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
morphimMr A B : B \subset D -> f @* (A * B) = f @* A * f @* B.	Proof. move=> sBD; apply: invg_inj. by rewrite invMg -!morphimV invMg morphimMl // -invGid invSg. Qed.	Lemma	morphimMr	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "invGid", "invMg", "invSg", "invg_inj", "morphimMl", "morphimV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpreMl R S : R \subset f @* D -> f @^-1 (R S) = f @^-1 R f @*^-1 S.	Proof. move=> sRfD; apply/setP=> x; rewrite !inE. apply/andP/imset2P=> [[Dx] \| [y z]]; last first. rewrite !inE => /andP[Dy Rfy] /andP[Dz Rfz] ->. by rewrite ?(groupM, morphM, imset2_f). case/imset2P=> fy fz Rfy Rfz def_fx. have /morphimP[y Dy _ def_fy]: fy \in f @* D := subsetP sRfD fy Rfy. exists y (y^-1 * x); la...	Lemma	morphpreMl	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "groupM", "groupV", "imset2P", "imset2_f", "inE", "last", "morphM", "morphV", "morphimP", "mulKVg", "mulKg", "setP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimJ A x : x \in D -> f @* (A :^ x) = f @* A :^ f x.	Proof. move=> Dx; rewrite !conjsgE morphimMl ?(morphimMr, sub1set, groupV) //. by rewrite !(morphim_set1, groupV, morphV). Qed.	Lemma	morphimJ	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "conjsgE", "groupV", "morphV", "morphimMl", "morphimMr", "morphim_set1", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpreJ R x : x \in D -> f @^-1 (R :^ f x) = f @^-1 R :^ x.	Proof. move=> Dx; apply/setP=> y; rewrite conjIg !inE conjGid // !mem_conjg inE. by case Dy: (y \in D); rewrite // morphJ ?(morphV, groupV). Qed.	Lemma	morphpreJ	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "conjGid", "conjIg", "groupV", "inE", "mem_conjg", "morphJ", "morphV", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_class x A : x \in D -> A \subset D -> f @* (x ^: A) = f x ^: f @* A.	Proof. move=> Dx sAD; rewrite !morphimEsub ?class_subG // /class -!imset_comp. by apply: eq_in_imset => y Ay /=; rewrite morphJ // (subsetP sAD). Qed.	Lemma	morphim_class	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "class", "class_subG", "eq_in_imset", "imset_comp", "morphJ", "morphimEsub", "sAD", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
classes_morphim A : A \subset D -> classes (f @* A) = [set f @* xA \| xA in classes A].	Proof. move=> sAD; rewrite morphimEsub // /classes -!imset_comp. apply: eq_in_imset => x /(subsetP sAD) Dx /=. by rewrite morphim_class ?morphimEsub. Qed.	Lemma	classes_morphim	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "classes", "eq_in_imset", "imset_comp", "morphimEsub", "morphim_class", "sAD", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimT : f @* setT = f @* D.	Proof. by rewrite -morphimIdom setIT. Qed.	Lemma	morphimT	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphimIdom", "setIT", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimU A B : f @* (A :\|: B) = f @* A :\|: f @* B.	Proof. by rewrite -imsetU -setIUr. Qed.	Lemma	morphimU	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "imsetU", "setIUr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimI A B : f @* (A :&: B) \subset f @* A :&: f @* B.	Proof. by rewrite subsetI // ?morphimS ?(subsetIl, subsetIr). Qed.	Lemma	morphimI	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphimS", "subsetI", "subsetIl", "subsetIr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre0 : f @*^-1 set0 = set0.	Proof. by rewrite morphpreE preimset0 setI0. Qed.	Lemma	morphpre0	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphpreE", "preimset0", "set0", "setI0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpreT : f @*^-1 setT = D.	Proof. by rewrite morphpreE preimsetT setIT. Qed.	Lemma	morphpreT	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphpreE", "preimsetT", "setIT", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpreU R S : f @^-1 (R :\|: S) = f @^-1 R :\|: f @*^-1 S.	Proof. by rewrite -setIUr -preimsetU. Qed.	Lemma	morphpreU	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "preimsetU", "setIUr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpreI R S : f @^-1 (R :&: S) = f @^-1 R :&: f @*^-1 S.	Proof. by rewrite -setIIr -preimsetI. Qed.	Lemma	morphpreI	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "preimsetI", "setIIr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpreD R S : f @^-1 (R :\: S) = f @^-1 R :\: f @*^-1 S.	Proof. by apply/setP=> x /[!inE]; case: (x \in D). Qed.	Lemma	morphpreD	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "inE", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
kerP x : x \in D -> reflect (f x = 1) (x \in 'ker f).	Proof. by move=> Dx; rewrite 2!inE Dx; apply: set1P. Qed.	Lemma	kerP	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "inE", "ker", "set1P" ]	kernel, domain properties	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dom_ker : {subset 'ker f <= D}.	Proof. by move=> x /morphpreP[]. Qed.	Lemma	dom_ker	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "morphpreP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mker x : x \in 'ker f -> f x = 1.	Proof. by move=> Kx; apply/kerP=> //; apply: dom_ker. Qed.	Lemma	mker	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "dom_ker", "ker", "kerP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mkerl x y : x \in 'ker f -> y \in D -> f (x * y) = f y.	Proof. by move=> Kx Dy; rewrite morphM // ?(dom_ker, mker Kx, mul1g). Qed.	Lemma	mkerl	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "dom_ker", "ker", "mker", "morphM", "mul1g" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mkerr x y : x \in D -> y \in 'ker f -> f (x * y) = f x.	Proof. by move=> Dx Ky; rewrite morphM // ?(dom_ker, mker Ky, mulg1). Qed.	Lemma	mkerr	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "dom_ker", "ker", "mker", "morphM", "mulg1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcoset_kerP x y : x \in D -> y \in D -> reflect (f x = f y) (x \in 'ker f :* y).	Proof. move=> Dx Dy; rewrite mem_rcoset !inE groupM ?morphM ?groupV //=. by rewrite morphV // -eq_mulgV1; apply: eqP. Qed.	Lemma	rcoset_kerP	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "eq_mulgV1", "groupM", "groupV", "inE", "ker", "mem_rcoset", "morphM", "morphV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_rcoset x y : x \in D -> y \in D -> f x = f y -> exists2 z, z \in 'ker f & x = z * y.	Proof. by move=> Dx Dy eqfxy; apply/rcosetP; apply/rcoset_kerP. Qed.	Lemma	ker_rcoset	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "ker", "rcosetP", "rcoset_kerP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_norm : D \subset 'N('ker f).	Proof. apply/subsetP=> x Dx /[1!inE]; apply/subsetP=> _ /imsetP[y Ky ->]. by rewrite !inE groupJ ?morphJ // ?dom_ker //= mker ?conj1g. Qed.	Lemma	ker_norm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "conj1g", "dom_ker", "groupJ", "imsetP", "inE", "ker", "mker", "morphJ", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_normal : 'ker f <\| D.	Proof. by rewrite /(_ <\| D) subsetIl ker_norm. Qed.	Lemma	ker_normal	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "ker_norm", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimGI G A : 'ker f \subset G -> f @* (G :&: A) = f @* G :&: f @* A.	Proof. move=> sKG; apply/eqP; rewrite eqEsubset morphimI setIC. apply/subsetP=> _ /setIP[/morphimP[x Dx Ax ->] /morphimP[z Dz Gz]]. case/ker_rcoset=> {Dz}// y Ky def_x. have{z Gz y Ky def_x} Gx: x \in G by rewrite def_x groupMl // (subsetP sKG). by rewrite imset_f ?inE // Dx Gx Ax. Qed.	Lemma	morphimGI	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "eqEsubset", "groupMl", "imset_f", "inE", "ker", "ker_rcoset", "morphimI", "morphimP", "sKG", "setIC", "setIP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimIG A G : 'ker f \subset G -> f @* (A :&: G) = f @* A :&: f @* G.	Proof. by move=> sKG; rewrite setIC morphimGI // setIC. Qed.	Lemma	morphimIG	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "morphimGI", "sKG", "setIC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimD A B : f @* A :\: f @* B \subset f @* (A :\: B).	Proof. rewrite subDset -morphimU morphimS //. by rewrite setDE setUIr setUCr setIT subsetUr. Qed.	Lemma	morphimD	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphimS", "morphimU", "setDE", "setIT", "setUCr", "setUIr", "subDset", "subsetUr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimDG A G : 'ker f \subset G -> f @* (A :\: G) = f @* A :\: f @* G.	Proof. move=> sKG; apply/eqP; rewrite eqEsubset morphimD andbT !setDE subsetI. rewrite morphimS ?subsetIl // -[~: f @* G]setU0 -subDset setDE setCK. by rewrite -morphimIG //= setIAC -setIA setICr setI0 morphim0. Qed.	Lemma	morphimDG	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "eqEsubset", "ker", "morphim0", "morphimD", "morphimIG", "morphimS", "sKG", "setCK", "setDE", "setI0", "setIA", "setIAC", "setICr", "setU0", "subDset", "subsetI", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimD1 A : (f @* A)^# \subset f @* A^#.	Proof. by rewrite -!set1gE -morphim1 morphimD. Qed.	Lemma	morphimD1	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphim1", "morphimD", "set1gE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_groupset M : group_set (f @*^-1 M).	Proof. apply/group_setP; split=> [\|x y]; rewrite !inE ?(morph1, group1) //. by case/andP=> Dx Mfx /andP[Dy Mfy]; rewrite morphM ?groupM. Qed.	Lemma	morphpre_groupset	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "group1", "groupM", "group_set", "group_setP", "inE", "morph1", "morphM", "split" ]	group structure preservation	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_groupset G : group_set (f @* G).	Proof. apply/group_setP; split=> [\|_ _ /morphimP[x Dx Gx ->] /morphimP[y Dy Gy ->]]. by rewrite -morph1 imset_f ?group1. by rewrite -morphM ?imset_f ?inE ?groupM. Qed.	Lemma	morphim_groupset	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "group1", "groupM", "group_set", "group_setP", "imset_f", "inE", "morph1", "morphM", "morphimP", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_group fPh M	:= @group _ (morphpre fPh M) (morphpre_groupset M).	Canonical	morphpre_group	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "group", "morphpre", "morphpre_groupset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_group fPh G	:= @group _ (morphim fPh G) (morphim_groupset G).	Canonical	morphim_group	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "group", "morphim", "morphim_groupset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_group fPh : {group aT}	:= Eval hnf in [group of ker fPh].	Canonical	ker_group	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "group", "ker" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_dom_groupset : group_set (f @: D).	Proof. by rewrite -morphimEdom groupP. Qed.	Lemma	morph_dom_groupset	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "groupP", "group_set", "morphimEdom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_dom_group	:= group morph_dom_groupset.	Canonical	morph_dom_group	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "group", "morph_dom_groupset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpreMr R S : S \subset f @* D -> f @^-1 (R S) = f @^-1 R f @*^-1 S.	Proof. move=> sSfD; apply: invg_inj. by rewrite invMg -!morphpreV invMg morphpreMl // -invSg invgK invGid. Qed.	Lemma	morphpreMr	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "invGid", "invMg", "invSg", "invgK", "invg_inj", "morphpreMl", "morphpreV" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimK A : A \subset D -> f @^-1 (f @ A) = 'ker f * A.	