHK", "sKG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientMidr A : A * H / H = A / H.	Proof. by rewrite [_ /_]morphimMr ?normG //= -!quotientE trivg_quotient mulg1. Qed.	Lemma	quotientMidr	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimMr", "mulg1", "normG", "quotientE", "trivg_quotient" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientMidl A : H * A / H = A / H.	Proof. by rewrite [_ /_]morphimMl ?normG //= -!quotientE trivg_quotient mul1g. Qed.	Lemma	quotientMidl	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimMl", "mul1g", "normG", "quotientE", "trivg_quotient" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientYidr G : G \subset 'N(H) -> G <*> H / H = G / H.	Proof. move=> nHG; rewrite -genM_join quotient_gen ?mul_subG ?normG //. by rewrite quotientMidr genGid. Qed.	Lemma	quotientYidr	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "genGid", "genM_join", "mul_subG", "nHG", "normG", "quotientMidr", "quotient_gen" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotientYidl G : G \subset 'N(H) -> H <*> G / H = G / H.	Proof. by move=> nHG; rewrite joingC quotientYidr. Qed.	Lemma	quotientYidl	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "joingC", "nHG", "quotientYidr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(nHG : G \subset 'N(H)) (tiHG : H :&: G = 1).		Hypotheses	nHG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_isom : isom G (G / H) (restrm nHG (coset H)).	Proof. by apply/isomP; rewrite ker_restrm setIC ker_coset tiHG im_restrm. Qed.	Lemma	quotient_isom	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "im_restrm", "isom", "isomP", "ker_coset", "ker_restrm", "nHG", "restrm", "setIC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient_isog : isog G (G / H).	Proof. exact: isom_isog quotient_isom. Qed.	Lemma	quotient_isog	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "isog", "isom_isog", "quotient_isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"A / H"	:= (quotient_group A H) : Group_scope.	Notation	A / H	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "quotient_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coset1_injm : 'injm (@coset gT 1).	Proof. by rewrite ker_coset /=. Qed.	Lemma	coset1_injm	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "gT", "ker_coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient1_isom : isom A (A / 1) (coset 1).	Proof. by apply: sub_isom coset1_injm; rewrite ?norms1. Qed.	Lemma	quotient1_isom	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "coset", "coset1_injm", "isom", "norms1", "sub_isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotient1_isog : isog A (A / 1).	Proof. by apply: isom_isog quotient1_isom; apply: norms1. Qed.	Lemma	quotient1_isog	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "isog", "isom_isog", "norms1", "quotient1_isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(nsHG : H <\| G).		Hypotheses	nsHG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sHG : H \subset G	:= normal_sub nsHG.	Let	sHG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "normal_sub", "nsHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nHG : G \subset 'N(H)	:= normal_norm nsHG.	Let	nHG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "normal_norm", "nsHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nfHfG : f @* G \subset 'N(f @* H)	:= morphim_norms f nHG.	Let	nfHfG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphim_norms", "nHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fH	:= (coset (f @* H) \o f).	Notation	fH	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotm_dom_proof : G \subset 'dom fH.	Proof. by rewrite -sub_morphim_pre. Qed.	Lemma	quotm_dom_proof	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "dom", "fH", "sub_morphim_pre" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fH_G	:= (restrm quotm_dom_proof fH).	Notation	fH_G	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "fH", "quotm_dom_proof", "restrm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotm_ker_proof : 'ker (coset H) \subset 'ker fH_G.	Proof. by rewrite ker_restrm ker_comp !ker_coset morphpreIdom morphimK ?mulG_subr. Qed.	Lemma	quotm_ker_proof	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "fH_G", "ker", "ker_comp", "ker_coset", "ker_restrm", "morphimK", "morphpreIdom", "mulG_subr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotm	:= factm quotm_ker_proof nHG.	Definition	quotm	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "factm", "nHG", "quotm_ker_proof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotm_morphism	:= [morphism G / H of quotm].	Canonical	quotm_morphism	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphism", "quotm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotmE x : x \in G -> quotm (coset H x) = coset (f @* H) (f x).	Proof. exact: factmE. Qed.	