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_normal_preM : 'ker f <| f @*^-1 M.
Proof. by rewrite /normal ker_sub_pre subIset ?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_pre
| |
morphpreSKR 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphpreSK
| |
sub_morphim_preA 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
sub_morphim_pre
| |
morphpre_properR 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
|
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_proper
| |
sub_morphpre_imR 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
sub_morphpre_im
| |
ker_trivg_morphimA :
(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
|
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_trivg_morphim
| |
morphimSKA 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimSK
| |
morphimSGKA 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimSGK
| |
ltn_morphimA : [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; last first.
by rewrite -(mker kx) mem_morphim ?(dom_ker kx) // inE x1.
by rewrite (leq_trans (leq_imset_card _ _)) ?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
|
ltn_morphim
| |
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
|
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_inj
| |
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
|
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_injG
| |
morphim_injG 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
|
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_inj
| |
morphim_genA : A \subset D -> f @* <<A>> = <<f @* A>>.
Proof.
move=> sAD; apply/eqP.
rewrite eqEsubset andbC gen_subG morphimS; last exact: subset_gen.
by rewrite sub_morphim_pre gen_subG // -sub_morphim_pre // subset_gen.
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_gen
| |
morphim_cyclex : x \in D -> f @* <[x]> = <[f x]>.
Proof. by move=> Dx; rewrite morphim_gen (sub1set, morphim_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_cycle
| |
morphimYA 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
morphimY
| |
morphpre_genR :
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; last 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
|
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_gen
| |
morphimRA B :
A \subset D -> B \subset D -> f @* [~: A, B] = [~: f @* A, f @* B].
Proof.
move/subsetP=> sAD /subsetP sBD.
rewrite morphim_gen; last first; 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); rewrite ?(imset_f, morphR) // ?(inE, Dx, Dy).
case/morphimP=> x Dx Ax ->{fx}; case/morphimP=> y Dy By ->{fy} -> {fz}.
by exists [~ x, y]; rewrite ?(inE, morphR, groupR, imset2_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
|
morphimR
| |
morphim_normA : f @* 'N(A) \subset 'N(f @* A).
Proof.
apply/subsetP=> fx /morphimP[x Dx Nx -> {fx}].
by rewrite inE -morphimJ ?(normP Nx).
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_norm
| |
morphim_normsA 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
|
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_norms
| |
morphim_subnormA B : f @* 'N_A(B) \subset 'N_(f @* A)(f @* B).
Proof. exact: subset_trans (morphimI A _) (setIS _ (morphim_norm B)). 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_subnorm
| |
morphim_normalA B : A <| B -> f @* A <| f @* B.
Proof. by case/andP=> sAB nAB; rewrite /(_ <| _) morphimS // morphim_norms. 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_normal
| |
morphim_cent1x : x \in D -> f @* 'C[x] \subset 'C[f x].
Proof. by move=> Dx; rewrite -(morphim_set1 Dx) morphim_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
|
morphim_cent1
| |
morphim_cent1sA 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
|
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_cent1s
| |
morphim_subcent1A 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
|
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_subcent1
| |
morphim_centA : 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
|
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_cent
| |
morphim_centsA 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
|
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_cents
| |
morphim_subcentA B : f @* 'C_A(B) \subset 'C_(f @* A)(f @* B).
Proof. exact: subset_trans (morphimI A _) (setIS _ (morphim_cent B)). 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_subcent
| |
morphim_abelianA : abelian A -> abelian (f @* A).
Proof. exact: morphim_cents. 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_abelian
| |
morphpre_normR : f @*^-1 'N(R) \subset 'N(f @*^-1 R).
Proof.
by apply/subsetP=> x /[!inE] /andP[Dx Nfx]; rewrite -morphpreJ ?morphpreS.
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_norm
| |
morphpre_normsR 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
|
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_norms
| |
morphpre_normalR 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
|
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_normal
| |
morphpre_subnormR S : f @*^-1 'N_R(S) \subset 'N_(f @*^-1 R)(f @*^-1 S).
