fact
stringlengths 8
1.54k
| type
stringclasses 19
values | library
stringclasses 8
values | imports
listlengths 1
10
| filename
stringclasses 98
values | symbolic_name
stringlengths 1
42
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
ker_dprodm: 'ker dprodm = [set a * b^-1 | a in H, b in K & fH a == fK b].
Proof. exact: ker_cprodm. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
ker_dprodm
| |
injm_dprodm:
'injm dprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == 1].
Proof.
rewrite injm_cprodm -(morphimIdom fH K).
by case/dprodP: eqHK_G => _ _ _ ->; rewrite morphim1.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
injm_dprodm
| |
isog_dprodA B G C D L :
A \x B = G -> C \x D = L -> isog A C -> isog B D -> isog G L.
Proof.
move=> defG {C D} /dprodP[[C D -> ->] defL cCD trCD].
case/dprodP: defG (defG) => {A B} [[A B -> ->] defG _ _] dG defC defD.
case/isogP: defC defL cCD trCD => fA injfA <-{C}.
case/isogP: defD => fB injfB <-{D} defL cCD trCD.
apply/isogP; exists (dprodm_morphism dG cCD).
by rewrite injm_dprodm injfA injfB trCD eqxx.
by rewrite /= -{2}defG morphim_dprodm.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
isog_dprod
| |
xsdprodm_dom1: DgH \subset 'dom fsH.
Proof. by rewrite ['dom _]morphpre_invm. Qed.
Local Notation gH := (restrm xsdprodm_dom1 fsH).
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
xsdprodm_dom1
| |
xsdprodm_dom2: DgK \subset 'dom fsK.
Proof. by rewrite ['dom _]morphpre_invm. Qed.
Local Notation gK := (restrm xsdprodm_dom2 fsK).
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
xsdprodm_dom2
| |
im_sdprodm1: gH @* DgH = fH @* H.
Proof. by rewrite morphim_restrm setIid morphim_comp im_invm. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
im_sdprodm1
| |
im_sdprodm2: gK @* DgK = fK @* K.
Proof. by rewrite morphim_restrm setIid morphim_comp im_invm. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
im_sdprodm2
| |
xsdprodm_act: {in DgH & DgK, morph_act 'J 'J gH gK}.
Proof.
move=> fh fk; case/morphimP=> h _ Hh ->{fh}; case/morphimP=> k _ Kk ->{fk}.
by rewrite /= -sdpair_act // /restrm /= !invmE ?actf ?gact_stable.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
xsdprodm_act
| |
xsdprodm:= sdprodm (sdprod_sdpair to) xsdprodm_act.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
xsdprodm
| |
xsdprod_morphism:= [morphism of xsdprodm].
|
Canonical
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
xsdprod_morphism
| |
im_xsdprodm: xsdprodm @* setT = fH @* H * fK @* K.
Proof. by rewrite -im_sdpair morphim_sdprodm // im_sdprodm1 im_sdprodm2. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
im_xsdprodm
| |
injm_xsdprodm:
'injm xsdprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == 1].
Proof.
rewrite injm_sdprodm im_sdprodm1 im_sdprodm2 !subG1 /=.
rewrite (ker_restrm xsdprodm_dom1) (ker_restrm xsdprodm_dom2) /= !ker_comp.
rewrite !morphpre_invm !morphimIim.
by rewrite !morphim_injm_eq1 ?subsetIl ?injm_sdpair1 ?injm_sdpair2.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
injm_xsdprodm
| |
mulgm: gT * gT -> _ := uncurry mul.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
mulgm
| |
imset_mulgm(A B : {set gT}) : mulgm @: setX A B = A * B.
Proof. by rewrite -curry_imset2X. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
imset_mulgm
| |
mulgmPH1 H2 G : reflect (H1 \x H2 = G) (misom (setX H1 H2) G mulgm).