Lemma	quotmE	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "factmE", "quotm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_quotm A : quotm @* (A / H) = f @* A / f @* H.	Proof. by rewrite morphim_factm [LHS]morphim_restrm morphim_comp morphimIdom. Qed.	Lemma	morphim_quotm	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimIdom", "morphim_comp", "morphim_factm", "morphim_restrm", "quotm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_quotm Abar : quotm @^-1 (Abar / f @ H) = f @*^-1 Abar / H.	Proof. rewrite morphpre_factm morphpre_restrm morphpre_comp /=. rewrite morphpreIdom -[Abar / _]quotientInorm quotientK ?subsetIr //=. rewrite morphpreMl ?morphimS // morphimK // [_ * H]normC ?subIset ?nHG //. rewrite -quotientE -mulgA quotientMidl /= setIC -morphpreIim setIA. by rewrite (setIidPl nfHfG) morphpreIim -m...	Lemma	morphpre_quotm	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphimK", "morphimS", "morphpreIdom", "morphpreIim", "morphpreMl", "morphpre_comp", "morphpre_factm", "morphpre_restrm", "mul1g", "mulgA", "nHG", "nfHfG", "normC", "quotientE", "quotientInorm", "quotientK", "quotientMidl", "quotm", "setIA", "setIC", "setIidPl", "sub1G", ...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_quotm : 'ker quotm = 'ker f / H.	Proof. by rewrite -morphpre_quotm /quotient morphim1. Qed.	Lemma	ker_quotm	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "ker", "morphim1", "morphpre_quotm", "quotient", "quotm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_quotm : 'injm f -> 'injm quotm.	Proof. by move/trivgP=> /= kf1; rewrite ker_quotm kf1 quotientE morphim1. Qed.	Lemma	injm_quotm	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "ker_quotm", "morphim1", "quotientE", "quotm", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(eqGH : G :=: H).		Hypothesis	eqGH	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_qisom_proof : 'N(H) \subset 'N(G).	Proof. by rewrite eqGH. Qed.	Lemma	im_qisom_proof	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "eqGH" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qisom_ker_proof : 'ker (coset G) \subset 'ker (coset H).	Proof. by rewrite eqGH. Qed.	Lemma	qisom_ker_proof	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "eqGH", "ker" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qisom_restr_proof : setT \subset 'N(H) / G.	Proof. by rewrite eqGH im_quotient. Qed.	Lemma	qisom_restr_proof	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "eqGH", "im_quotient", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qisom	:= restrm qisom_restr_proof (factm qisom_ker_proof im_qisom_proof).	Definition	qisom	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "factm", "im_qisom_proof", "qisom_ker_proof", "qisom_restr_proof", "restrm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qisom_morphism	:= Eval hnf in [morphism of qisom].	Canonical	qisom_morphism	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphism", "qisom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qisomE x : qisom (coset G x) = coset H x.	Proof. case Nx: (x \in 'N(H)); first exact: factmE. by rewrite !coset_default ?eqGH ?morph1. Qed.	Lemma	qisomE	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "coset_default", "eqGH", "factmE", "morph1", "qisom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_qisom Gx : val (qisom Gx) = val Gx.	Proof. by case: (cosetP Gx) => x Nx ->{Gx}; rewrite qisomE /= !val_coset -?eqGH. Qed.	Lemma	val_qisom	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "cosetP", "eqGH", "qisom", "qisomE", "val", "val_coset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_qisom A : qisom @* (A / G) = A / H.	Proof. by rewrite morphim_restrm setTI morphim_factm. Qed.	Lemma	morphim_qisom	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphim_factm", "morphim_restrm", "qisom", "setTI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_qisom A : qisom @*^-1 (A / H) = A / G.	Proof. rewrite morphpre_restrm setTI morphpre_factm eqGH. by rewrite morphpreK // im_coset subsetT. Qed.	Lemma	morphpre_qisom	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "eqGH", "im_coset", "morphpreK", "morphpre_factm", "morphpre_restrm", "qisom", "setTI", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_qisom : 'injm qisom.	Proof. by rewrite -quotient1 -morphpre_qisom morphpreS ?sub1G. Qed.	Lemma	injm_qisom	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "morphpreS", "morphpre_qisom", "qisom", "quotient1", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_qisom : qisom @* setT = setT.	Proof. by rewrite -{2}im_quotient morphim_qisom eqGH im_quotient. Qed.	Lemma	im_qisom	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "eqGH", "im_quotient", "morphim_qisom", "qisom", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qisom_isom : isom setT setT qisom.	Proof. by apply/isomP; rewrite injm_qisom im_qisom. Qed.	