Proof. move=> sAD; apply/setP=> x; rewrite !inE. apply/idP/idP=> [/andP[Dx /morphimP[y Dy Ay eqxy]] \| /imset2P[z y Kz Ay ->{x}]]. rewrite -(mulgKV y x) mem_mulg // !inE !(groupM, morphM, groupV) //. by rewrite morphV //= eqxy mulgV. have [Dy Dz]: y \in D /\ z \in D by rewrite (subsetP sAD) // dom_ker. by rewrite gr...	Lemma	morphimK	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "dom_ker", "groupM", "groupV", "imset2P", "imset_f", "inE", "ker", "mem_mulg", "mker", "morphM", "morphV", "morphimP", "mul1g", "mulgKV", "mulgV", "sAD", "setP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimGK G : 'ker f \subset G -> G \subset D -> f @^-1 (f @ G) = G.	Proof. by move=> sKG sGD; rewrite morphimK // mulSGid. Qed.	Lemma	morphimGK	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "morphimK", "mulSGid", "sGD", "sKG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_set1 x : x \in D -> f @^-1 [set f x] = 'ker f : x.	Proof. by move=> Dx; rewrite -morphim_set1 // morphimK ?sub1set. Qed.	Lemma	morphpre_set1	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "ker", "morphimK", "morphim_set1", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpreK R : R \subset f @* D -> f @* (f @*^-1 R) = R.	Proof. move=> sRfD; apply/setP=> y; apply/morphimP/idP=> [[x _] \| Ry]. by rewrite !inE; case/andP=> _ Rfx ->. have /morphimP[x Dx _ defy]: y \in f @* D := subsetP sRfD y Ry. by exists x; rewrite // !inE Dx -defy. Qed.	Lemma	morphpreK	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "inE", "morphimP", "setP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_ker : f @* 'ker f = 1.	Proof. by rewrite morphpreK ?sub1G. Qed.	Lemma	morphim_ker	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "morphpreK", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_sub_pre M : 'ker f \subset f @*^-1 M.	Proof. by rewrite morphpreS ?sub1G. Qed.	Lemma	ker_sub_pre	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "morphpreS", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_normal_pre M : 'ker f <\| f @*^-1 M.	Proof. by rewrite /normal ker_sub_pre subIset ?ker_norm. Qed.	Lemma	ker_normal_pre	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "ker_norm", "ker_sub_pre", "normal", "subIset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpreSK R S : R \subset f @* D -> (f @^-1 R \subset f @^-1 S) = (R \subset S).	Proof. move=> sRfD; apply/idP/idP=> [sf'RS\|]; last exact: morphpreS. suffices: R \subset f @* D :&: S by rewrite subsetI sRfD. rewrite -(morphpreK sRfD) -[_ :&: S]morphpreK (morphimS, subsetIl) //. by rewrite morphpreI morphimGK ?subsetIl // setIA setIid. Qed.	Lemma	morphpreSK	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "last", "morphimGK", "morphimS", "morphpreI", "morphpreK", "morphpreS", "setIA", "setIid", "subsetI", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_morphim_pre A R : A \subset D -> (f @* A \subset R) = (A \subset f @*^-1 R).	Proof. move=> sAD; rewrite -morphpreSK (morphimS, morphimK) //. apply/idP/idP; first by apply: subset_trans; apply: mulG_subr. by move/(mulgS ('ker f)); rewrite -morphpreMl ?(sub1G, mul1g). Qed.	Lemma	sub_morphim_pre	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "ker", "morphimK", "morphimS", "morphpreMl", "morphpreSK", "mul1g", "mulG_subr", "mulgS", "sAD", "sub1G", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_proper R S : R \subset f @* D -> S \subset f @* D -> (f @^-1 R \proper f @^-1 S) = (R \proper S).	Proof. by move=> dQ dR; rewrite /proper !morphpreSK. Qed.	Lemma	morphpre_proper	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphpreSK", "proper" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_morphpre_im R G : 'ker f \subset G -> G \subset D -> R \subset f @* D -> (f @^-1 R \subset G) = (R \subset f @ G).	Proof. by symmetry; rewrite -morphpreSK ?morphimGK. Qed.	Lemma	sub_morphpre_im	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "morphimGK", "morphpreSK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_trivg_morphim A : (A \subset 'ker f) = (A \subset D) && (f @* A \subset [1]).	Proof. case sAD: (A \subset D); first by rewrite sub_morphim_pre. by rewrite subsetI sAD. Qed.	Lemma	ker_trivg_morphim	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "sAD", "sub_morphim_pre", "subsetI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimSK A B : A \subset D -> (f @* A \subset f @* B) = (A \subset 'ker f * B).	Proof. move=> sAD; transitivity (A \subset 'ker f * (D :&: B)). by rewrite -morphimK ?subsetIl // -sub_morphim_pre // /morphim setIA setIid. by rewrite setIC group_modl (subsetIl, subsetI) // andbC sAD. Qed.	Lemma	morphimSK	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "group_modl", "ker", "morphim", "morphimK", "sAD", "setIA", "setIC", "setIid", "sub_morphim_pre", "subsetI", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimSGK A G : A \subset D -> 'ker f \subset G -> (f @* A \subset f @* G) = (A \subset G).	Proof. by move=> sGD skfK; rewrite morphimSK // mulSGid. Qed.	Lemma	morphimSGK	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "morphimSK", "mulSGid", "sGD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltn_morphim A : [1] \proper 'ker_A f -> #\|f @* A\| < #\|A\|.	Proof. case/properP; rewrite sub1set => /setIP[A1 _] [x /setIP[Ax kx] x1]. rewrite (cardsD1 1 A) A1 ltnS -{1}(setD1K A1) morphimU morphim1. rewrite (setUidPr _) ?sub1set. by rewrite -(mker kx) mem_morphim ?(dom_ker kx) // inE x1. by rewrite (leq_trans (leq_imset_card _ _)) ?subset_leq_card ?subsetIr. Qed.	Lemma	ltn_morphim	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "cardsD1", "dom_ker", "inE", "leq_imset_card", "leq_trans", "ltnS", "mem_morphim", "mker", "morphim1", "morphimU", "proper", "properP", "setD1K", "setIP", "setUidPr", "sub1set", "subsetIr", "subset_leq_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_inj : {in [pred R : {set rT} \| R \subset f @* D] &, injective (fun R => f @*^-1 R)}.	Proof. exact: can_in_inj morphpreK. Qed.	Lemma	morphpre_inj	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphpreK" ]	injectivity of image and preimage	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_injG : {in [pred G : {group aT} \| 'ker f \subset G & G \subset D] &, injective (fun G => f @* G)}.	Proof. move=> G H /andP[sKG sGD] /andP[sKH sHD] eqfGH. by apply: val_inj; rewrite /= -(morphimGK sKG sGD) eqfGH morphimGK. Qed.	Lemma	morphim_injG	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "aT", "apply", "group", "ker", "morphimGK", "sGD", "sHD", "sKG", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_inj G H : ('ker f \subset G) && (G \subset D) -> ('ker f \subset H) && (H \subset D) -> f @* G = f @* H -> G :=: H.	Proof. by move=> nsGf nsHf /morphim_injG->. Qed.	Lemma	morphim_inj	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "morphim_injG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_gen A : A \subset D -> f @* <<A>> = <<f @* A>>.	Proof. move=> sAD; apply/eqP. rewrite eqEsubset andbC gen_subG morphimS; first exact: subset_gen. by rewrite sub_morphim_pre gen_subG // -sub_morphim_pre // subset_gen. Qed.	Lemma	morphim_gen	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "eqEsubset", "gen_subG", "morphimS", "sAD", "sub_morphim_pre", "subset_gen" ]	commutation with generated groups and cycles	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_cycle x : x \in D -> f @* <[x]> = <[f x]>.	Proof. by move=> Dx; rewrite morphim_gen (sub1set, morphim_set1). Qed.	Lemma	morphim_cycle	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "morphim_gen", "morphim_set1", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimY A B : A \subset D -> B \subset D -> f @* (A <> B) = f @ A <> f @ B.	Proof. by move=> sAD sBD; rewrite morphim_gen ?morphimU // subUset sAD. Qed.	Lemma	morphimY	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphimU", "morphim_gen", "sAD", "subUset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_gen R : 1 \in R -> R \subset f @* D -> f @^-1 <<R>> = <<f @^-1 R>>.	Proof. move=> R1 sRfD; apply/eqP. rewrite eqEsubset andbC gen_subG morphpreS; first exact: subset_gen. rewrite -{1}(morphpreK sRfD) -morphim_gen ?subsetIl // morphimGK //=. by rewrite sub_gen // setIS // preimsetS ?sub1set. by rewrite gen_subG subsetIl. Qed.	Lemma	morphpre_gen	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "R1", "apply", "eqEsubset", "gen_subG", "morphimGK", "morphim_gen", "morphpreK", "morphpreS", "preimsetS", "setIS", "sub1set", "sub_gen", "subsetIl", "subset_gen" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphimR A B : A \subset D -> B \subset D -> f @* [~: A, B] = [~: f @* A, f @* B].	Proof. move/subsetP=> sAD /subsetP sBD. rewrite morphim_gen; last congr <<_>>. by apply/subsetP=> _ /imset2P[x y Ax By ->]; rewrite groupR; auto. apply/setP=> fz; apply/morphimP/imset2P=> [[z _] \| [fx fy]]. case/imset2P=> x y Ax By -> -> {z fz}. have Dx := sAD x Ax; have Dy := sBD y By. by exists (f x) (f y); r...	Lemma	morphimR	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "groupR", "imset2P", "imset2_f", "imset_f", "inE", "last", "morphR", "morphimP", "morphim_gen", "sAD", "setP", "subsetP" ]	commutator, normaliser, normal, center properties	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_norm A : f @* 'N(A) \subset 'N(f @* A).	Proof. apply/subsetP=> fx /morphimP[x Dx Nx -> {fx}]. by rewrite inE -morphimJ ?(normP Nx). Qed.	Lemma	morphim_norm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "inE", "morphimJ", "morphimP", "normP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_norms A B : A \subset 'N(B) -> f @* A \subset 'N(f @* B).	Proof. by move=> nBA; apply: subset_trans (morphim_norm B); apply: morphimS. Qed.	Lemma	morphim_norms	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "morphimS", "morphim_norm", "nBA", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_subnorm A B : f @* 'N_A(B) \subset 'N_(f @* A)(f @* B).	Proof. exact: subset_trans (morphimI A _) (setIS _ (morphim_norm B)). Qed.	Lemma	morphim_subnorm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphimI", "morphim_norm", "setIS", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_normal A B : A <\| B -> f @* A <\| f @* B.	Proof. by case/andP=> sAB nAB; rewrite /(_ <\| _) morphimS // morphim_norms. Qed.	Lemma	morphim_normal	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphimS", "morphim_norms" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_cent1 x : x \in D -> f @* 'C[x] \subset 'C[f x].	Proof. by move=> Dx; rewrite -(morphim_set1 Dx) morphim_norm. Qed.	Lemma	morphim_cent1	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "morphim_norm", "morphim_set1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_cent1s A x : x \in D -> A \subset 'C[x] -> f @* A \subset 'C[f x].	Proof. by move=> Dx cAx; apply: subset_trans (morphim_cent1 Dx); apply: morphimS. Qed.	Lemma	morphim_cent1s	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "morphimS", "morphim_cent1", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_subcent1 A x : x \in D -> f @* 'C_A[x] \subset 'C_(f @* A)[f x].	Proof. by move=> Dx; rewrite -(morphim_set1 Dx) morphim_subnorm. Qed.	Lemma	morphim_subcent1	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "morphim_set1", "morphim_subnorm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_cent A : f @* 'C(A) \subset 'C(f @* A).	