Proof. by rewrite morphpreI setIS ?morphpre_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
|
morphpre_subnorm
| |
morphim_normGG :
'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
|
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_normG
| |
morphim_subnormGA 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
|
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_subnormG
| |
morphpre_cent1x : 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
|
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_cent1
| |
morphpre_cent1sR 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
|
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_cent1s
| |
morphpre_subcent1R 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
|
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_subcent1
| |
morphpre_centA : '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
|
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_cent
| |
morphpre_centsA 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
|
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_cents
| |
morphpre_subcentR 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
|
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_subcent
| |
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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injmP
| |
card_im_injm: (#|f @* D| == #|D|) = 'injm f.
Proof. by rewrite morphimEdom (sameP imset_injP injmP). 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
|
card_im_injm
| |
ker_injm: 'ker f = 1.
Proof. exact/trivgP. 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_injm
| |
injmKA : A \subset D -> f @*^-1 (f @* A) = A.
Proof. by move=> sAD; rewrite morphimK // ker_injm // 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
|
injmK
| |
injm_morphim_injA 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_morphim_inj
| |
card_injmA : A \subset D -> #|f @* A| = #|A|.
Proof.
move=> sAD; rewrite morphimEsub // card_in_imset //.
exact: (sub_in2 (subsetP sAD) (injmP injf)).
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
|
card_injm
| |
order_injmx : x \in D -> #[f x] = #[x].
Proof.
by move=> Dx; rewrite orderE -morphim_cycle // card_injm ?cycle_subG.
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
|
order_injm
| |
injm1x : x \in D -> f x = 1 -> x = 1.
Proof. by move=> Dx; move/(kerP Dx); rewrite ker_injm; move/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
|
injm1
| |
morph_injm_eq1x : x \in D -> (f x == 1) = (x == 1).
Proof. by move=> Dx; rewrite -morph1 (inj_in_eq (injmP injf)) ?group1. 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_injm_eq1
| |
injmSKA B :
A \subset D -> (f @* A \subset f @* B) = (A \subset B).
Proof. by move=> sAD; rewrite morphimSK // ker_injm 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
|
injmSK
| |
sub_morphpre_injmR 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
sub_morphpre_injm
| |
injm_eqA B : A \subset D -> B \subset D -> (f @* A == f @* B) = (A == B).
Proof. by move=> sAD sBD; rewrite !eqEsubset !injmSK. 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
|
injm_eq
| |
morphim_injm_eq1A : A \subset D -> (f @* A == 1) = (A == 1).
Proof. by move=> sAD; rewrite -morphim1 injm_eq ?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_injm_eq1
| |
injmIA 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injmI
| |
injmD1A : f @* A^# = (f @* A)^#.
Proof. by have:= morphimDG A injf; rewrite morphim1. 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
|
injmD1
| |
nclasses_injmA : 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
nclasses_injm
| |
injm_normA : 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_norm
| |
injm_normsA 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
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_norms
| |
injm_normalA B :
A \subset D -> B \subset D -> (f @* A <| f @* B) = (A <| B).
Proof. by move=> sAD sBD; rewrite /normal injmSK ?injm_norms. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_normal
| |
injm_subnormA B : B \subset D -> f @* 'N_A(B) = 'N_(f @* A)(f @* B).
Proof. by move=> sBD; rewrite injmI injm_norm // setICA setIA morphimIim. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_subnorm
| |
injm_cent1x : x \in D -> f @* 'C[x] = 'C_(f @* D)[f x].
Proof. by move=> Dx; rewrite injm_norm ?morphim_set1 ?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
|
injm_cent1
| |
injm_subcent1A x : x \in D -> f @* 'C_A[x] = 'C_(f @* A)[f x].
Proof. by move=> Dx; rewrite injm_subnorm ?morphim_set1 ?sub1set. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_subcent1
| |
injm_centA : A \subset D -> f @* 'C(A) = 'C_(f @* D)(f @* A).