Lemma	qisom_isom	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "im_qisom", "injm_qisom", "isom", "isomP", "qisom", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qisom_isog : [set: coset_of G] \isog [set: coset_of H].	Proof. exact: isom_isog qisom_isom. Qed.	Lemma	qisom_isog	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset_of", "isog", "isom_isog", "qisom_isom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qisom_inj : injective qisom.	Proof. by move=> x y; apply: (injmP injm_qisom); rewrite inE. Qed.	Lemma	qisom_inj	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "inE", "injmP", "injm_qisom", "qisom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_qisom_inj : injective (fun Gx => qisom @* Gx).	Proof. by move=> Gx Gy; apply: injm_morphim_inj; rewrite (injm_qisom, subsetT). Qed.	Lemma	morphim_qisom_inj	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "injm_morphim_inj", "injm_qisom", "qisom", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
first_isom (G : {group aT}) (f : {morphism G >-> rT}) : {g : {morphism G / 'ker f >-> rT} \| 'injm g & forall A : {set aT}, g @* (A / 'ker f) = f @* A}.	Proof. have nkG := ker_norm f. have skk: 'ker (coset ('ker f)) \subset 'ker f by rewrite ker_coset. exists (factm_morphism skk nkG) => /=; last exact: morphim_factm. by rewrite ker_factm -quotientE trivg_quotient. Qed.	Lemma	first_isom	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "aT", "coset", "factm_morphism", "group", "ker", "ker_coset", "ker_factm", "ker_norm", "last", "morphim_factm", "morphism", "quotientE", "trivg_quotient" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sHG : H \subset G.		Hypothesis	sHG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
first_isog : (G / 'ker f) \isog (f @* G).	Proof. by case: (first_isom f) => g injg im_g; apply/isogP; exists g; rewrite ?im_g. Qed.	Lemma	first_isog	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "first_isom", "isog", "isogP", "ker" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
first_isom_loc : {g : {morphism H / 'ker_H f >-> rT} \| 'injm g & forall A : {set aT}, A \subset H -> g @* (A / 'ker_H f) = f @* A}.	Proof. case: (first_isom (restrm_morphism sHG f)). rewrite ker_restrm => g injg im_g; exists g => // A sAH. by rewrite im_g morphim_restrm (setIidPr sAH). Qed.	Lemma	first_isom_loc	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "aT", "first_isom", "ker_restrm", "morphim_restrm", "morphism", "restrm_morphism", "sHG", "setIidPr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
first_isog_loc : (H / 'ker_H f) \isog (f @* H).	Proof. by case: first_isom_loc => g injg im_g; apply/isogP; exists g; rewrite ?im_g. Qed.	Lemma	first_isog_loc	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "first_isom_loc", "isog", "isogP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nKH : H \subset 'N(K).		Hypothesis	nKH	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
second_isom : {f : {morphism H / (K :&: H) >-> coset_of K} \| 'injm f & forall A : {set gT}, A \subset H -> f @* (A / (K :&: H)) = A / K}.	Proof. have ->: K :&: H = 'ker_H (coset K) by rewrite ker_coset setIC. exact: first_isom_loc. Qed.	Lemma	second_isom	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "coset_of", "first_isom_loc", "gT", "ker_coset", "morphism", "setIC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
second_isog : H / (K :&: H) \isog H / K.	Proof. by rewrite setIC -{1 3}(ker_coset K); apply: first_isog_loc. Qed.	Lemma	second_isog	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "first_isog_loc", "isog", "ker_coset", "setIC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
weak_second_isog : H / (K :&: H) \isog H * K / K.	Proof. by rewrite quotientMidr; apply: second_isog. Qed.	Lemma	weak_second_isog	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "apply", "isog", "quotientMidr", "second_isog" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
homg_quotientS (A : {set gT}) : A \subset 'N(H) -> A \subset 'N(K) -> H \subset K -> A / K \homg A / H.	Proof. rewrite -!(gen_subG A) /=; set L := <<A>> => nHL nKL sKH. have sub_ker: 'ker (restrm nHL (coset H)) \subset 'ker (restrm nKL (coset K)). by rewrite !ker_restrm !ker_coset setIS. have sAL: A \subset L := subset_gen A; rewrite -(setIidPr sAL). rewrite -[_ / H](morphim_restrm nHL) -[_ / K](morphim_restrm nKL) /=....	Lemma	homg_quotientS	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[ "coset", "gT", "gen_subG", "homg", "ker", "ker_coset", "ker_restrm", "morphimS", "morphim_factm", "morphim_homg", "morphim_restrm", "restrm", "setIS", "setIidPr", "subset_gen", "subxx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sHK : H \subset K.		Hypothesis	sHK	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
snHG : H <\| G.		Hypothesis	snHG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
snKG : K <\| G.		Hypothesis	snKG	finite_group	finite_group/quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpre_set1_coset xbar : coset H @*^-1 [set xbar] = xbar.