Proof. apply/bigcapsP=> fx; case/morphimP=> x Dx Ax ->{fx}. by apply: subset_trans (morphim_cent1 Dx); apply: morphimS; apply: bigcap_inf. Qed.	Lemma	morphim_cent	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "bigcap_inf", "bigcapsP", "morphimP", "morphimS", "morphim_cent1", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_cents A B : A \subset 'C(B) -> f @* A \subset 'C(f @* B).	Proof. by move=> cBA; apply: subset_trans (morphim_cent B); apply: morphimS. Qed.	Lemma	morphim_cents	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "morphimS", "morphim_cent", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_subcent A B : f @* 'C_A(B) \subset 'C_(f @* A)(f @* B).	Proof. exact: subset_trans (morphimI A _) (setIS _ (morphim_cent B)). Qed.	Lemma	morphim_subcent	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphimI", "morphim_cent", "setIS", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_abelian A : abelian A -> abelian (f @* A).	Proof. exact: morphim_cents. Qed.	Lemma	morphim_abelian	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "abelian", "morphim_cents" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_norm R : f @^-1 'N(R) \subset 'N(f @^-1 R).	Proof. by apply/subsetP=> x /[!inE] /andP[Dx Nfx]; rewrite -morphpreJ ?morphpreS. Qed.	Lemma	morphpre_norm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "inE", "morphpreJ", "morphpreS", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_norms R S : R \subset 'N(S) -> f @^-1 R \subset 'N(f @^-1 S).	Proof. by move=> nSR; apply: subset_trans (morphpre_norm S); apply: morphpreS. Qed.	Lemma	morphpre_norms	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "morphpreS", "morphpre_norm", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_normal R S : R \subset f @* D -> S \subset f @* D -> (f @^-1 R <\| f @^-1 S) = (R <\| S).	Proof. move=> sRfD sSfD; apply/idP/andP=> [\|[sRS nSR]]. by move/morphim_normal; rewrite !morphpreK //; case/andP. by rewrite /(_ <\| _) (subset_trans _ (morphpre_norm _)) morphpreS. Qed.	Lemma	morphpre_normal	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "morphim_normal", "morphpreK", "morphpreS", "morphpre_norm", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_subnorm R S : f @^-1 'N_R(S) \subset 'N_(f @^-1 R)(f @*^-1 S).	Proof. by rewrite morphpreI setIS ?morphpre_norm. Qed.	Lemma	morphpre_subnorm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphpreI", "morphpre_norm", "setIS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_normG G : 'ker f \subset G -> G \subset D -> f @* 'N(G) = 'N_(f @* D)(f @* G).	Proof. move=> sKG sGD; apply/eqP; rewrite eqEsubset -{1}morphimIdom morphim_subnorm. rewrite -(morphpreK (subsetIl _ _)) morphimS //= morphpreI subIset // orbC. by rewrite -{2}(morphimGK sKG sGD) morphpre_norm. Qed.	Lemma	morphim_normG	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "eqEsubset", "ker", "morphimGK", "morphimIdom", "morphimS", "morphim_subnorm", "morphpreI", "morphpreK", "morphpre_norm", "sGD", "sKG", "subIset", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_subnormG A G : 'ker f \subset G -> G \subset D -> f @* 'N_A(G) = 'N_(f @* A)(f @* G).	Proof. move=> sKB sBD; rewrite morphimIG ?normsG // morphim_normG //. by rewrite setICA setIA morphimIim. Qed.	Lemma	morphim_subnormG	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "morphimIG", "morphimIim", "morphim_normG", "normsG", "setIA", "setICA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_cent1 x : x \in D -> 'C_D[x] \subset f @*^-1 'C[f x].	Proof. move=> Dx; rewrite -sub_morphim_pre ?subsetIl //. by apply: subset_trans (morphim_cent1 Dx); rewrite morphimS ?subsetIr. Qed.	Lemma	morphpre_cent1	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "morphimS", "morphim_cent1", "sub_morphim_pre", "subsetIl", "subsetIr", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_cent1s R x : x \in D -> R \subset f @* D -> f @*^-1 R \subset 'C[x] -> R \subset 'C[f x].	Proof. by move=> Dx sRfD; move/(morphim_cent1s Dx); rewrite morphpreK. Qed.	Lemma	morphpre_cent1s	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "morphim_cent1s", "morphpreK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_subcent1 R x : x \in D -> 'C_(f @^-1 R)[x] \subset f @^-1 'C_R[f x].	Proof. move=> Dx; rewrite -morphpreIdom -setIA setICA morphpreI setIS //. exact: morphpre_cent1. Qed.	Lemma	morphpre_subcent1	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "morphpreI", "morphpreIdom", "morphpre_cent1", "setIA", "setICA", "setIS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_cent A : 'C_D(A) \subset f @^-1 'C(f @ A).	Proof. rewrite -sub_morphim_pre ?subsetIl // morphimGI ?(subsetIl, subIset) // orbC. by rewrite (subset_trans (morphim_cent _)). Qed.	Lemma	morphpre_cent	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphimGI", "morphim_cent", "subIset", "sub_morphim_pre", "subsetIl", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_cents A R : R \subset f @* D -> f @^-1 R \subset 'C(A) -> R \subset 'C(f @ A).	Proof. by move=> sRfD; move/morphim_cents; rewrite morphpreK. Qed.	Lemma	morphpre_cents	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "morphim_cents", "morphpreK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphpre_subcent R A : 'C_(f @^-1 R)(A) \subset f @^-1 'C_R(f @* A).	Proof. by rewrite -morphpreIdom -setIA setICA morphpreI setIS //; apply: morphpre_cent. Qed.	Lemma	morphpre_subcent	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "morphpreI", "morphpreIdom", "morphpre_cent", "setIA", "setICA", "setIS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injmP : reflect {in D &, injective f} ('injm f).	Proof. apply: (iffP subsetP) => [injf x y Dx Dy \| injf x /= Kx]. by case/ker_rcoset=> // z /injf/set1P->; rewrite mul1g. have Dx := dom_ker Kx; apply/set1P/injf => //. by apply/rcoset_kerP; rewrite // mulg1. Qed.	Lemma	injmP	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "apply", "dom_ker", "injf", "ker_rcoset", "mul1g", "mulg1", "rcoset_kerP", "set1P", "subsetP" ]	local injectivity properties	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_im_injm : (#\|f @* D\| == #\|D\|) = 'injm f.	Proof. by rewrite morphimEdom (sameP imset_injP injmP). Qed.	Lemma	card_im_injm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "imset_injP", "injmP", "morphimEdom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_injm : 'ker f = 1.	Proof. exact/trivgP. Qed.	Lemma	ker_injm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker", "trivgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injmK A : A \subset D -> f @^-1 (f @ A) = A.	Proof. by move=> sAD; rewrite morphimK // ker_injm // mul1g. Qed.	Lemma	injmK	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker_injm", "morphimK", "mul1g", "sAD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_morphim_inj A B : A \subset D -> B \subset D -> f @* A = f @* B -> A = B.	Proof. by move=> sAD sBD eqAB; rewrite -(injmK sAD) eqAB injmK. Qed.	Lemma	injm_morphim_inj	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injmK", "sAD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_injm A : A \subset D -> #\|f @* A\| = #\|A\|.	Proof. move=> sAD; rewrite morphimEsub // card_in_imset //. exact: (sub_in2 (subsetP sAD) (injmP injf)). Qed.	Lemma	card_injm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "card_in_imset", "injf", "injmP", "morphimEsub", "sAD", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
order_injm x : x \in D -> #[f x] = #[x].	Proof. by move=> Dx; rewrite orderE -morphim_cycle // card_injm ?cycle_subG. Qed.	Lemma	order_injm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "card_injm", "cycle_subG", "morphim_cycle", "orderE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm1 x : x \in D -> f x = 1 -> x = 1.	Proof. by move=> Dx; move/(kerP Dx); rewrite ker_injm; move/set1P. Qed.	Lemma	injm1	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "kerP", "ker_injm", "set1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morph_injm_eq1 x : x \in D -> (f x == 1) = (x == 1).	Proof. by move=> Dx; rewrite -morph1 (inj_in_eq (injmP injf)) ?group1. Qed.	Lemma	morph_injm_eq1	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "Dx", "group1", "inj_in_eq", "injf", "injmP", "morph1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injmSK A B : A \subset D -> (f @* A \subset f @* B) = (A \subset B).	Proof. by move=> sAD; rewrite morphimSK // ker_injm mul1g. Qed.	Lemma	injmSK	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "ker_injm", "morphimSK", "mul1g", "sAD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_morphpre_injm R A : A \subset D -> R \subset f @* D -> (f @^-1 R \subset A) = (R \subset f @ A).	Proof. by move=> sAD sRfD; rewrite -morphpreSK ?injmK. Qed.	Lemma	sub_morphpre_injm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injmK", "morphpreSK", "sAD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_eq A B : A \subset D -> B \subset D -> (f @* A == f @* B) = (A == B).	Proof. by move=> sAD sBD; rewrite !eqEsubset !injmSK. Qed.	Lemma	injm_eq	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "eqEsubset", "injmSK", "sAD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_injm_eq1 A : A \subset D -> (f @* A == 1) = (A == 1).	Proof. by move=> sAD; rewrite -morphim1 injm_eq ?sub1G. Qed.	Lemma	morphim_injm_eq1	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injm_eq", "morphim1", "sAD", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injmI A B : f @* (A :&: B) = f @* A :&: f @* B.	Proof. rewrite -morphimIdom setIIr -4!(injmK (subsetIl D _), =^~ morphimIdom). by rewrite -morphpreI morphpreK // subIset ?morphim_sub. Qed.	Lemma	injmI	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injmK", "morphimIdom", "morphim_sub", "morphpreI", "morphpreK", "setIIr", "subIset", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injmD1 A : f @* A^# = (f @* A)^#.	Proof. by have:= morphimDG A injf; rewrite morphim1. Qed.	Lemma	injmD1	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injf", "morphim1", "morphimDG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nclasses_injm A : A \subset D -> #\|classes (f @* A)\| = #\|classes A\|.	Proof. move=> sAD; rewrite classes_morphim // card_in_imset //. move=> _ _ /imsetP[x Ax ->] /imsetP[y Ay ->]. by apply: injm_morphim_inj; rewrite // class_subG ?(subsetP sAD). Qed.	Lemma	nclasses_injm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "card_in_imset", "class_subG", "classes", "classes_morphim", "imsetP", "injm_morphim_inj", "sAD", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_norm A : A \subset D -> f @* 'N(A) = 'N_(f @* D)(f @* A).	Proof. move=> sAD; apply/eqP; rewrite -morphimIdom eqEsubset morphim_subnorm. rewrite -sub_morphpre_injm ?subsetIl // morphpreI injmK // setIS //. by rewrite -{2}(injmK sAD) morphpre_norm. Qed.	Lemma	injm_norm	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "apply", "eqEsubset", "injmK", "morphimIdom", "morphim_subnorm", "morphpreI", "morphpre_norm", "sAD", "setIS", "sub_morphpre_injm", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_norms A B : A \subset D -> B \subset D -> (f @* A \subset 'N(f @* B)) = (A \subset 'N(B)).	Proof. by move=> sAD sBD; rewrite -injmSK // injm_norm // subsetI morphimS. Qed.	Lemma	injm_norms	finite_group	finite_group/morphism.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "finfun", "bigop", "finset", "fingroup" ]	[ "injmSK", "injm_norm", "morphimS", "sAD", "subsetI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimMr A B : B \subset D -> f @* (A * B) = f @* A * f @* B.

