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 |
|---|---|---|---|---|---|---|
dffun_morphismi := @Morphism _ _ setT _ (in2W (@dffunM i)).
|
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
|
dffun_morphism
| |
injm_dfung1i : 'injm (@dfung1 i).
Proof.
apply/subsetP => x /morphpreP[_ /set1P /ffunP/=/(_ i)].
by rewrite !(ffunE, dfung1_id) => ->; apply: set11.
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_dfung1
| |
group_set_dfwithH i (G : {group gT i}) j :
group_set (dfwith (H : forall k, {set gT k}) (G : {set _}) j).
Proof.
have [<-|ij] := eqVneq i j; first by rewrite !dfwith_in// groupP.
by rewrite !dfwith_out // groupP.
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
|
group_set_dfwith
| |
group_dfwithH i G j := Group (@group_set_dfwith H i G j).
|
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
|
group_dfwith
| |
group_dfwithEH i G j : @group_dfwith H i G j = dfwith H G j.
Proof.
by apply/val_inj; have [<-|nij]/= := eqVneq i j;
[rewrite !dfwith_in|rewrite !dfwith_out].
Qed.
Fact set1gXn_key : unit. Proof. by []. 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
|
group_dfwithE
| |
set1gXn{i} (H : {set gT i}) : {set {dffun forall i : I, gT i}} :=
locked_with set1gXn_key (setXn (dfwith (fun i0 : I => [1 gT _]%g) H)).
|
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
|
set1gXn
| |
set1gXnE{i} (H : {set gT i}) :
set1gXn H = setXn (dfwith (fun i0 : I => [1 gT _]%g) H).
Proof. by rewrite /set1gXn unlock. 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
|
set1gXnE
| |
set1gXnP{i} (H : {set gT i}) x :
reflect (exists2 h, h \in H & x = dfung1 h) (x \in set1gXn H).
Proof.
rewrite set1gXnE/=; apply: (iffP setXnP) => [xP|[h hH ->] j]; last first.
by rewrite ffunE; case: dfwithP => [|k ?]; rewrite (dfwith_in, dfwith_out).
exists (x i); first by have := xP i; rewrite dfwith_in.
apply/ffunP => j; have := xP j; rewrite ffunE.
case: dfwithP => // [xiH|k neq_ik]; first by rewrite dfwith_in.
by move=> /set1gP->; rewrite dfwith_out.
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
|
set1gXnP
| |
morphim_dfung1i (G : {set gT i}) : @dfung1 i @* G = set1gXn G.
Proof.
by rewrite morphimEsub//=; apply/setP=> /= x; apply/imsetP/set1gXnP.
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
|
morphim_dfung1
| |
morphim_dffunXni H : dffun_morphism i @* setXn H = H i.
Proof.
apply/eqP; rewrite eqEsubset morphimE setTI /=.
apply/andP; split; apply/subsetP=> x.
by case/imsetP => x0 /[1!inE] /forallP/(_ i)/= ? ->.
move=> Hx1; apply/imsetP; exists (dfung1 x); last by rewrite dfung1_id.
by rewrite in_setXn; apply/forallP => j /[!ffunE]; case: dfwithP.
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
|
morphim_dffunXn
| |
set1gXn_group_set{i} (H : {group gT i}) : group_set (set1gXn H).
Proof. by rewrite set1gXnE; exact: group_setXn. 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
|
set1gXn_group_set
| |
groupXn1{i} (H : {group gT i}) := Group (set1gXn_group_set H).
|
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
|
groupXn1
| |
setXn_prodH : \prod_i set1gXn (H i) = setXn H.
Proof.
apply/setP => /= x; apply/prodsgP /setXnP => [[/= f fH {x}-> i]|xH /=].
rewrite prodg_ffun group_prod// => j _.
by have /set1gXnP[x xH ->] := fH j isT; rewrite ffunE; case: dfwithP.
exists (fun i => dfung1 (x i)) => [i _|]; first by apply/set1gXnP; exists (x i).
apply/ffunP => i; rewrite prodg_ffun (big_only1 i) ?dfung1_id//.
by move=> j ij _; rewrite dfung1_dflt.
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
|
setXn_prod
| |
set1gXn_commute(H : forall i, {group gT i}) i j :
commute (set1gXn (H i)) (set1gXn (H j)).