Proof. by case: (cosetP xbar) => x Nx ->; rewrite cosetpre_set1 ?val_coset. Qed.

Lemma

cosetpre_set1_coset

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "cosetP", "cosetpre_set1", "val_coset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpreK C : coset H @*^-1 C / H = C.

Proof. by rewrite /quotient morphpreK ?sub_im_coset. Qed.

Lemma

cosetpreK

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "morphpreK", "quotient", "sub_im_coset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivg_quotient : H / H = 1.

Proof. by rewrite -[X in X / _]ker_coset /quotient morphim_ker. Qed.

Lemma

trivg_quotient

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "ker_coset", "morphim_ker", "quotient" ]

Variant of morhphim_ker

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientS1 G : G \subset H -> G / H = 1.

Proof. by move=> sGH; apply/trivgP; rewrite -trivg_quotient quotientS. Qed.

Lemma

quotientS1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "quotientS", "sGH", "trivgP", "trivg_quotient" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_cosetpre M : H \subset coset H @*^-1 M.

Proof. by rewrite -{1}ker_coset; apply: ker_sub_pre. Qed.

Lemma

sub_cosetpre

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "ker_coset", "ker_sub_pre" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_proper G K : H <| G -> H <| K -> (G / H \proper K / H) = (G \proper K).

Proof. by move=> nHG nHK; rewrite -cosetpre_proper ?quotientGK. Qed.

Lemma

quotient_proper

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "cosetpre_proper", "nHG", "nHK", "proper", "quotientGK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

normal_cosetpre M : H <| coset H @*^-1 M.

Proof. by rewrite -{1}ker_coset; apply: ker_normal_pre. Qed.

Lemma

normal_cosetpre

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "ker_coset", "ker_normal_pre" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpreSK C D : (coset H @*^-1 C \subset coset H @*^-1 D) = (C \subset D).

Proof. by rewrite morphpreSK ?sub_im_coset. Qed.

Lemma

cosetpreSK

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "morphpreSK", "sub_im_coset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_quotient_pre A C : A \subset 'N(H) -> (A / H \subset C) = (A \subset coset H @*^-1 C).

Proof. exact: sub_morphim_pre. Qed.

Lemma

sub_quotient_pre

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "sub_morphim_pre" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_cosetpre_quo C G : H <| G -> (coset H @*^-1 C \subset G) = (C \subset G / H).

Proof. by move=> nHG; rewrite -cosetpreSK quotientGK. Qed.

Lemma

sub_cosetpre_quo

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "cosetpreSK", "nHG", "quotientGK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_sub1 A : A \subset 'N(H) -> (A / H \subset [1]) = (A \subset H).

Proof. by move=> nHA /=; rewrite -[gval H in RHS]ker_coset ker_trivg_morphim nHA. Qed.

Lemma

quotient_sub1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "ker_coset", "ker_trivg_morphim" ]

Variant of ker_trivg_morphim.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientSK A B : A \subset 'N(H) -> (A / H \subset B / H) = (A \subset H * B).

Proof. by move=> nHA; rewrite morphimSK ?ker_coset. Qed.

Lemma

quotientSK

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "ker_coset", "morphimSK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientSGK A G : A \subset 'N(H) -> H \subset G -> (A / H \subset G / H) = (A \subset G).

Proof. by rewrite -{2}ker_coset; apply: morphimSGK. Qed.

Lemma

quotientSGK

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "ker_coset", "morphimSGK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_injG : {in [pred G : {group gT} | H <| G] &, injective (fun G => G / H)}.

Proof. by rewrite /normal -{1}ker_coset; apply: morphim_injG. Qed.

Lemma

quotient_injG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "gT", "group", "ker_coset", "morphim_injG", "normal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_inj G1 G2 : H <| G1 -> H <| G2 -> G1 / H = G2 / H -> G1 :=: G2.

Proof. by rewrite /normal -[in mem H]ker_coset; apply: morphim_inj. Qed.

Lemma

quotient_inj

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "G1", "apply", "ker_coset", "morphim_inj", "normal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_neq1 A : H <| A -> (A / H != 1) = (H \proper A).

Proof. case/andP=> sHA nHA; rewrite /proper sHA -trivg_quotient eqEsubset andbC. by rewrite quotientS //= quotientSGK. Qed.

Lemma

quotient_neq1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "eqEsubset", "proper", "quotientS", "quotientSGK", "trivg_quotient" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_gen A : A \subset 'N(H) -> <<A>> / H = <<A / H>>.

Proof. exact: morphim_gen. Qed.

Lemma

quotient_gen

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphim_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpre_gen C : 1 \in C -> coset H @*^-1 <<C>> = <<coset H @*^-1 C>>.

Proof. by move=> C1; rewrite morphpre_gen ?sub_im_coset. Qed.

Lemma

cosetpre_gen

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "morphpre_gen", "sub_im_coset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientR A B : A \subset 'N(H) -> B \subset 'N(H) -> [~: A, B] / H = [~: A / H, B / H].

Proof. exact: morphimR. Qed.

Lemma

quotientR

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimR" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_norm A : 'N(A) / H \subset 'N(A / H).

Proof. exact: morphim_norm. Qed.

Lemma

quotient_norm

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphim_norm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_norms A B : A \subset 'N(B) -> A / H \subset 'N(B / H).

Proof. exact: morphim_norms. Qed.

Lemma

quotient_norms

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphim_norms" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_subnorm A B : 'N_A(B) / H \subset 'N_(A / H)(B / H).

Proof. exact: morphim_subnorm. Qed.

Lemma

quotient_subnorm

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphim_subnorm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_normal A B : A <| B -> A / H <| B / H.

Proof. exact: morphim_normal. Qed.

Lemma

quotient_normal

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphim_normal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_cent1 x : 'C[x] / H \subset 'C[coset H x].

Proof. case Nx: (x \in 'N(H)); first exact: morphim_cent1. by rewrite coset_default // cent11T subsetT. Qed.