Proof. move=> sBD; apply: invg_inj. by rewrite invMg -!morphimV invMg morphimMl // -invGid invSg. Qed.

Lemma

morphimMr

finite_group

finite_group/morphism.v

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

[ "apply", "invGid", "invMg", "invSg", "invg_inj", "morphimMl", "morphimV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpreMl R S : R \subset f @* D -> f @*^-1 (R * S) = f @*^-1 R * f @*^-1 S.

Proof. move=> sRfD; apply/setP=> x; rewrite !inE. apply/andP/imset2P=> [[Dx] | [y z]]; last first. rewrite !inE => /andP[Dy Rfy] /andP[Dz Rfz] ->. by rewrite ?(groupM, morphM, imset2_f). case/imset2P=> fy fz Rfy Rfz def_fx. have /morphimP[y Dy _ def_fy]: fy \in f @* D := subsetP sRfD fy Rfy. exists y (y^-1 * x); la...

Lemma

morphpreMl

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "groupM", "groupV", "imset2P", "imset2_f", "inE", "last", "morphM", "morphV", "morphimP", "mulKVg", "mulKg", "setP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimJ A x : x \in D -> f @* (A :^ x) = f @* A :^ f x.

Proof. move=> Dx; rewrite !conjsgE morphimMl ?(morphimMr, sub1set, groupV) //. by rewrite !(morphim_set1, groupV, morphV). Qed.

Lemma

morphimJ

finite_group

finite_group/morphism.v

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

[ "Dx", "conjsgE", "groupV", "morphV", "morphimMl", "morphimMr", "morphim_set1", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpreJ R x : x \in D -> f @*^-1 (R :^ f x) = f @*^-1 R :^ x.

Proof. move=> Dx; apply/setP=> y; rewrite conjIg !inE conjGid // !mem_conjg inE. by case Dy: (y \in D); rewrite // morphJ ?(morphV, groupV). Qed.