Proof.
have [-> //|neqij] := eqVneq j i.
apply/centC/centsP => _ /set1gXnP [hi hiH ->] _ /set1gXnP [hj hjH ->].
apply/ffunP => k; rewrite !ffunE.
by case: dfwithP => [|?]; rewrite ?mulg1 ?mul1g// dfwith_out// mulg1 mul1g.
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
|
set1gXn_commute
| |
setXn_dprodH : \big[dprod/1]_i set1gXn (H i) = setXn H.
Proof.
rewrite -setXn_prod//=.
suff -> : \big[dprod/1]_i groupXn1 (H i) = (\prod_i groupXn1 (H i))%G.
by rewrite comm_prodG//=; apply: in2W; apply: set1gXn_commute.
apply/eqP; apply/bigdprodYP => i //= _; rewrite subsetD.
apply/andP; split.
rewrite comm_prodG; last by apply: in2W; apply: set1gXn_commute.
apply/centsP => _ /prodsgP[/= h_ h_P ->] _ /set1gXnP [h hH ->].
apply/ffunP => j; rewrite !ffunE/=.
rewrite (big_morph _ (@dffunM j) (_ : _ = 1)) ?ffunE//.
case: dfwithP => {j} [|? ?]; last by rewrite mulg1 mul1g.
rewrite big1 ?mulg1 ?mul1g// => j neq_ji.
by have /set1gXnP[? _ ->] := h_P j neq_ji; rewrite ffunE dfwith_out.
rewrite -setI_eq0 -subset0; apply/subsetP => /= x; rewrite !inE.
rewrite comm_prodG; last by apply: in2W; apply: set1gXn_commute.
move=> /and3P[+ + /set1gXnP [h _ x_h]]; rewrite {x}x_h.
move=> /prodsgP[x_ x_P /ffunP/(_ i)]; rewrite ffunE dfwith_in => {h}->.
apply: contra_neqT => _; apply/ffunP => j; rewrite !ffunE/=.
case: dfwithP => // {j}; rewrite (big_morph _ (@dffunM i) (_ : _ = 1)) ?ffunE//.
rewrite big1// => j neq_ji.
by have /set1gXnP[g gH /ffunP->] := x_P _ neq_ji; rewrite ffunE dfwith_out.
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
|
setXn_dprod
| |
isog_setXni (G : {group gT i}) : G \isog set1gXn G.
Proof.
apply/(@isogP _ _ G); exists [morphism of restrm (subsetT G) (@dfung1 i)].
by rewrite injm_restrm ?injm_dfung1.
by rewrite morphim_restrm morphim_dfung1 setIid.
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_setXn
| |
setXn_genH : (forall i, 1 \in H i) ->
<<setXn H>> = setXn (fun i => <<H i>>).
Proof.
move=> H1; apply/eqP; rewrite eqEsubset gen_subG setXnS/=; last first.
by move=> ?; rewrite subset_gen.
rewrite -[in X in X \subset _]setXn_prod; under eq_bigr do
rewrite -morphim_dfung1 morphim_gen ?subsetT// morphim_dfung1.
rewrite prod_subG// => i; rewrite genS // set1gXnE setXnS // => j.
by case: dfwithP => // k _; rewrite sub1set.
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
|
setXn_gen
| |
groupX0(gT : 'I_0 -> finGroupType) (G : forall i, {group gT i}) :
setXn G = 1%g.
Proof.
by apply/setP => ?; apply/setXnP/set1P => [_|_ []//]; apply/ffunP => -[].
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
|
groupX0
| |
sdprod_by(to : groupAction D R) : predArgType :=
SdPair (ax : aT * rT) of ax \in setX D R.
|
Inductive
|
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
|
sdprod_by
| |
pair_of_sdto (u : sdprod_by to) := let: SdPair ax _ := u in ax.
Variable to : groupAction D R.
|
Coercion
|
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
|
pair_of_sd
| |
sdT:= (sdprod_by to).
|
Notation
|
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
|
sdT
| |
sdval:= (@pair_of_sd to).
HB.instance Definition _ := [isSub for sdval].
#[hnf] HB.instance Definition _ := [Finite of sdT by <:].
|
Notation
|
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
|
sdval
| |
sdprod_one:= SdPair to (group1 _).
|
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
|
sdprod_one
| |
sdprod_inv_proof(u : sdT) : (u.1^-1, to u.2^-1 u.1^-1) \in setX D R.