Lemma

quotient_cent1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "cent11T", "coset", "coset_default", "morphim_cent1", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_cent1s A x : A \subset 'C[x] -> A / H \subset 'C[coset H x].

Proof. by move=> sAC; apply: subset_trans (quotientS sAC) (quotient_cent1 x). Qed.

Lemma

quotient_cent1s

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "quotientS", "quotient_cent1", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_subcent1 A x : 'C_A[x] / H \subset 'C_(A / H)[coset H x].

Proof. exact: subset_trans (quotientI _ _) (setIS _ (quotient_cent1 x)). Qed.

Lemma

quotient_subcent1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "quotientI", "quotient_cent1", "setIS", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_cent A : 'C(A) / H \subset 'C(A / H).

Proof. exact: morphim_cent. Qed.

Lemma

quotient_cent

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphim_cent" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_cents A B : A \subset 'C(B) -> A / H \subset 'C(B / H).

Proof. exact: morphim_cents. Qed.

Lemma

quotient_cents

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphim_cents" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_abelian A : abelian A -> abelian (A / H).

Proof. exact: morphim_abelian. Qed.

Lemma

quotient_abelian

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "abelian", "morphim_abelian" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_subcent A B : 'C_A(B) / H \subset 'C_(A / H)(B / H).

Proof. exact: morphim_subcent. Qed.

Lemma

quotient_subcent

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphim_subcent" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

norm_quotient_pre A C : A \subset 'N(H) -> A / H \subset 'N(C) -> A \subset 'N(coset H @*^-1 C).

Proof. by move/sub_quotient_pre=> -> /subset_trans-> //; apply: morphpre_norm. Qed.

Lemma

norm_quotient_pre

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "morphpre_norm", "sub_quotient_pre", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpre_normal C D : (coset H @*^-1 C <| coset H @*^-1 D) = (C <| D).

Proof. by rewrite morphpre_normal ?sub_im_coset. Qed.

Lemma

cosetpre_normal

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "morphpre_normal", "sub_im_coset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_normG G : H <| G -> 'N(G) / H = 'N(G / H).

Proof. case/andP=> sHG nHG. by rewrite [_ / _]morphim_normG ?ker_coset // im_coset setTI. Qed.

Lemma

quotient_normG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "im_coset", "ker_coset", "morphim_normG", "nHG", "sHG", "setTI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_subnormG A G : H <| G -> 'N_A(G) / H = 'N_(A / H)(G / H).

Proof. by case/andP=> sHG nHG; rewrite -morphim_subnormG ?ker_coset. Qed.

Lemma

quotient_subnormG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "ker_coset", "morphim_subnormG", "nHG", "sHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpre_cent1 x : 'C_('N(H))[x] \subset coset H @*^-1 'C[coset H x].

Proof. case Nx: (x \in 'N(H)); first by rewrite morphpre_cent1. by rewrite coset_default // cent11T morphpreT subsetIl. Qed.

Lemma

cosetpre_cent1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "cent11T", "coset", "coset_default", "morphpreT", "morphpre_cent1", "subsetIl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpre_cent1s C x : coset H @*^-1 C \subset 'C[x] -> C \subset 'C[coset H x].

Proof. move=> sC; rewrite -cosetpreSK; apply: subset_trans (cosetpre_cent1 x). by rewrite subsetI subsetIl. Qed.

Lemma

cosetpre_cent1s

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "cosetpreSK", "cosetpre_cent1", "subsetI", "subsetIl", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpre_subcent1 C x : 'C_(coset H @*^-1 C)[x] \subset coset H @*^-1 'C_C[coset H x].

Proof. by rewrite -morphpreIdom -setIA setICA morphpreI setIS // cosetpre_cent1. Qed.

Lemma

cosetpre_subcent1

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "cosetpre_cent1", "morphpreI", "morphpreIdom", "setIA", "setICA", "setIS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpre_cent A : 'C_('N(H))(A) \subset coset H @*^-1 'C(A / H).

Proof. exact: morphpre_cent. Qed.

Lemma

cosetpre_cent

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "morphpre_cent" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpre_cents A C : coset H @*^-1 C \subset 'C(A) -> C \subset 'C(A / H).

Proof. by apply: morphpre_cents; rewrite ?sub_im_coset. Qed.

Lemma

cosetpre_cents

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "morphpre_cents", "sub_im_coset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cosetpre_subcent C A : 'C_(coset H @*^-1 C)(A) \subset coset H @*^-1 'C_C(A / H).

Proof. exact: morphpre_subcent. Qed.

Lemma

cosetpre_subcent

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "morphpre_subcent" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

restrm_quotientE G A (nHG : G \subset 'N(H)) : A \subset G -> restrm nHG (coset H) @* A = A / H.

Proof. exact: restrmEsub. Qed.

Lemma

restrm_quotientE

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "nHG", "restrm", "restrmEsub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nHG : H <| G.