Lemma

morphpreJ

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "conjGid", "conjIg", "groupV", "inE", "mem_conjg", "morphJ", "morphV", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_class x A : x \in D -> A \subset D -> f @* (x ^: A) = f x ^: f @* A.

Proof. move=> Dx sAD; rewrite !morphimEsub ?class_subG // /class -!imset_comp. by apply: eq_in_imset => y Ay /=; rewrite morphJ // (subsetP sAD). Qed.

Lemma

morphim_class

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "class", "class_subG", "eq_in_imset", "imset_comp", "morphJ", "morphimEsub", "sAD", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

classes_morphim A : A \subset D -> classes (f @* A) = [set f @* xA | xA in classes A].

Proof. move=> sAD; rewrite morphimEsub // /classes -!imset_comp. apply: eq_in_imset => x /(subsetP sAD) Dx /=. by rewrite morphim_class ?morphimEsub. Qed.

Lemma

classes_morphim

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "classes", "eq_in_imset", "imset_comp", "morphimEsub", "morphim_class", "sAD", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimT : f @* setT = f @* D.

Proof. by rewrite -morphimIdom setIT. Qed.

Lemma

morphimT

finite_group

finite_group/morphism.v

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

[ "morphimIdom", "setIT", "setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimU A B : f @* (A :|: B) = f @* A :|: f @* B.

Proof. by rewrite -imsetU -setIUr. Qed.

Lemma

morphimU

finite_group

finite_group/morphism.v

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

[ "imsetU", "setIUr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimI A B : f @* (A :&: B) \subset f @* A :&: f @* B.

Proof. by rewrite subsetI // ?morphimS ?(subsetIl, subsetIr). Qed.

Lemma

morphimI

finite_group

finite_group/morphism.v

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

[ "morphimS", "subsetI", "subsetIl", "subsetIr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre0 : f @*^-1 set0 = set0.

Proof. by rewrite morphpreE preimset0 setI0. Qed.

Lemma

morphpre0

finite_group

finite_group/morphism.v

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

[ "morphpreE", "preimset0", "set0", "setI0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpreT : f @*^-1 setT = D.

Proof. by rewrite morphpreE preimsetT setIT. Qed.

Lemma

morphpreT

finite_group

finite_group/morphism.v

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

[ "morphpreE", "preimsetT", "setIT", "setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpreU R S : f @*^-1 (R :|: S) = f @*^-1 R :|: f @*^-1 S.

Proof. by rewrite -setIUr -preimsetU. Qed.

Lemma

morphpreU

finite_group

finite_group/morphism.v

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

[ "preimsetU", "setIUr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpreI R S : f @*^-1 (R :&: S) = f @*^-1 R :&: f @*^-1 S.

Proof. by rewrite -setIIr -preimsetI. Qed.

Lemma

morphpreI

finite_group

finite_group/morphism.v

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

[ "preimsetI", "setIIr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpreD R S : f @*^-1 (R :\: S) = f @*^-1 R :\: f @*^-1 S.

Proof. by apply/setP=> x /[!inE]; case: (x \in D). Qed.

Lemma

morphpreD

finite_group

finite_group/morphism.v

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

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

kerP x : x \in D -> reflect (f x = 1) (x \in 'ker f).

Proof. by move=> Dx; rewrite 2!inE Dx; apply: set1P. Qed.

Lemma

kerP

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "inE", "ker", "set1P" ]

kernel, domain properties

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dom_ker : {subset 'ker f <= D}.

Proof. by move=> x /morphpreP[]. Qed.

Lemma

dom_ker

finite_group

finite_group/morphism.v

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

[ "ker", "morphpreP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mker x : x \in 'ker f -> f x = 1.

Proof. by move=> Kx; apply/kerP=> //; apply: dom_ker. Qed.

Lemma

mker

finite_group

finite_group/morphism.v

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

[ "apply", "dom_ker", "ker", "kerP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mkerl x y : x \in 'ker f -> y \in D -> f (x * y) = f y.

Proof. by move=> Kx Dy; rewrite morphM // ?(dom_ker, mker Kx, mul1g). Qed.

Lemma

mkerl

finite_group

finite_group/morphism.v

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

[ "dom_ker", "ker", "mker", "morphM", "mul1g" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mkerr x y : x \in D -> y \in 'ker f -> f (x * y) = f x.

Proof. by move=> Dx Ky; rewrite morphM // ?(dom_ker, mker Ky, mulg1). Qed.

Lemma

mkerr

finite_group

finite_group/morphism.v

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

[ "Dx", "dom_ker", "ker", "mker", "morphM", "mulg1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcoset_kerP x y : x \in D -> y \in D -> reflect (f x = f y) (x \in 'ker f :* y).

Proof. move=> Dx Dy; rewrite mem_rcoset !inE groupM ?morphM ?groupV //=. by rewrite morphV // -eq_mulgV1; apply: eqP. Qed.

Lemma

rcoset_kerP

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "eq_mulgV1", "groupM", "groupV", "inE", "ker", "mem_rcoset", "morphM", "morphV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_rcoset x y : x \in D -> y \in D -> f x = f y -> exists2 z, z \in 'ker f & x = z * y.

Proof. by move=> Dx Dy eqfxy; apply/rcosetP; apply/rcoset_kerP. Qed.

Lemma

ker_rcoset

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "ker", "rcosetP", "rcoset_kerP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_norm : D \subset 'N('ker f).

Proof. apply/subsetP=> x Dx /[1!inE]; apply/subsetP=> _ /imsetP[y Ky ->]. by rewrite !inE groupJ ?morphJ // ?dom_ker //= mker ?conj1g. Qed.

Lemma

ker_norm

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "conj1g", "dom_ker", "groupJ", "imsetP", "inE", "ker", "mker", "morphJ", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_normal : 'ker f <| D.

Proof. by rewrite /(_ <| D) subsetIl ker_norm. Qed.

Lemma

ker_normal

finite_group

finite_group/morphism.v

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

[ "ker", "ker_norm", "subsetIl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimGI G A : 'ker f \subset G -> f @* (G :&: A) = f @* G :&: f @* A.

Proof. move=> sKG; apply/eqP; rewrite eqEsubset morphimI setIC. apply/subsetP=> _ /setIP[/morphimP[x Dx Ax ->] /morphimP[z Dz Gz]]. case/ker_rcoset=> {Dz}// y Ky def_x. have{z Gz y Ky def_x} Gx: x \in G by rewrite def_x groupMl // (subsetP sKG). by rewrite imset_f ?inE // Dx Gx Ax. Qed.

Lemma

morphimGI

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "eqEsubset", "groupMl", "imset_f", "inE", "ker", "ker_rcoset", "morphimI", "morphimP", "sKG", "setIC", "setIP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimIG A G : 'ker f \subset G -> f @* (A :&: G) = f @* A :&: f @* G.

Proof. by move=> sKG; rewrite setIC morphimGI // setIC. Qed.

Lemma

morphimIG

finite_group

finite_group/morphism.v

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

[ "ker", "morphimGI", "sKG", "setIC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimD A B : f @* A :\: f @* B \subset f @* (A :\: B).

Proof. rewrite subDset -morphimU morphimS //. by rewrite setDE setUIr setUCr setIT subsetUr. Qed.

Lemma

morphimD

finite_group

finite_group/morphism.v

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

[ "morphimS", "morphimU", "setDE", "setIT", "setUCr", "setUIr", "subDset", "subsetUr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimDG A G : 'ker f \subset G -> f @* (A :\: G) = f @* A :\: f @* G.

Proof. move=> sKG; apply/eqP; rewrite eqEsubset morphimD andbT !setDE subsetI. rewrite morphimS ?subsetIl // -[~: f @* G]setU0 -subDset setDE setCK. by rewrite -morphimIG //= setIAC -setIA setICr setI0 morphim0. Qed.

Lemma

morphimDG

finite_group

finite_group/morphism.v

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

[ "apply", "eqEsubset", "ker", "morphim0", "morphimD", "morphimIG", "morphimS", "sKG", "setCK", "setDE", "setI0", "setIA", "setIAC", "setICr", "setU0", "subDset", "subsetI", "subsetIl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimD1 A : (f @* A)^# \subset f @* A^#.

Proof. by rewrite -!set1gE -morphim1 morphimD. Qed.

Lemma

morphimD1

finite_group

finite_group/morphism.v

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

[ "morphim1", "morphimD", "set1gE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_groupset M : group_set (f @*^-1 M).

Proof. apply/group_setP; split=> [|x y]; rewrite !inE ?(morph1, group1) //. by case/andP=> Dx Mfx /andP[Dy Mfy]; rewrite morphM ?groupM. Qed.

Lemma

morphpre_groupset

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "group1", "groupM", "group_set", "group_setP", "inE", "morph1", "morphM", "split" ]

group structure preservation

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_groupset G : group_set (f @* G).

Proof. apply/group_setP; split=> [|_ _ /morphimP[x Dx Gx ->] /morphimP[y Dy Gy ->]]. by rewrite -morph1 imset_f ?group1. by rewrite -morphM ?imset_f ?inE ?groupM. Qed.