Proof.
by case: u => [[a x]] /= /setXP[Da Rx]; rewrite inE gact_stable !groupV ?Da.
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
|
sdprod_inv_proof
| |
sdprod_invu := SdPair to (sdprod_inv_proof u).
|
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
|
sdprod_inv
| |
sdprod_mul_proof(u v : sdT) :
(u.1 * v.1, to u.2 v.1 * v.2) \in setX D R.
Proof.
case: u v => [[a x] /= /setXP[Da Rx]] [[b y] /= /setXP[Db Ry]].
by rewrite inE !groupM //= 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
|
sdprod_mul_proof
| |
sdprod_mulu v := SdPair to (sdprod_mul_proof u v).
|
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
|
sdprod_mul
| |
sdprod_mul1g: left_id sdprod_one sdprod_mul.
Proof.
move=> u; apply: val_inj; case: u => [[a x] /=]; case/setXP=> Da _.
by rewrite gact1 // !mul1g.
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
|
sdprod_mul1g
| |
sdprod_mulVg: left_inverse sdprod_one sdprod_inv sdprod_mul.
Proof.
move=> u; apply: val_inj; case: u => [[a x] /=]; case/setXP=> Da _.
by rewrite actKVin ?mulVg.
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
|
sdprod_mulVg
| |
sdprod_mulgA: associative sdprod_mul.
Proof.
move=> u v w; apply: val_inj; case: u => [[a x]] /=; case/setXP=> Da Rx.
case: v w => [[b y]] /=; case/setXP=> Db Ry [[c z]] /=; case/setXP=> Dc Rz.
by rewrite !(actMin to) // gactM ?gact_stable // !mulgA.
Qed.
HB.instance Definition _ := Finite_isGroup.Build sdT
sdprod_mulgA sdprod_mul1g sdprod_mulVg.
|
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
|
sdprod_mulgA
| |
sdprod_groupType: finGroupType := sdT.
|
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
|
sdprod_groupType
| |
sdpair1x := insubd sdprod_one (1, x) : sdT.
|
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
|
sdpair1
| |
sdpair2a := insubd sdprod_one (a, 1) : sdT.
|
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
|
sdpair2
| |
sdpair1_morphM: {in R &, {morph sdpair1 : x y / x * y}}.
Proof.
move=> x y Rx Ry; apply: val_inj.
by rewrite /= !val_insubd !inE !group1 !groupM ?Rx ?Ry //= mulg1 act1.
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
|
sdpair1_morphM
| |
sdpair2_morphM: {in D &, {morph sdpair2 : a b / a * b}}.
Proof.
move=> a b Da Db; apply: val_inj.
by rewrite /= !val_insubd !inE !group1 !groupM ?Da ?Db //= mulg1 gact1.
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
|
sdpair2_morphM
| |
sdpair1_morphism:= Morphism sdpair1_morphM.
|
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
|
sdpair1_morphism
| |
sdpair2_morphism:= Morphism sdpair2_morphM.
|
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
|
sdpair2_morphism
| |
injm_sdpair1: 'injm sdpair1.
Proof.
apply/subsetP=> x /setIP[Rx].
by rewrite !inE -val_eqE val_insubd inE Rx group1 /=; case/andP.
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_sdpair1
| |
injm_sdpair2: 'injm sdpair2.
Proof.
apply/subsetP=> a /setIP[Da].
by rewrite !inE -val_eqE val_insubd inE Da group1 /=; case/andP.
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_sdpair2
| |
sdpairE(u : sdT) : u = sdpair2 u.1 * sdpair1 u.2.
Proof.
apply: val_inj; case: u => [[a x] /= /setXP[Da Rx]].
by rewrite !val_insubd !inE Da Rx !(group1, gact1) // mulg1 mul1g.
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
|
sdpairE
| |
sdpair_act: {in R & D,
forall x a, sdpair1 (to x a) = sdpair1 x ^ sdpair2 a}.
Proof.
move=> x a Rx Da; apply: val_inj.
rewrite /= !val_insubd !inE !group1 gact_stable ?Da ?Rx //=.
by rewrite !mul1g mulVg invg1 mulg1 actKVin ?mul1g.