Proof.
apply: (iffP misomP) => [[pM /isomP[injf /= <-]] | ].
have /dprodP[_ /= defX cH12] := setX_dprod H1 H2.
rewrite -{4}defX {}defX => /(congr1 (fun A => morphm pM @* A)).
move/(morphimS (morphm_morphism pM)): cH12 => /=.
have sH1H: setX H1 1 \subset setX H1 H2 by rewrite setXS ?sub1G.
have sH2H: setX 1 H2 \subset setX H1 H2 by rewrite setXS ?sub1G.
rewrite morphim1 injm_cent ?injmI //= subsetI => /andP[_].
by rewrite !morphimEsub //= !imset_mulgm mulg1 mul1g; apply: dprodE.
case/dprodP=> _ defG cH12 trH12.
have fM: morphic (setX H1 H2) mulgm.
apply/morphicP=> [[x1 x2] [y1 y2] /setXP[_ Hx2] /setXP[Hy1 _]].
by rewrite /= mulgA -(mulgA x1) -(centsP cH12 x2 _ y1) ?mulgA//.
exists fM; apply/isomP; split; last by rewrite morphimEsub //= imset_mulgm.
apply/subsetP=> [[x1 x2]]; rewrite !inE /= andbC -eq_invg_mul.
case: eqP => //= <-; rewrite groupV -in_setI trH12 => /set1P->.
by rewrite invg1 eqxx.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype bigop finset fingroup morphism",
"From mathcomp Require Import quotient action finfun"
] |
fingroup/gproduct.v
|
mulgmP
| |
morphism(D : {set aT}) : Type := Morphism {
mfun :> aT -> FinGroup.sort rT;
_ : {in D &, {morph mfun : x y / x * y}}
}.
|
Structure
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphism
| |
morphism_forD of phant rT := morphism D.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphism_for
| |
clone_morphismD f :=
let: Morphism _ fM := f
return {type of @Morphism D for f} -> morphism_for D (Phant rT)
in fun k => k fM.
Variables (D A : {set aT}) (R : {set rT}) (x : aT) (y : rT) (f : aT -> rT).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
clone_morphism
| |
morphim_spec: Prop := MorphimSpec z & z \in D & z \in A & y = f z.
|
Variant
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_spec
| |
morphimP: reflect morphim_spec (y \in f @: (D :&: A)).
Proof.
apply: (iffP imsetP) => [] [z]; first by case/setIP; exists z.
by exists z; first apply/setIP.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimP
| |
morphpreP: reflect (x \in D /\ f x \in R) (x \in D :&: f @^-1: R).
Proof. by rewrite !inE; apply: andP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreP
| |
morphM: {in D &, {morph f : x y / x * y}}.
Proof. by case f. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphM
| |
morPhantom:= (phantom (aT -> rT)).
|
Notation
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morPhantom
| |
MorPhantom:= Phantom (aT -> rT).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
MorPhantom
| |
domof morPhantom f := D.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
dom
| |
morphimof morPhantom f := fun A => f @: (D :&: A).
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim
| |
morphpreof morPhantom f := fun R : {set rT} => D :&: f @^-1: R.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpre
| |
kermph := morphpre mph 1.
|
Definition
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ker
| |
morph1: f 1 = 1.
Proof. by apply: (mulgI (f 1)); rewrite -morphM ?mulg1. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morph1
| |
morph_prodI r (P : pred I) F :
(forall i, P i -> F i \in D) ->
f (\prod_(i <- r | P i) F i) = \prod_( i <- r | P i) f (F i).
Proof.
move=> D_F; elim/(big_load (fun x => x \in D)): _.
elim/big_rec2: _ => [|i _ x Pi [Dx <-]]; first by rewrite morph1.
by rewrite groupM ?morphM // D_F.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morph_prod
| |
morphV: {in D, {morph f : x / x^-1}}.
Proof.
move=> x Dx; apply: (mulgI (f x)).
by rewrite -morphM ?groupV // !mulgV morph1.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphV
| |
morphJ: {in D &, {morph f : x y / x ^ y}}.
Proof. by move=> * /=; rewrite !morphM ?morphV // ?groupM ?groupV. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphJ
| |
morphXn : {in D, {morph f : x / x ^+ n}}.
Proof.
by elim: n => [|n IHn] x Dx; rewrite ?morph1 // !expgS morphM ?(groupX, IHn).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphX
| |
morphR: {in D &, {morph f : x y / [~ x, y]}}.
Proof. by move=> * /=; rewrite morphM ?(groupV, groupJ) // morphJ ?morphV. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphR
| |
morphimEA : f @* A = f @: (D :&: A). Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimE
| |
morphpreER : f @*^-1 R = D :&: f @^-1: R. Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreE
| |
kerE: 'ker f = f @*^-1 1. Proof. by []. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
kerE
| |
morphimEsubA : A \subset D -> f @* A = f @: A.
Proof. by move=> sAD; rewrite /morphim (setIidPr sAD). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimEsub
| |
morphimEdom: f @* D = f @: D.