Proof.
move=> sAD; apply/eqP; rewrite -morphimIdom eqEsubset morphim_subcent.
apply/subsetP=> fx; case/setIP; case/morphimP=> x Dx _ ->{fx} cAfx.
rewrite mem_morphim // inE Dx -sub1set centsC cent_set1 -injmSK //.
by rewrite injm_cent1 // subsetI morphimS // -cent_set1 centsC sub1set.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_cent
| |
injm_centsA B :
A \subset D -> B \subset D -> (f @* A \subset 'C(f @* B)) = (A \subset 'C(B)).
Proof. by move=> sAD sBD; rewrite -injmSK // injm_cent // subsetI morphimS. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_cents
| |
injm_subcentA B : B \subset D -> f @* 'C_A(B) = 'C_(f @* A)(f @* B).
Proof. by move=> sBD; rewrite injmI injm_cent // setICA setIA morphimIim. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
injm_subcent
| |
injm_abelianA : A \subset D -> abelian (f @* A) = abelian A.
Proof.
by move=> sAD; rewrite /abelian -subsetIidl -injm_subcent // injmSK ?subsetIidl.
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
|
injm_abelian
| |
eq_morphim(g : {morphism D >-> rT}):
{in D, f =1 g} -> forall A, f @* A = g @* A.
Proof.
by move=> efg A; apply: eq_in_imset; apply: sub_in1 efg => x /setIP[].
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
eq_morphim
| |
eq_in_morphimB A (g : {morphism B >-> rT}) :
D :&: A = B :&: A -> {in A, f =1 g} -> f @* A = g @* A.
Proof.
move=> eqDBA eqAfg; rewrite /morphim /= eqDBA.
by apply: eq_in_imset => x /setIP[_]/eqAfg.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
eq_in_morphim
| |
idmof {set gT} := fun x : gT => x : FinGroup.sort gT.
|
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
|
idm
| |
idm_morphMA : {in A & , {morph idm A : x y / x * y}}.
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
|
idm_morphM
| |
idm_morphismA := Morphism (@idm_morphM A).
|
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
|
idm_morphism
| |
injm_idmG : 'injm (idm G).
Proof. by apply/injmP=> x y _ _. 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
|
injm_idm
| |
ker_idmG : 'ker (idm G) = 1.
Proof. by apply/trivgP; apply: injm_idm. 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_idm
| |
morphim_idmA B : B \subset A -> idm A @* B = B.
Proof.
rewrite /morphim /= /idm => /setIidPr->.
by apply/setP=> x; apply/imsetP/idP=> [[y By ->]|Bx]; last exists x.
Qed.
|
Lemma
|
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_idm
| |
morphpre_idmA B : idm A @*^-1 B = A :&: B.
Proof. by apply/setP=> x; rewrite !inE. 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_idm
| |
im_idmA : idm A @* A = A.
Proof. exact: morphim_idm. 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
|
im_idm
| |
restrmof A \subset D := @id (aT -> FinGroup.sort 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
|
restrm
| |
restrm_morphism:=
@Morphism aT rT A fA (sub_in2 (subsetP sAD) (morphM f)).
|
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
|
restrm_morphism
| |
morphim_restrmB : fA @* B = f @* (A :&: B).
Proof. by rewrite {2}/morphim setIA (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
|
morphim_restrm
| |
restrmEsubB : B \subset A -> fA @* B = f @* B.
Proof. by rewrite morphim_restrm => /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
|
restrmEsub
| |
im_restrm: fA @* A = f @* A.
Proof. exact: restrmEsub. 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
|
im_restrm
| |
morphpre_restrmR : fA @*^-1 R = A :&: f @*^-1 R.
Proof. by rewrite setIA (setIidPl 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
|
morphpre_restrm
| |
ker_restrm: 'ker fA = 'ker_A f.
Proof. exact: morphpre_restrm. 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_restrm
| |
injm_restrm: 'injm f -> 'injm fA.