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
|
sdpair_act
| |
sdpair_setact(G : {set rT}) a : G \subset R -> a \in D ->
sdpair1 @* (to^~ a @: G) = (sdpair1 @* G) :^ sdpair2 a.
Proof.
move=> sGR Da; have GtoR := subsetP sGR; apply/eqP.
rewrite eqEcard cardJg !(card_injm injm_sdpair1) //; last first.
by apply/subsetP=> _ /imsetP[x Gx ->]; rewrite gact_stable ?GtoR.
rewrite (card_imset _ (act_inj _ _)) leqnn andbT.
apply/subsetP=> _ /morphimP[xa Rxa /imsetP[x Gx def_xa ->]].
rewrite mem_conjg -morphV // -sdpair_act ?groupV // def_xa actKin //.
by rewrite mem_morphim ?GtoR.
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
|
sdpair_setact
| |
im_sdpair_norm: sdpair2 @* D \subset 'N(sdpair1 @* R).
Proof.
apply/subsetP=> _ /morphimP[a _ Da ->].
rewrite inE -sdpair_setact // morphimS //.
by apply/subsetP=> _ /imsetP[x Rx ->]; rewrite 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
|
im_sdpair_norm
| |
im_sdpair_TI: (sdpair1 @* R) :&: (sdpair2 @* D) = 1.
Proof.
apply/trivgP; apply/subsetP=> _ /setIP[/morphimP[x _ Rx ->]].
case/morphimP=> a _ Da /eqP; rewrite inE -!val_eqE.
by rewrite !val_insubd !inE Da Rx !group1 /eq_op /= eqxx; case/andP.
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_sdpair_TI
| |
im_sdpair: (sdpair1 @* R) * (sdpair2 @* D) = setT.
Proof.
apply/eqP; rewrite -subTset -(normC im_sdpair_norm).
apply/subsetP=> /= u _; rewrite [u]sdpairE.
by case: u => [[a x] /= /setXP[Da Rx]]; rewrite mem_mulg ?mem_morphim.
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_sdpair
| |
sdprod_sdpair: sdpair1 @* R ><| sdpair2 @* D = setT.
Proof. by rewrite sdprodE ?(im_sdpair_norm, im_sdpair, im_sdpair_TI). Qed.
Variables (A : {set aT}) (G : {set rT}).
|
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
|
sdprod_sdpair
| |
gacentEsd: 'C_(|to)(A) = sdpair1 @*^-1 'C(sdpair2 @* A).
Proof.
apply/setP=> x; apply/idP/idP.
case/setIP=> Rx /afixP cDAx; rewrite mem_morphpre //.
apply/centP=> _ /morphimP[a Da Aa ->]; red.
by rewrite conjgC -sdpair_act // cDAx // inE Da.
case/morphpreP=> Rx cAx; rewrite inE Rx; apply/afixP=> a /setIP[Da Aa].
apply: (injmP injm_sdpair1); rewrite ?gact_stable /= ?sdpair_act //=.
by rewrite /conjg (centP cAx) ?mulKg ?mem_morphim.
Qed.
Hypotheses (sAD : A \subset D) (sGR : G \subset R).
|
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
|
gacentEsd
| |
astabEsd: 'C(G | to) = sdpair2 @*^-1 'C(sdpair1 @* G).
Proof.
have ssGR := subsetP sGR; apply/setP=> a; apply/idP/idP=> [cGa|].
rewrite mem_morphpre ?(astab_dom cGa) //.
apply/centP=> _ /morphimP[x Rx Gx ->]; symmetry.
by rewrite conjgC -sdpair_act ?(astab_act cGa) ?(astab_dom cGa).
case/morphpreP=> Da cGa; rewrite !inE Da; apply/subsetP=> x Gx; rewrite inE.
apply/eqP; apply: (injmP injm_sdpair1); rewrite ?gact_stable ?ssGR //=.
by rewrite sdpair_act ?ssGR // /conjg -(centP cGa) ?mulKg ?mem_morphim ?ssGR.
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
|
astabEsd
| |
astabsEsd: 'N(G | to) = sdpair2 @*^-1 'N(sdpair1 @* G).