Hypothesis

nHG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inv_quotient_spec (P : pred {group gT}) : Prop

:= InvQuotientSpec K of Kbar :=: K / H & H \subset K & P K.

Variant

inv_quotient_spec

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "gT", "group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inv_quotientS : Kbar \subset G / H -> inv_quotient_spec (fun K => K \subset G).

Proof. move=> sKH; exists (coset H @*^-1 Kbar); first by rewrite cosetpreK. by rewrite sub_cosetpre. by rewrite sub_cosetpre_quo. Qed.

Lemma

inv_quotientS

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "cosetpreK", "inv_quotient_spec", "sub_cosetpre", "sub_cosetpre_quo" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inv_quotientN : Kbar <| G / H -> inv_quotient_spec (fun K => K <| G).

Proof. move=> nKbar; case/inv_quotientS: (normal_sub nKbar) => K defKbar sHK sKG. exists K => //; rewrite defKbar -cosetpre_normal !quotientGK // in nKbar. exact: normalS nHG. Qed.

Lemma

inv_quotientN

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "cosetpre_normal", "inv_quotientS", "inv_quotient_spec", "nHG", "normalS", "normal_sub", "quotientGK", "sHK", "sKG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientMidr A : A * H / H = A / H.

Proof. by rewrite [_ /_]morphimMr ?normG //= -!quotientE trivg_quotient mulg1. Qed.

Lemma

quotientMidr

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimMr", "mulg1", "normG", "quotientE", "trivg_quotient" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientMidl A : H * A / H = A / H.

Proof. by rewrite [_ /_]morphimMl ?normG //= -!quotientE trivg_quotient mul1g. Qed.

Lemma

quotientMidl

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimMl", "mul1g", "normG", "quotientE", "trivg_quotient" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientYidr G : G \subset 'N(H) -> G <*> H / H = G / H.

Proof. move=> nHG; rewrite -genM_join quotient_gen ?mul_subG ?normG //. by rewrite quotientMidr genGid. Qed.

Lemma

quotientYidr

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "genGid", "genM_join", "mul_subG", "nHG", "normG", "quotientMidr", "quotient_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotientYidl G : G \subset 'N(H) -> H <*> G / H = G / H.

Proof. by move=> nHG; rewrite joingC quotientYidr. Qed.

Lemma

quotientYidl

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "joingC", "nHG", "quotientYidr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(nHG : G \subset 'N(H)) (tiHG : H :&: G = 1).

Hypotheses

nHG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_isom : isom G (G / H) (restrm nHG (coset H)).

Proof. by apply/isomP; rewrite ker_restrm setIC ker_coset tiHG im_restrm. Qed.

Lemma

quotient_isom

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "im_restrm", "isom", "isomP", "ker_coset", "ker_restrm", "nHG", "restrm", "setIC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient_isog : isog G (G / H).

Proof. exact: isom_isog quotient_isom. Qed.

Lemma

quotient_isog

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "isog", "isom_isog", "quotient_isom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"A / H"

:= (quotient_group A H) : Group_scope.

Notation

A / H

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "quotient_group" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coset1_injm : 'injm (@coset gT 1).

Proof. by rewrite ker_coset /=. Qed.

Lemma

coset1_injm

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "gT", "ker_coset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient1_isom : isom A (A / 1) (coset 1).

Proof. by apply: sub_isom coset1_injm; rewrite ?norms1. Qed.

Lemma

quotient1_isom

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "coset", "coset1_injm", "isom", "norms1", "sub_isom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotient1_isog : isog A (A / 1).

Proof. by apply: isom_isog quotient1_isom; apply: norms1. Qed.

Lemma

quotient1_isog

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "isog", "isom_isog", "norms1", "quotient1_isom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(nsHG : H <| G).

Hypotheses

nsHG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sHG : H \subset G

:= normal_sub nsHG.

Let

sHG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "normal_sub", "nsHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nHG : G \subset 'N(H)

:= normal_norm nsHG.

Let

nHG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "normal_norm", "nsHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nfHfG : f @* G \subset 'N(f @* H)

:= morphim_norms f nHG.

Let

nfHfG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphim_norms", "nHG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fH

:= (coset (f @* H) \o f).

Notation

fH

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotm_dom_proof : G \subset 'dom fH.

Proof. by rewrite -sub_morphim_pre. Qed.

Lemma

quotm_dom_proof

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "dom", "fH", "sub_morphim_pre" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fH_G

:= (restrm quotm_dom_proof fH).

Notation

fH_G

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "fH", "quotm_dom_proof", "restrm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotm_ker_proof : 'ker (coset H) \subset 'ker fH_G.

Proof. by rewrite ker_restrm ker_comp !ker_coset morphpreIdom morphimK ?mulG_subr. Qed.

Lemma

quotm_ker_proof

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "fH_G", "ker", "ker_comp", "ker_coset", "ker_restrm", "morphimK", "morphpreIdom", "mulG_subr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotm

:= factm quotm_ker_proof nHG.