Proof. by apply: subset_trans; rewrite ker_restrm 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
|
injm_restrm
| |
restrmP(f : {morphism D >-> rT}) : A \subset 'dom f ->
{g : {morphism A >-> rT} | [/\ g = f :> (aT -> rT), 'ker g = 'ker_A f,
forall R, g @*^-1 R = A :&: f @*^-1 R
& forall B, B \subset A -> g @* B = f @* B]}.
Proof.
move=> sAD; exists (restrm_morphism sAD f).
split=> // [|R|B sBA]; first 1 [exact: ker_restrm | exact: morphpre_restrm].
by rewrite morphim_restrm (setIidPr sBA).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
restrmP
| |
domP(f : {morphism D >-> rT}) : 'dom f = A ->
{g : {morphism A >-> rT} | [/\ g = f :> (aT -> rT), 'ker g = 'ker f,
forall R, g @*^-1 R = f @*^-1 R
& forall B, g @* B = f @* B]}.
Proof. by move <-; exists f. Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
domP
| |
trivmof {set aT} & aT := 1 : FinGroup.sort 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
|
trivm
| |
trivm_morphM(A : {set aT}) : {in A &, {morph trivm A : x y / x * y}}.
Proof. by move=> x y /=; rewrite 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
|
trivm_morphM
| |
triv_morphA := Morphism (@trivm_morphM A).
|
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
|
triv_morph
| |
morphim_trivm(G H : {group aT}) : trivm G @* H = 1.
Proof.
apply/setP=> /= y; rewrite inE; apply/idP/eqP=> [|->]; first by case/morphimP.
by apply/morphimP; exists (1 : aT); rewrite /= ?group1.
Qed.
|
Lemma
|
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_trivm
| |
ker_trivm(G : {group aT}) : 'ker (trivm G) = G.
Proof. by apply/setIidPl/subsetP=> x _; rewrite !inE /=. 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_trivm
| |
comp_morphM: {in f @*^-1 H &, {morph gof: x y / x * y}}.
Proof.
by move=> x y; rewrite /= !inE => /andP[? ?] /andP[? ?]; rewrite !morphM.
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
comp_morphM
| |
comp_morphism:= Morphism comp_morphM.
|
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
|
comp_morphism
| |
ker_comp: 'ker gof = f @*^-1 'ker g.
Proof. by apply/setP=> x; rewrite !inE andbA. 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_comp
| |
injm_comp: 'injm f -> 'injm g -> 'injm gof.
Proof. by move=> injf; rewrite ker_comp; move/trivgP=> ->. 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
|
injm_comp
| |
morphim_comp(A : {set gT}) : gof @* A = g @* (f @* A).
Proof.
apply/setP=> z; apply/morphimP/morphimP=> [[x]|[y Hy fAy ->{z}]].
rewrite !inE => /andP[Gx Hfx]; exists (f x) => //.
by apply/morphimP; exists x.
by case/morphimP: fAy Hy => x Gx Ax ->{y} Hfx; exists x; rewrite ?inE ?Gx.
Qed.
|
Lemma
|
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_comp
| |
morphpre_comp(C : {set rT}) : gof @*^-1 C = f @*^-1 (g @*^-1 C).
Proof. by apply/setP=> z; rewrite !inE andbA. 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_comp
| |
factmof 'ker q \subset 'ker f & G \subset H :=
fun x => f (repr (q @*^-1 [set x])).
Hypothesis sKqKf : 'ker q \subset 'ker f.
Hypothesis sGH : G \subset H.
|
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
|
factm
| |
ff:= (factm sKqKf sGH).
|
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
|
ff
| |
factmEx : x \in G -> ff (q x) = f x.
Proof.
rewrite /ff => Gx; have Hx := subsetP sGH x Gx.
have /mem_repr: x \in q @*^-1 [set q x] by rewrite !inE Hx /=.
case/morphpreP; move: (repr _) => y Hy /set1P.
by case/ker_rcoset=> // z Kz ->; rewrite mkerl ?(subsetP sKqKf).
Qed.
|
Lemma
|
fingroup
|
[
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup"
] |
fingroup/morphism.v
|
factmE
|
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