Proof.
apply/setP=> a; apply/idP/idP=> [nGa|].
have Da := astabs_dom nGa; rewrite mem_morphpre // inE sub_conjg.
apply/subsetP=> _ /morphimP[x Rx Gx ->].
by rewrite mem_conjgV -sdpair_act // mem_morphim ?gact_stable ?astabs_act.
case/morphpreP=> Da nGa; rewrite !inE Da; apply/subsetP=> x Gx.
have Rx := subsetP sGR _ Gx; have Rxa: to x a \in R by rewrite gact_stable.
rewrite inE -sub1set -(injmSK injm_sdpair1) ?morphim_set1 ?sub1set //=.
by rewrite sdpair_act ?memJ_norm ?mem_morphim.
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
|
astabsEsd
| |
actsEsd: [acts A, on G | to] = (sdpair2 @* A \subset 'N(sdpair1 @* G)).
Proof. by rewrite sub_morphim_pre -?astabsEsd. 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
|
actsEsd
| |
pprodmof B \subset 'N(A) & {in A & B, morph_act 'J 'J fA fB}
& {in A :&: B, fA =1 fB} :=
fun x => fA (divgr A B x) * fB (remgr A B x).
|
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
|
pprodm
| |
pprodmEx a : x \in H -> a \in K -> f (x * a) = fH x * fK a.
Proof.
move=> Hx Ka; have: x * a \in H * K by rewrite mem_mulg.
rewrite -remgrP inE /f rcoset_sym mem_rcoset /divgr -mulgA groupMl //.
case/andP; move: (remgr H K _) => b Hab Kb; rewrite morphM // -mulgA.
have Kab: a * b^-1 \in K by rewrite groupM ?groupV.
by congr (_ * _); rewrite eqfHK 1?inE ?Hab // -morphM // mulgKV.
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
|
pprodmE
| |
pprodmEl: {in H, f =1 fH}.
Proof. by move=> x Hx; rewrite -(mulg1 x) pprodmE // morph1 !mulg1. 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
|
pprodmEl
| |
pprodmEr: {in K, f =1 fK}.
Proof. by move=> a Ka; rewrite -(mul1g a) pprodmE // morph1 !mul1g. 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
|
pprodmEr
| |
pprodmM: {in H <*> K &, {morph f: x y / x * y}}.
Proof.
move=> xa yb; rewrite norm_joinEr //.
move=> /imset2P[x a Ha Ka ->{xa}] /imset2P[y b Hy Kb ->{yb}].
have Hya: y ^ a^-1 \in H by rewrite -mem_conjg (normsP nHK).
rewrite mulgA -(mulgA x) (conjgCV a y) (mulgA x) -mulgA !pprodmE 1?groupMl //.
by rewrite morphM // actf ?groupV ?morphV // morphM // !mulgA mulgKV invgK.
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
|
pprodmM
| |
pprodm_morphism:= Morphism pprodmM.
|
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
|
pprodm_morphism
| |
morphim_pprodmA B :
A \subset H -> B \subset K -> f @* (A * B) = fH @* A * fK @* B.
Proof.
move=> sAH sBK; rewrite [f @* _]morphimEsub /=; last first.
by rewrite norm_joinEr // mulgSS.
apply/setP=> y; apply/imsetP/idP=> [[_ /mulsgP[x a Ax Ba ->] ->{y}] |].
have Hx := subsetP sAH x Ax; have Ka := subsetP sBK a Ba.
by rewrite pprodmE // imset2_f ?mem_morphim.
case/mulsgP=> _ _ /morphimP[x Hx Ax ->] /morphimP[a Ka Ba ->] ->{y}.
by exists (x * a); rewrite ?mem_mulg ?pprodmE.
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
|
morphim_pprodm
| |
morphim_pprodmlA : A \subset H -> f @* A = fH @* A.
Proof.
by move=> sAH; rewrite -{1}(mulg1 A) morphim_pprodm ?sub1G // morphim1 mulg1.
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
|
morphim_pprodml
| |
morphim_pprodmrB : B \subset K -> f @* B = fK @* B.
Proof.
by move=> sBK; rewrite -{1}(mul1g B) morphim_pprodm ?sub1G // morphim1 mul1g.
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
|
morphim_pprodmr
| |
ker_pprodm: 'ker f = [set x * a^-1 | x in H, a in K & fH x == fK a].