Proof. exact: morphimEsub. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimEdom
| |
morphimIdomA : f @* (D :&: A) = f @* A.
Proof. by rewrite /morphim setIA setIid. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimIdom
| |
morphpreIdomR : D :&: f @*^-1 R = f @*^-1 R.
Proof. by rewrite /morphim setIA setIid. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreIdom
| |
morphpreIimR : f @*^-1 (f @* D :&: R) = f @*^-1 R.
Proof.
apply/setP=> x; rewrite morphimEdom !inE.
by case Dx: (x \in D); rewrite // imset_f.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreIim
| |
morphimIimA : f @* D :&: f @* A = f @* A.
Proof. by apply/setIidPr; rewrite imsetS // setIid subsetIl. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimIim
| |
mem_morphimA x : x \in D -> x \in A -> f x \in f @* A.
Proof. by move=> Dx Ax; apply/morphimP; exists x. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
mem_morphim
| |
mem_morphpreR x : x \in D -> f x \in R -> x \in f @*^-1 R.
Proof. by move=> Dx Rfx; apply/morphpreP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
mem_morphpre
| |
morphimSA B : A \subset B -> f @* A \subset f @* B.
Proof. by move=> sAB; rewrite imsetS ?setIS. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimS
| |
morphim_subA : f @* A \subset f @* D.
Proof. by rewrite imsetS // setIid subsetIl. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_sub
| |
leq_morphimA : #|f @* A| <= #|A|.
Proof.
by apply: (leq_trans (leq_imset_card _ _)); rewrite subset_leq_card ?subsetIr.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
leq_morphim
| |
morphpreSR S : R \subset S -> f @*^-1 R \subset f @*^-1 S.
Proof. by move=> sRS; rewrite setIS ?preimsetS. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreS
| |
morphpre_subR : f @*^-1 R \subset D.
Proof. exact: subsetIl. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpre_sub
| |
morphim_setIpreA R : f @* (A :&: f @*^-1 R) = f @* A :&: R.
Proof.
apply/setP=> fa; apply/morphimP/setIP=> [[a Da] | [/morphimP[a Da Aa ->] Rfa]].
by rewrite !inE Da /= => /andP[Aa Rfa] ->; rewrite mem_morphim.
by exists a; rewrite // !inE Aa Da.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_setIpre
| |
morphim0: f @* set0 = set0.
Proof. by rewrite morphimE setI0 imset0. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim0
| |
morphim_eq0A : A \subset D -> (f @* A == set0) = (A == set0).
Proof. by rewrite imset_eq0 => /setIidPr->. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_eq0
| |
morphim_set1x : x \in D -> f @* [set x] = [set f x].
Proof. by rewrite /morphim -sub1set => /setIidPr->; apply: imset_set1. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_set1
| |
morphim1: f @* 1 = 1.
Proof. by rewrite morphim_set1 ?morph1. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim1
| |
morphimVA : f @* A^-1 = (f @* A)^-1.
Proof.
wlog suffices: A / f @* A^-1 \subset (f @* A)^-1.
by move=> IH; apply/eqP; rewrite eqEsubset IH -invSg invgK -{1}(invgK A) IH.
apply/subsetP=> _ /morphimP[x Dx Ax' ->]; rewrite !inE in Ax' *.
by rewrite -morphV // imset_f // inE groupV Dx.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimV
| |
morphpreVR : f @*^-1 R^-1 = (f @*^-1 R)^-1.
Proof.
apply/setP=> x; rewrite !inE groupV; case Dx: (x \in D) => //=.
by rewrite morphV.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreV
| |
morphimMlA B : A \subset D -> f @* (A * B) = f @* A * f @* B.
Proof.
move=> sAD; rewrite /morphim setIC -group_modl // (setIidPr sAD).
apply/setP=> fxy; apply/idP/idP.
case/imsetP=> _ /imset2P[x y Ax /setIP[Dy By] ->] ->{fxy}.
by rewrite morphM // (subsetP sAD, imset2_f) // imset_f // inE By.
case/imset2P=> _ _ /imsetP[x Ax ->] /morphimP[y Dy By ->] ->{fxy}.
by rewrite -morphM // (subsetP sAD, imset_f) // mem_mulg // inE By.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimMl
| |
morphimMrA 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimMr
| |
morphpreMlR 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); last by rewrite mulKVg.
by rewrite !inE Dy -def_fy.
by rewrite !inE groupM ?(morphM, morphV, groupV) // def_fx -def_fy mulKg.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreMl
| |
morphimJA 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimJ
| |
morphpreJR 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreJ
| |
morphim_classx 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_class
| |
classes_morphimA :
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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
classes_morphim
| |
morphimT: f @* setT = f @* D.