Lemma

morphim_groupset

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "group1", "groupM", "group_set", "group_setP", "imset_f", "inE", "morph1", "morphM", "morphimP", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_group fPh M

:= @group _ (morphpre fPh M) (morphpre_groupset M).

Canonical

morphpre_group

finite_group

finite_group/morphism.v

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

[ "group", "morphpre", "morphpre_groupset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_group fPh G

:= @group _ (morphim fPh G) (morphim_groupset G).

Canonical

morphim_group

finite_group

finite_group/morphism.v

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

[ "group", "morphim", "morphim_groupset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_group fPh : {group aT}

:= Eval hnf in [group of ker fPh].

Canonical

ker_group

finite_group

finite_group/morphism.v

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

[ "aT", "group", "ker" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morph_dom_groupset : group_set (f @: D).

Proof. by rewrite -morphimEdom groupP. Qed.

Lemma

morph_dom_groupset

finite_group

finite_group/morphism.v

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

[ "groupP", "group_set", "morphimEdom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morph_dom_group

:= group morph_dom_groupset.

Canonical

morph_dom_group

finite_group

finite_group/morphism.v

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

[ "group", "morph_dom_groupset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpreMr R S : S \subset f @* D -> f @*^-1 (R * S) = f @*^-1 R * f @*^-1 S.

Proof. move=> sSfD; apply: invg_inj. by rewrite invMg -!morphpreV invMg morphpreMl // -invSg invgK invGid. Qed.

Lemma

morphpreMr

finite_group

finite_group/morphism.v

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

[ "apply", "invGid", "invMg", "invSg", "invgK", "invg_inj", "morphpreMl", "morphpreV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimK A : A \subset D -> f @*^-1 (f @* A) = 'ker f * A.

Proof. move=> sAD; apply/setP=> x; rewrite !inE. apply/idP/idP=> [/andP[Dx /morphimP[y Dy Ay eqxy]] | /imset2P[z y Kz Ay ->{x}]]. rewrite -(mulgKV y x) mem_mulg // !inE !(groupM, morphM, groupV) //. by rewrite morphV //= eqxy mulgV. have [Dy Dz]: y \in D /\ z \in D by rewrite (subsetP sAD) // dom_ker. by rewrite gr...

Lemma

morphimK

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "dom_ker", "groupM", "groupV", "imset2P", "imset_f", "inE", "ker", "mem_mulg", "mker", "morphM", "morphV", "morphimP", "mul1g", "mulgKV", "mulgV", "sAD", "setP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimGK G : 'ker f \subset G -> G \subset D -> f @*^-1 (f @* G) = G.

Proof. by move=> sKG sGD; rewrite morphimK // mulSGid. Qed.

Lemma

morphimGK

finite_group

finite_group/morphism.v

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

[ "ker", "morphimK", "mulSGid", "sGD", "sKG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_set1 x : x \in D -> f @*^-1 [set f x] = 'ker f :* x.

Proof. by move=> Dx; rewrite -morphim_set1 // morphimK ?sub1set. Qed.

Lemma

morphpre_set1

finite_group

finite_group/morphism.v

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

[ "Dx", "ker", "morphimK", "morphim_set1", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpreK R : R \subset f @* D -> f @* (f @*^-1 R) = R.

Proof. move=> sRfD; apply/setP=> y; apply/morphimP/idP=> [[x _] | Ry]. by rewrite !inE; case/andP=> _ Rfx ->. have /morphimP[x Dx _ defy]: y \in f @* D := subsetP sRfD y Ry. by exists x; rewrite // !inE Dx -defy. Qed.

Lemma

morphpreK

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "inE", "morphimP", "setP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_ker : f @* 'ker f = 1.

Proof. by rewrite morphpreK ?sub1G. Qed.

Lemma

morphim_ker

finite_group

finite_group/morphism.v

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

[ "ker", "morphpreK", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_sub_pre M : 'ker f \subset f @*^-1 M.

Proof. by rewrite morphpreS ?sub1G. Qed.

Lemma

ker_sub_pre

finite_group

finite_group/morphism.v

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

[ "ker", "morphpreS", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_normal_pre M : 'ker f <| f @*^-1 M.

Proof. by rewrite /normal ker_sub_pre subIset ?ker_norm. Qed.

Lemma

ker_normal_pre

finite_group

finite_group/morphism.v

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

[ "ker", "ker_norm", "ker_sub_pre", "normal", "subIset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpreSK R S : R \subset f @* D -> (f @*^-1 R \subset f @*^-1 S) = (R \subset S).

Proof. move=> sRfD; apply/idP/idP=> [sf'RS|]; last exact: morphpreS. suffices: R \subset f @* D :&: S by rewrite subsetI sRfD. rewrite -(morphpreK sRfD) -[_ :&: S]morphpreK (morphimS, subsetIl) //. by rewrite morphpreI morphimGK ?subsetIl // setIA setIid. Qed.

Lemma

morphpreSK

finite_group

finite_group/morphism.v

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

[ "apply", "last", "morphimGK", "morphimS", "morphpreI", "morphpreK", "morphpreS", "setIA", "setIid", "subsetI", "subsetIl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_morphim_pre A R : A \subset D -> (f @* A \subset R) = (A \subset f @*^-1 R).

Proof. move=> sAD; rewrite -morphpreSK (morphimS, morphimK) //. apply/idP/idP; first by apply: subset_trans; apply: mulG_subr. by move/(mulgS ('ker f)); rewrite -morphpreMl ?(sub1G, mul1g). Qed.

Lemma

sub_morphim_pre

finite_group

finite_group/morphism.v

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

[ "apply", "ker", "morphimK", "morphimS", "morphpreMl", "morphpreSK", "mul1g", "mulG_subr", "mulgS", "sAD", "sub1G", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_proper R S : R \subset f @* D -> S \subset f @* D -> (f @*^-1 R \proper f @*^-1 S) = (R \proper S).

Proof. by move=> dQ dR; rewrite /proper !morphpreSK. Qed.

Lemma

morphpre_proper

finite_group

finite_group/morphism.v

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

[ "morphpreSK", "proper" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_morphpre_im R G : 'ker f \subset G -> G \subset D -> R \subset f @* D -> (f @*^-1 R \subset G) = (R \subset f @* G).

Proof. by symmetry; rewrite -morphpreSK ?morphimGK. Qed.

Lemma

sub_morphpre_im

finite_group

finite_group/morphism.v

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

[ "ker", "morphimGK", "morphpreSK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_trivg_morphim A : (A \subset 'ker f) = (A \subset D) && (f @* A \subset [1]).

Proof. case sAD: (A \subset D); first by rewrite sub_morphim_pre. by rewrite subsetI sAD. Qed.

Lemma

ker_trivg_morphim

finite_group

finite_group/morphism.v

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

[ "ker", "sAD", "sub_morphim_pre", "subsetI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. move=> sAD; transitivity (A \subset 'ker f * (D :&: B)). by rewrite -morphimK ?subsetIl // -sub_morphim_pre // /morphim setIA setIid. by rewrite setIC group_modl (subsetIl, subsetI) // andbC sAD. Qed.

Lemma

morphimSK

finite_group

finite_group/morphism.v

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

[ "group_modl", "ker", "morphim", "morphimK", "sAD", "setIA", "setIC", "setIid", "sub_morphim_pre", "subsetI", "subsetIl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimSGK A G : A \subset D -> 'ker f \subset G -> (f @* A \subset f @* G) = (A \subset G).

Proof. by move=> sGD skfK; rewrite morphimSK // mulSGid. Qed.

Lemma

morphimSGK

finite_group

finite_group/morphism.v

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

[ "ker", "morphimSK", "mulSGid", "sGD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltn_morphim A : [1] \proper 'ker_A f -> #|f @* A| < #|A|.

Proof. case/properP; rewrite sub1set => /setIP[A1 _] [x /setIP[Ax kx] x1]. rewrite (cardsD1 1 A) A1 ltnS -{1}(setD1K A1) morphimU morphim1. rewrite (setUidPr _) ?sub1set. by rewrite -(mker kx) mem_morphim ?(dom_ker kx) // inE x1. by rewrite (leq_trans (leq_imset_card _ _)) ?subset_leq_card ?subsetIr. Qed.

Lemma

ltn_morphim

finite_group

finite_group/morphism.v

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

[ "cardsD1", "dom_ker", "inE", "leq_imset_card", "leq_trans", "ltnS", "mem_morphim", "mker", "morphim1", "morphimU", "proper", "properP", "setD1K", "setIP", "setUidPr", "sub1set", "subsetIr", "subset_leq_card" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_inj : {in [pred R : {set rT} | R \subset f @* D] &, injective (fun R => f @*^-1 R)}.

Proof. exact: can_in_inj morphpreK. Qed.

Lemma

morphpre_inj

finite_group

finite_group/morphism.v

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

[ "morphpreK" ]

injectivity of image and preimage

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_injG : {in [pred G : {group aT} | 'ker f \subset G & G \subset D] &, injective (fun G => f @* G)}.

Proof. move=> G H /andP[sKG sGD] /andP[sKH sHD] eqfGH. by apply: val_inj; rewrite /= -(morphimGK sKG sGD) eqfGH morphimGK. Qed.