Proof.
apply/setP=> y; rewrite 3!inE {1}norm_joinEr //=.
apply/andP/imset2P=> [[/mulsgP[x a Hx Ka ->{y}]]|[x a Hx]].
rewrite pprodmE // => fxa1.
by exists x a^-1; rewrite ?invgK // inE groupVr ?morphV // eq_mulgV1 invgK.
case/setIdP=> Kx /eqP fx ->{y}.
by rewrite imset2_f ?pprodmE ?groupV ?morphV // fx mulgV.
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_pprodm
| |
injm_pprodm:
'injm f = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == fH @* K].
Proof.
apply/idP/and3P=> [injf | [injfH injfK]].
rewrite eq_sym -{1}morphimIdom -(morphim_pprodml (subsetIl _ _)) injmI //.
rewrite morphim_pprodml // morphim_pprodmr //=; split=> //.
apply/injmP=> x y Hx Hy /=; rewrite -!pprodmEl //.
by apply: (injmP injf); rewrite ?mem_gen ?inE ?Hx ?Hy.
apply/injmP=> a b Ka Kb /=; rewrite -!pprodmEr //.
by apply: (injmP injf); rewrite ?mem_gen //; apply/setUP; right.
move/eqP=> fHK; rewrite ker_pprodm; apply/subsetP=> y.
case/imset2P=> x a Hx /setIdP[Ka /eqP fxa] ->.
have: fH x \in fH @* K by rewrite -fHK inE {2}fxa !mem_morphim.
case/morphimP=> z Hz Kz /(injmP injfH) def_x.
rewrite def_x // eqfHK ?inE ?Hz // in fxa.
by rewrite def_x // (injmP injfK _ _ Kz Ka fxa) mulgV set11.
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_pprodm
| |
sdprodm_norm: K \subset 'N(H).
Proof. by case/sdprodP: eqHK_G. 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
|
sdprodm_norm
| |
sdprodm_sub: G \subset H <*> K.
Proof. by case/sdprodP: eqHK_G => _ <- nHK _; rewrite norm_joinEr. 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
|
sdprodm_sub
| |
sdprodm_eqf: {in H :&: K, fH =1 fK}.
Proof.
by case/sdprodP: eqHK_G => _ _ _ -> _ /set1P->; rewrite !morph1.
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
|
sdprodm_eqf
| |
sdprodm:=
restrm sdprodm_sub (pprodm sdprodm_norm actf sdprodm_eqf).
|
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
|
sdprodm
| |
sdprodm_morphism:= Eval hnf in [morphism of sdprodm].
|
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
|
sdprodm_morphism
| |
sdprodmEa b : a \in H -> b \in K -> sdprodm (a * b) = fH a * fK b.
Proof. exact: pprodmE. 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
|
sdprodmE
| |
sdprodmEla : a \in H -> sdprodm a = fH a.
Proof. exact: pprodmEl. 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
|
sdprodmEl
| |
sdprodmErb : b \in K -> sdprodm b = fK b.
Proof. exact: pprodmEr. 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
|
sdprodmEr
| |
morphim_sdprodmA B :
A \subset H -> B \subset K -> sdprodm @* (A * B) = fH @* A * fK @* B.
Proof.
move=> sAH sBK; rewrite /sdprodm morphim_restrm /= (setIidPr _) ?morphim_pprodm //.
by case/sdprodP: eqHK_G => _ <- _ _; apply: mulgSS.
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
|
morphim_sdprodm
| |
im_sdprodm: sdprodm @* G = fH @* H * fK @* K.
Proof. by rewrite -morphim_sdprodm //; case/sdprodP: eqHK_G => _ ->. 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_sdprodm
| |
morphim_sdprodmlA : A \subset H -> sdprodm @* A = fH @* A.
Proof.
by move=> sHA; rewrite -{1}(mulg1 A) morphim_sdprodm ?sub1G // morphim1 mulg1.
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
|
morphim_sdprodml
| |
morphim_sdprodmrB : B \subset K -> sdprodm @* B = fK @* B.
Proof.
by move=> sBK; rewrite -{1}(mul1g B) morphim_sdprodm ?sub1G // morphim1 mul1g.
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
|
morphim_sdprodmr
| |
ker_sdprodm:
'ker sdprodm = [set a * b^-1 | a in H, b in K & fH a == fK b].