Proof. by rewrite -morphimIdom setIT. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimT
| |
morphimUA B : f @* (A :|: B) = f @* A :|: f @* B.
Proof. by rewrite -imsetU -setIUr. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimU
| |
morphimIA B : f @* (A :&: B) \subset f @* A :&: f @* B.
Proof. by rewrite subsetI // ?morphimS ?(subsetIl, subsetIr). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimI
| |
morphpre0: f @*^-1 set0 = set0.
Proof. by rewrite morphpreE preimset0 setI0. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpre0
| |
morphpreT: f @*^-1 setT = D.
Proof. by rewrite morphpreE preimsetT setIT. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreT
| |
morphpreUR S : f @*^-1 (R :|: S) = f @*^-1 R :|: f @*^-1 S.
Proof. by rewrite -setIUr -preimsetU. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreU
| |
morphpreIR S : f @*^-1 (R :&: S) = f @*^-1 R :&: f @*^-1 S.
Proof. by rewrite -setIIr -preimsetI. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreI
| |
morphpreDR S : f @*^-1 (R :\: S) = f @*^-1 R :\: f @*^-1 S.
Proof. by apply/setP=> x /[!inE]; case: (x \in D). Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreD
| |
kerPx : x \in D -> reflect (f x = 1) (x \in 'ker f).
Proof. by move=> Dx; rewrite 2!inE Dx; apply: set1P. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
kerP
| |
dom_ker: {subset 'ker f <= D}.
Proof. by move=> x /morphpreP[]. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
dom_ker
| |
mkerx : x \in 'ker f -> f x = 1.
Proof. by move=> Kx; apply/kerP=> //; apply: dom_ker. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
mker
| |
mkerlx 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
mkerl
| |
mkerrx 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
mkerr
| |
rcoset_kerPx 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
rcoset_kerP
| |
ker_rcosetx 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ker_rcoset
| |
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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ker_norm
| |
ker_normal: 'ker f <| D.
Proof. by rewrite /(_ <| D) subsetIl ker_norm. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ker_normal
| |
morphimGIG 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimGI
| |
morphimIGA G : 'ker f \subset G -> f @* (A :&: G) = f @* A :&: f @* G.
Proof. by move=> sKG; rewrite setIC morphimGI // setIC. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimIG
| |
morphimDA B : f @* A :\: f @* B \subset f @* (A :\: B).
Proof.
rewrite subDset -morphimU morphimS //.
by rewrite setDE setUIr setUCr setIT subsetUr.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimD
| |
morphimDGA 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimDG
| |
morphimD1A : (f @* A)^# \subset f @* A^#.
Proof. by rewrite -!set1gE -morphim1 morphimD. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimD1
| |
morphpre_groupsetM : 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpre_groupset
| |
morphim_groupsetG : 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_groupset
| |
morphpre_groupfPh M :=
@group _ (morphpre fPh M) (morphpre_groupset M).
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpre_group
| |
morphim_groupfPh G := @group _ (morphim fPh G) (morphim_groupset G).
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_group
| |
ker_groupfPh : {group aT} := Eval hnf in [group of ker fPh].
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ker_group
| |
morph_dom_groupset: group_set (f @: D).
Proof. by rewrite -morphimEdom groupP. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morph_dom_groupset
| |
morph_dom_group:= group morph_dom_groupset.
|
Canonical
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morph_dom_group
| |
morphpreMrR 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreMr
| |
morphimKA : 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 groupM // morphM // mker // mul1g imset_f // inE Dy.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimK
| |
morphimGKG : 'ker f \subset G -> G \subset D -> f @*^-1 (f @* G) = G.
Proof. by move=> sKG sGD; rewrite morphimK // mulSGid. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimGK
| |
morphpre_set1x : x \in D -> f @*^-1 [set f x] = 'ker f :* x.
Proof. by move=> Dx; rewrite -morphim_set1 // morphimK ?sub1set. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpre_set1
| |
morphpreKR : 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreK
| |
morphim_ker: f @* 'ker f = 1.
Proof. by rewrite morphpreK ?sub1G. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphim_ker
| |
ker_sub_preM : 'ker f \subset f @*^-1 M.
Proof. by rewrite morphpreS ?sub1G. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
ker_sub_pre
|
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