Definition

quotm

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "factm", "nHG", "quotm_ker_proof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotm_morphism

:= [morphism G / H of quotm].

Canonical

quotm_morphism

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphism", "quotm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotmE x : x \in G -> quotm (coset H x) = coset (f @* H) (f x).

Proof. exact: factmE. Qed.

Lemma

quotmE

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "factmE", "quotm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_quotm A : quotm @* (A / H) = f @* A / f @* H.

Proof. by rewrite morphim_factm [LHS]morphim_restrm morphim_comp morphimIdom. Qed.

Lemma

morphim_quotm

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimIdom", "morphim_comp", "morphim_factm", "morphim_restrm", "quotm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_quotm Abar : quotm @*^-1 (Abar / f @* H) = f @*^-1 Abar / H.

Proof. rewrite morphpre_factm morphpre_restrm morphpre_comp /=. rewrite morphpreIdom -[Abar / _]quotientInorm quotientK ?subsetIr //=. rewrite morphpreMl ?morphimS // morphimK // [_ * H]normC ?subIset ?nHG //. rewrite -quotientE -mulgA quotientMidl /= setIC -morphpreIim setIA. by rewrite (setIidPl nfHfG) morphpreIim -m...

Lemma

morphpre_quotm

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphimK", "morphimS", "morphpreIdom", "morphpreIim", "morphpreMl", "morphpre_comp", "morphpre_factm", "morphpre_restrm", "mul1g", "mulgA", "nHG", "nfHfG", "normC", "quotientE", "quotientInorm", "quotientK", "quotientMidl", "quotm", "setIA", "setIC", "setIidPl", "sub1G", ...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_quotm : 'ker quotm = 'ker f / H.

Proof. by rewrite -morphpre_quotm /quotient morphim1. Qed.

Lemma

ker_quotm

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "ker", "morphim1", "morphpre_quotm", "quotient", "quotm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_quotm : 'injm f -> 'injm quotm.

Proof. by move/trivgP=> /= kf1; rewrite ker_quotm kf1 quotientE morphim1. Qed.

Lemma

injm_quotm

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "ker_quotm", "morphim1", "quotientE", "quotm", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(eqGH : G :=: H).

Hypothesis

eqGH

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_qisom_proof : 'N(H) \subset 'N(G).

Proof. by rewrite eqGH. Qed.

Lemma

im_qisom_proof

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "eqGH" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qisom_ker_proof : 'ker (coset G) \subset 'ker (coset H).

Proof. by rewrite eqGH. Qed.

Lemma

qisom_ker_proof

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "eqGH", "ker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qisom_restr_proof : setT \subset 'N(H) / G.

Proof. by rewrite eqGH im_quotient. Qed.

Lemma

qisom_restr_proof

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "eqGH", "im_quotient", "setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qisom

:= restrm qisom_restr_proof (factm qisom_ker_proof im_qisom_proof).

Definition

qisom

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "factm", "im_qisom_proof", "qisom_ker_proof", "qisom_restr_proof", "restrm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qisom_morphism

:= Eval hnf in [morphism of qisom].

Canonical

qisom_morphism

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphism", "qisom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qisomE x : qisom (coset G x) = coset H x.

Proof. case Nx: (x \in 'N(H)); first exact: factmE. by rewrite !coset_default ?eqGH ?morph1. Qed.

Lemma

qisomE

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "coset_default", "eqGH", "factmE", "morph1", "qisom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

val_qisom Gx : val (qisom Gx) = val Gx.

Proof. by case: (cosetP Gx) => x Nx ->{Gx}; rewrite qisomE /= !val_coset -?eqGH. Qed.

Lemma

val_qisom

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "cosetP", "eqGH", "qisom", "qisomE", "val", "val_coset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_qisom A : qisom @* (A / G) = A / H.

Proof. by rewrite morphim_restrm setTI morphim_factm. Qed.

Lemma

morphim_qisom

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphim_factm", "morphim_restrm", "qisom", "setTI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_qisom A : qisom @*^-1 (A / H) = A / G.

Proof. rewrite morphpre_restrm setTI morphpre_factm eqGH. by rewrite morphpreK // im_coset subsetT. Qed.

Lemma

morphpre_qisom

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "eqGH", "im_coset", "morphpreK", "morphpre_factm", "morphpre_restrm", "qisom", "setTI", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_qisom : 'injm qisom.

Proof. by rewrite -quotient1 -morphpre_qisom morphpreS ?sub1G. Qed.

Lemma

injm_qisom

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "morphpreS", "morphpre_qisom", "qisom", "quotient1", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_qisom : qisom @* setT = setT.

Proof. by rewrite -{2}im_quotient morphim_qisom eqGH im_quotient. Qed.

Lemma

im_qisom

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "eqGH", "im_quotient", "morphim_qisom", "qisom", "setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qisom_isom : isom setT setT qisom.

Proof. by apply/isomP; rewrite injm_qisom im_qisom. Qed.