Lemma

morphim_injG

finite_group

finite_group/morphism.v

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

[ "aT", "apply", "group", "ker", "morphimGK", "sGD", "sHD", "sKG", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_inj G H : ('ker f \subset G) && (G \subset D) -> ('ker f \subset H) && (H \subset D) -> f @* G = f @* H -> G :=: H.

Proof. by move=> nsGf nsHf /morphim_injG->. Qed.

Lemma

morphim_inj

finite_group

finite_group/morphism.v

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

[ "ker", "morphim_injG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_gen A : A \subset D -> f @* <<A>> = <<f @* A>>.

Proof. move=> sAD; apply/eqP. rewrite eqEsubset andbC gen_subG morphimS; first exact: subset_gen. by rewrite sub_morphim_pre gen_subG // -sub_morphim_pre // subset_gen. Qed.

Lemma

morphim_gen

finite_group

finite_group/morphism.v

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

[ "apply", "eqEsubset", "gen_subG", "morphimS", "sAD", "sub_morphim_pre", "subset_gen" ]

commutation with generated groups and cycles

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_cycle x : x \in D -> f @* <[x]> = <[f x]>.

Proof. by move=> Dx; rewrite morphim_gen (sub1set, morphim_set1). Qed.

Lemma

morphim_cycle

finite_group

finite_group/morphism.v

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

[ "Dx", "morphim_gen", "morphim_set1", "sub1set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by move=> sAD sBD; rewrite morphim_gen ?morphimU // subUset sAD. Qed.

Lemma

morphimY

finite_group

finite_group/morphism.v

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

[ "morphimU", "morphim_gen", "sAD", "subUset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_gen R : 1 \in R -> R \subset f @* D -> f @*^-1 <<R>> = <<f @*^-1 R>>.

Proof. move=> R1 sRfD; apply/eqP. rewrite eqEsubset andbC gen_subG morphpreS; first exact: subset_gen. rewrite -{1}(morphpreK sRfD) -morphim_gen ?subsetIl // morphimGK //=. by rewrite sub_gen // setIS // preimsetS ?sub1set. by rewrite gen_subG subsetIl. Qed.

Lemma

morphpre_gen

finite_group

finite_group/morphism.v

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

[ "R1", "apply", "eqEsubset", "gen_subG", "morphimGK", "morphim_gen", "morphpreK", "morphpreS", "preimsetS", "setIS", "sub1set", "sub_gen", "subsetIl", "subset_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphimR A B : A \subset D -> B \subset D -> f @* [~: A, B] = [~: f @* A, f @* B].

Proof. move/subsetP=> sAD /subsetP sBD. rewrite morphim_gen; last congr <<_>>. by apply/subsetP=> _ /imset2P[x y Ax By ->]; rewrite groupR; auto. apply/setP=> fz; apply/morphimP/imset2P=> [[z _] | [fx fy]]. case/imset2P=> x y Ax By -> -> {z fz}. have Dx := sAD x Ax; have Dy := sBD y By. by exists (f x) (f y); r...

Lemma

morphimR

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "groupR", "imset2P", "imset2_f", "imset_f", "inE", "last", "morphR", "morphimP", "morphim_gen", "sAD", "setP", "subsetP" ]

commutator, normaliser, normal, center properties

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_norm A : f @* 'N(A) \subset 'N(f @* A).

Proof. apply/subsetP=> fx /morphimP[x Dx Nx -> {fx}]. by rewrite inE -morphimJ ?(normP Nx). Qed.

Lemma

morphim_norm

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "inE", "morphimJ", "morphimP", "normP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_norms A B : A \subset 'N(B) -> f @* A \subset 'N(f @* B).

Proof. by move=> nBA; apply: subset_trans (morphim_norm B); apply: morphimS. Qed.

Lemma

morphim_norms

finite_group

finite_group/morphism.v

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

[ "apply", "morphimS", "morphim_norm", "nBA", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_subnorm A B : f @* 'N_A(B) \subset 'N_(f @* A)(f @* B).

Proof. exact: subset_trans (morphimI A _) (setIS _ (morphim_norm B)). Qed.

Lemma

morphim_subnorm

finite_group

finite_group/morphism.v

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

[ "morphimI", "morphim_norm", "setIS", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_normal A B : A <| B -> f @* A <| f @* B.

Proof. by case/andP=> sAB nAB; rewrite /(_ <| _) morphimS // morphim_norms. Qed.

Lemma

morphim_normal

finite_group

finite_group/morphism.v

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

[ "morphimS", "morphim_norms" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_cent1 x : x \in D -> f @* 'C[x] \subset 'C[f x].

Proof. by move=> Dx; rewrite -(morphim_set1 Dx) morphim_norm. Qed.

Lemma

morphim_cent1

finite_group

finite_group/morphism.v

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

[ "Dx", "morphim_norm", "morphim_set1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_cent1s A x : x \in D -> A \subset 'C[x] -> f @* A \subset 'C[f x].

Proof. by move=> Dx cAx; apply: subset_trans (morphim_cent1 Dx); apply: morphimS. Qed.

Lemma

morphim_cent1s

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "morphimS", "morphim_cent1", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_subcent1 A x : x \in D -> f @* 'C_A[x] \subset 'C_(f @* A)[f x].

Proof. by move=> Dx; rewrite -(morphim_set1 Dx) morphim_subnorm. Qed.

Lemma

morphim_subcent1

finite_group

finite_group/morphism.v

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

[ "Dx", "morphim_set1", "morphim_subnorm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_cent A : f @* 'C(A) \subset 'C(f @* A).

Proof. apply/bigcapsP=> fx; case/morphimP=> x Dx Ax ->{fx}. by apply: subset_trans (morphim_cent1 Dx); apply: morphimS; apply: bigcap_inf. Qed.

Lemma

morphim_cent

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "bigcap_inf", "bigcapsP", "morphimP", "morphimS", "morphim_cent1", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_cents A B : A \subset 'C(B) -> f @* A \subset 'C(f @* B).

Proof. by move=> cBA; apply: subset_trans (morphim_cent B); apply: morphimS. Qed.

Lemma

morphim_cents

finite_group

finite_group/morphism.v

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

[ "apply", "morphimS", "morphim_cent", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_subcent A B : f @* 'C_A(B) \subset 'C_(f @* A)(f @* B).

Proof. exact: subset_trans (morphimI A _) (setIS _ (morphim_cent B)). Qed.

Lemma

morphim_subcent

finite_group

finite_group/morphism.v

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

[ "morphimI", "morphim_cent", "setIS", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_abelian A : abelian A -> abelian (f @* A).

Proof. exact: morphim_cents. Qed.

Lemma

morphim_abelian

finite_group

finite_group/morphism.v

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

[ "abelian", "morphim_cents" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_norm R : f @*^-1 'N(R) \subset 'N(f @*^-1 R).

Proof. by apply/subsetP=> x /[!inE] /andP[Dx Nfx]; rewrite -morphpreJ ?morphpreS. Qed.

Lemma

morphpre_norm

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "inE", "morphpreJ", "morphpreS", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_norms R S : R \subset 'N(S) -> f @*^-1 R \subset 'N(f @*^-1 S).

Proof. by move=> nSR; apply: subset_trans (morphpre_norm S); apply: morphpreS. Qed.

Lemma

morphpre_norms

finite_group

finite_group/morphism.v

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

[ "apply", "morphpreS", "morphpre_norm", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_normal R S : R \subset f @* D -> S \subset f @* D -> (f @*^-1 R <| f @*^-1 S) = (R <| S).

Proof. move=> sRfD sSfD; apply/idP/andP=> [|[sRS nSR]]. by move/morphim_normal; rewrite !morphpreK //; case/andP. by rewrite /(_ <| _) (subset_trans _ (morphpre_norm _)) morphpreS. Qed.

Lemma

morphpre_normal

finite_group

finite_group/morphism.v

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

[ "apply", "morphim_normal", "morphpreK", "morphpreS", "morphpre_norm", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_subnorm R S : f @*^-1 'N_R(S) \subset 'N_(f @*^-1 R)(f @*^-1 S).

Proof. by rewrite morphpreI setIS ?morphpre_norm. Qed.

Lemma

morphpre_subnorm

finite_group

finite_group/morphism.v

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

[ "morphpreI", "morphpre_norm", "setIS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_normG G : 'ker f \subset G -> G \subset D -> f @* 'N(G) = 'N_(f @* D)(f @* G).

Proof. move=> sKG sGD; apply/eqP; rewrite eqEsubset -{1}morphimIdom morphim_subnorm. rewrite -(morphpreK (subsetIl _ _)) morphimS //= morphpreI subIset // orbC. by rewrite -{2}(morphimGK sKG sGD) morphpre_norm. Qed.

Lemma

morphim_normG

finite_group

finite_group/morphism.v

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

[ "apply", "eqEsubset", "ker", "morphimGK", "morphimIdom", "morphimS", "morphim_subnorm", "morphpreI", "morphpreK", "morphpre_norm", "sGD", "sKG", "subIset", "subsetIl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_subnormG A G : 'ker f \subset G -> G \subset D -> f @* 'N_A(G) = 'N_(f @* A)(f @* G).