Lemma

qisom_isom

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "im_qisom", "injm_qisom", "isom", "isomP", "qisom", "setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qisom_isog : [set: coset_of G] \isog [set: coset_of H].

Proof. exact: isom_isog qisom_isom. Qed.

Lemma

qisom_isog

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset_of", "isog", "isom_isog", "qisom_isom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qisom_inj : injective qisom.

Proof. by move=> x y; apply: (injmP injm_qisom); rewrite inE. Qed.

Lemma

qisom_inj

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "inE", "injmP", "injm_qisom", "qisom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_qisom_inj : injective (fun Gx => qisom @* Gx).

Proof. by move=> Gx Gy; apply: injm_morphim_inj; rewrite (injm_qisom, subsetT). Qed.

Lemma

morphim_qisom_inj

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "injm_morphim_inj", "injm_qisom", "qisom", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

first_isom (G : {group aT}) (f : {morphism G >-> rT}) : {g : {morphism G / 'ker f >-> rT} | 'injm g & forall A : {set aT}, g @* (A / 'ker f) = f @* A}.

Proof. have nkG := ker_norm f. have skk: 'ker (coset ('ker f)) \subset 'ker f by rewrite ker_coset. exists (factm_morphism skk nkG) => /=; last exact: morphim_factm. by rewrite ker_factm -quotientE trivg_quotient. Qed.

Lemma

first_isom

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "aT", "coset", "factm_morphism", "group", "ker", "ker_coset", "ker_factm", "ker_norm", "last", "morphim_factm", "morphism", "quotientE", "trivg_quotient" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sHG : H \subset G.

Hypothesis

sHG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

first_isog : (G / 'ker f) \isog (f @* G).

Proof. by case: (first_isom f) => g injg im_g; apply/isogP; exists g; rewrite ?im_g. Qed.

Lemma

first_isog

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "first_isom", "isog", "isogP", "ker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

first_isom_loc : {g : {morphism H / 'ker_H f >-> rT} | 'injm g & forall A : {set aT}, A \subset H -> g @* (A / 'ker_H f) = f @* A}.

Proof. case: (first_isom (restrm_morphism sHG f)). rewrite ker_restrm => g injg im_g; exists g => // A sAH. by rewrite im_g morphim_restrm (setIidPr sAH). Qed.

Lemma

first_isom_loc

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "aT", "first_isom", "ker_restrm", "morphim_restrm", "morphism", "restrm_morphism", "sHG", "setIidPr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

first_isog_loc : (H / 'ker_H f) \isog (f @* H).

Proof. by case: first_isom_loc => g injg im_g; apply/isogP; exists g; rewrite ?im_g. Qed.

Lemma

first_isog_loc

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "first_isom_loc", "isog", "isogP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nKH : H \subset 'N(K).

Hypothesis

nKH

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

second_isom : {f : {morphism H / (K :&: H) >-> coset_of K} | 'injm f & forall A : {set gT}, A \subset H -> f @* (A / (K :&: H)) = A / K}.

Proof. have ->: K :&: H = 'ker_H (coset K) by rewrite ker_coset setIC. exact: first_isom_loc. Qed.

Lemma

second_isom

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "coset_of", "first_isom_loc", "gT", "ker_coset", "morphism", "setIC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

second_isog : H / (K :&: H) \isog H / K.

Proof. by rewrite setIC -{1 3}(ker_coset K); apply: first_isog_loc. Qed.

Lemma

second_isog

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "first_isog_loc", "isog", "ker_coset", "setIC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

weak_second_isog : H / (K :&: H) \isog H * K / K.

Proof. by rewrite quotientMidr; apply: second_isog. Qed.

Lemma

weak_second_isog

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "apply", "isog", "quotientMidr", "second_isog" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

homg_quotientS (A : {set gT}) : A \subset 'N(H) -> A \subset 'N(K) -> H \subset K -> A / K \homg A / H.

Proof. rewrite -!(gen_subG A) /=; set L := <<A>> => nHL nKL sKH. have sub_ker: 'ker (restrm nHL (coset H)) \subset 'ker (restrm nKL (coset K)). by rewrite !ker_restrm !ker_coset setIS. have sAL: A \subset L := subset_gen A; rewrite -(setIidPr sAL). rewrite -[_ / H](morphim_restrm nHL) -[_ / K](morphim_restrm nKL) /=....

Lemma

homg_quotientS

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[ "coset", "gT", "gen_subG", "homg", "ker", "ker_coset", "ker_restrm", "morphimS", "morphim_factm", "morphim_homg", "morphim_restrm", "restrm", "setIS", "setIidPr", "subset_gen", "subxx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sHK : H \subset K.

Hypothesis

sHK

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

snHG : H <| G.

Hypothesis

snHG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

snKG : K <| G.

Hypothesis

snKG

finite_group

finite_group/quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "prime", "finset", "fingroup", "morphism", "automorphism" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d