Proof. move=> sKB sBD; rewrite morphimIG ?normsG // morphim_normG //. by rewrite setICA setIA morphimIim. Qed.

Lemma

morphim_subnormG

finite_group

finite_group/morphism.v

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

[ "ker", "morphimIG", "morphimIim", "morphim_normG", "normsG", "setIA", "setICA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_cent1 x : x \in D -> 'C_D[x] \subset f @*^-1 'C[f x].

Proof. move=> Dx; rewrite -sub_morphim_pre ?subsetIl //. by apply: subset_trans (morphim_cent1 Dx); rewrite morphimS ?subsetIr. Qed.

Lemma

morphpre_cent1

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "morphimS", "morphim_cent1", "sub_morphim_pre", "subsetIl", "subsetIr", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_cent1s R x : x \in D -> R \subset f @* D -> f @*^-1 R \subset 'C[x] -> R \subset 'C[f x].

Proof. by move=> Dx sRfD; move/(morphim_cent1s Dx); rewrite morphpreK. Qed.

Lemma

morphpre_cent1s

finite_group

finite_group/morphism.v

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

[ "Dx", "morphim_cent1s", "morphpreK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_subcent1 R x : x \in D -> 'C_(f @*^-1 R)[x] \subset f @*^-1 'C_R[f x].

Proof. move=> Dx; rewrite -morphpreIdom -setIA setICA morphpreI setIS //. exact: morphpre_cent1. Qed.

Lemma

morphpre_subcent1

finite_group

finite_group/morphism.v

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

[ "Dx", "morphpreI", "morphpreIdom", "morphpre_cent1", "setIA", "setICA", "setIS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_cent A : 'C_D(A) \subset f @*^-1 'C(f @* A).

Proof. rewrite -sub_morphim_pre ?subsetIl // morphimGI ?(subsetIl, subIset) // orbC. by rewrite (subset_trans (morphim_cent _)). Qed.

Lemma

morphpre_cent

finite_group

finite_group/morphism.v

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

[ "morphimGI", "morphim_cent", "subIset", "sub_morphim_pre", "subsetIl", "subset_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_cents A R : R \subset f @* D -> f @*^-1 R \subset 'C(A) -> R \subset 'C(f @* A).

Proof. by move=> sRfD; move/morphim_cents; rewrite morphpreK. Qed.

Lemma

morphpre_cents

finite_group

finite_group/morphism.v

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

[ "morphim_cents", "morphpreK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphpre_subcent R A : 'C_(f @*^-1 R)(A) \subset f @*^-1 'C_R(f @* A).

Proof. by rewrite -morphpreIdom -setIA setICA morphpreI setIS //; apply: morphpre_cent. Qed.

Lemma

morphpre_subcent

finite_group

finite_group/morphism.v

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

[ "apply", "morphpreI", "morphpreIdom", "morphpre_cent", "setIA", "setICA", "setIS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injmP : reflect {in D &, injective f} ('injm f).

Proof. apply: (iffP subsetP) => [injf x y Dx Dy | injf x /= Kx]. by case/ker_rcoset=> // z /injf/set1P->; rewrite mul1g. have Dx := dom_ker Kx; apply/set1P/injf => //. by apply/rcoset_kerP; rewrite // mulg1. Qed.

Lemma

injmP

finite_group

finite_group/morphism.v

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

[ "Dx", "apply", "dom_ker", "injf", "ker_rcoset", "mul1g", "mulg1", "rcoset_kerP", "set1P", "subsetP" ]

local injectivity properties

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_im_injm : (#|f @* D| == #|D|) = 'injm f.

Proof. by rewrite morphimEdom (sameP imset_injP injmP). Qed.

Lemma

card_im_injm

finite_group

finite_group/morphism.v

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

[ "imset_injP", "injmP", "morphimEdom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_injm : 'ker f = 1.

Proof. exact/trivgP. Qed.

Lemma

ker_injm

finite_group

finite_group/morphism.v

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

[ "ker", "trivgP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injmK A : A \subset D -> f @*^-1 (f @* A) = A.

Proof. by move=> sAD; rewrite morphimK // ker_injm // mul1g. Qed.

Lemma

injmK

finite_group

finite_group/morphism.v

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

[ "ker_injm", "morphimK", "mul1g", "sAD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_morphim_inj A B : A \subset D -> B \subset D -> f @* A = f @* B -> A = B.

Proof. by move=> sAD sBD eqAB; rewrite -(injmK sAD) eqAB injmK. Qed.

Lemma

injm_morphim_inj

finite_group

finite_group/morphism.v

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

[ "injmK", "sAD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_injm A : A \subset D -> #|f @* A| = #|A|.

Proof. move=> sAD; rewrite morphimEsub // card_in_imset //. exact: (sub_in2 (subsetP sAD) (injmP injf)). Qed.

Lemma

card_injm

finite_group

finite_group/morphism.v

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

[ "card_in_imset", "injf", "injmP", "morphimEsub", "sAD", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

order_injm x : x \in D -> #[f x] = #[x].

Proof. by move=> Dx; rewrite orderE -morphim_cycle // card_injm ?cycle_subG. Qed.

Lemma

order_injm

finite_group

finite_group/morphism.v

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

[ "Dx", "card_injm", "cycle_subG", "morphim_cycle", "orderE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm1 x : x \in D -> f x = 1 -> x = 1.

Proof. by move=> Dx; move/(kerP Dx); rewrite ker_injm; move/set1P. Qed.

Lemma

injm1

finite_group

finite_group/morphism.v

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

[ "Dx", "kerP", "ker_injm", "set1P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morph_injm_eq1 x : x \in D -> (f x == 1) = (x == 1).

Proof. by move=> Dx; rewrite -morph1 (inj_in_eq (injmP injf)) ?group1. Qed.

Lemma

morph_injm_eq1

finite_group

finite_group/morphism.v

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

[ "Dx", "group1", "inj_in_eq", "injf", "injmP", "morph1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injmSK A B : A \subset D -> (f @* A \subset f @* B) = (A \subset B).

Proof. by move=> sAD; rewrite morphimSK // ker_injm mul1g. Qed.

Lemma

injmSK

finite_group

finite_group/morphism.v

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

[ "ker_injm", "morphimSK", "mul1g", "sAD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_morphpre_injm R A : A \subset D -> R \subset f @* D -> (f @*^-1 R \subset A) = (R \subset f @* A).

Proof. by move=> sAD sRfD; rewrite -morphpreSK ?injmK. Qed.

Lemma

sub_morphpre_injm

finite_group

finite_group/morphism.v

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

[ "injmK", "morphpreSK", "sAD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_eq A B : A \subset D -> B \subset D -> (f @* A == f @* B) = (A == B).

Proof. by move=> sAD sBD; rewrite !eqEsubset !injmSK. Qed.

Lemma

injm_eq

finite_group

finite_group/morphism.v

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

[ "eqEsubset", "injmSK", "sAD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_injm_eq1 A : A \subset D -> (f @* A == 1) = (A == 1).

Proof. by move=> sAD; rewrite -morphim1 injm_eq ?sub1G. Qed.

Lemma

morphim_injm_eq1

finite_group

finite_group/morphism.v

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

[ "injm_eq", "morphim1", "sAD", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injmI A B : f @* (A :&: B) = f @* A :&: f @* B.

Proof. rewrite -morphimIdom setIIr -4!(injmK (subsetIl D _), =^~ morphimIdom). by rewrite -morphpreI morphpreK // subIset ?morphim_sub. Qed.

Lemma

injmI

finite_group

finite_group/morphism.v

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

[ "injmK", "morphimIdom", "morphim_sub", "morphpreI", "morphpreK", "setIIr", "subIset", "subsetIl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injmD1 A : f @* A^# = (f @* A)^#.

Proof. by have:= morphimDG A injf; rewrite morphim1. Qed.

Lemma

injmD1

finite_group

finite_group/morphism.v

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

[ "injf", "morphim1", "morphimDG" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nclasses_injm A : A \subset D -> #|classes (f @* A)| = #|classes A|.

Proof. move=> sAD; rewrite classes_morphim // card_in_imset //. move=> _ _ /imsetP[x Ax ->] /imsetP[y Ay ->]. by apply: injm_morphim_inj; rewrite // class_subG ?(subsetP sAD). Qed.

Lemma

nclasses_injm

finite_group

finite_group/morphism.v

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

[ "apply", "card_in_imset", "class_subG", "classes", "classes_morphim", "imsetP", "injm_morphim_inj", "sAD", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_norm A : A \subset D -> f @* 'N(A) = 'N_(f @* D)(f @* A).

Proof. move=> sAD; apply/eqP; rewrite -morphimIdom eqEsubset morphim_subnorm. rewrite -sub_morphpre_injm ?subsetIl // morphpreI injmK // setIS //. by rewrite -{2}(injmK sAD) morphpre_norm. Qed.

Lemma

injm_norm

finite_group

finite_group/morphism.v

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

[ "apply", "eqEsubset", "injmK", "morphimIdom", "morphim_subnorm", "morphpreI", "morphpre_norm", "sAD", "setIS", "sub_morphpre_injm", "subsetIl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

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

Lemma

injm_norms

finite_group

finite_group/morphism.v

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

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

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d