Proof.
rewrite ker_restrm (setIidPr _) ?subIset ?ker_pprodm //; apply/orP; left.
by case/sdprodP: eqHK_G => _ <- nHK _; rewrite norm_joinEr.
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_sdprodm
| |
injm_sdprodm:
'injm sdprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == 1].
Proof.
rewrite ker_sdprodm -(ker_pprodm sdprodm_norm actf sdprodm_eqf) injm_pprodm.
congr [&& _, _ & _ == _]; have [_ _ _ tiHK] := sdprodP eqHK_G.
by rewrite -morphimIdom tiHK 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_sdprodm
| |
cprodm_norm: K \subset 'N(H).
Proof. by rewrite cents_norm //; case/cprodP: eqHK_G. 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
|
cprodm_norm
| |
cprodm_sub: G \subset H <*> K.
Proof. by case/cprodP: eqHK_G => _ <- cHK; rewrite cent_joinEr. 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
|
cprodm_sub
| |
cprodm_actf: {in H & K, morph_act 'J 'J fH fK}.
Proof.
case/cprodP: eqHK_G => _ _ cHK a b Ha Kb /=.
by rewrite /conjg -(centsP cHK b) // -(centsP cfHK (fK b)) ?mulKg ?mem_morphim.
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
|
cprodm_actf
| |
cprodm:= restrm cprodm_sub (pprodm cprodm_norm cprodm_actf eqfHK).
|
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
|
cprodm
| |
cprodm_morphism:= Eval hnf in [morphism of cprodm].
|
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
|
cprodm_morphism
| |
cprodmEa b : a \in H -> b \in K -> cprodm (a * b) = fH a * fK b.
Proof. exact: pprodmE. 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
|
cprodmE
| |
cprodmEla : a \in H -> cprodm a = fH a.
Proof. exact: pprodmEl. 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
|
cprodmEl
| |
cprodmErb : b \in K -> cprodm b = fK b.
Proof. exact: pprodmEr. 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
|
cprodmEr
| |
morphim_cprodmA B :
A \subset H -> B \subset K -> cprodm @* (A * B) = fH @* A * fK @* B.
Proof.
move=> sAH sBK; rewrite [LHS]morphim_restrm /= (setIidPr _) ?morphim_pprodm //.
by case/cprodP: eqHK_G => _ <- _; apply: mulgSS.
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
|
morphim_cprodm
| |
im_cprodm: cprodm @* G = fH @* H * fK @* K.
Proof.
by have [_ defHK _] := cprodP eqHK_G; rewrite -{2}defHK morphim_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
|
im_cprodm
| |
morphim_cprodmlA : A \subset H -> cprodm @* A = fH @* A.
Proof.
by move=> sHA; rewrite -{1}(mulg1 A) morphim_cprodm ?sub1G // morphim1 mulg1.
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
|
morphim_cprodml
| |
morphim_cprodmrB : B \subset K -> cprodm @* B = fK @* B.
Proof.
by move=> sBK; rewrite -{1}(mul1g B) morphim_cprodm ?sub1G // morphim1 mul1g.
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
|
morphim_cprodmr
| |
ker_cprodm: 'ker cprodm = [set a * b^-1 | a in H, b in K & fH a == fK b].
Proof.
rewrite ker_restrm (setIidPr _) ?subIset ?ker_pprodm //; apply/orP; left.
by case/cprodP: eqHK_G => _ <- cHK; rewrite cent_joinEr.
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_cprodm
| |
injm_cprodm:
'injm cprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == fH @* K].
Proof.
by rewrite ker_cprodm -(ker_pprodm cprodm_norm cprodm_actf eqfHK) injm_pprodm.
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_cprodm
| |
dprodm_cprod: H \* K = G.
Proof.
by rewrite -eqHK_G /dprod; case/dprodP: eqHK_G => _ _ _ ->; rewrite subxx.
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
|
dprodm_cprod
| |
dprodm_eqf: {in H :&: K, fH =1 fK}.
Proof. by case/dprodP: eqHK_G => _ _ _ -> _ /set1P->; rewrite !morph1. 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
|
dprodm_eqf
| |
dprodm:= cprodm dprodm_cprod cfHK dprodm_eqf.
|
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
|
dprodm
| |
dprodm_morphism:= Eval hnf in [morphism of dprodm].
|
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
|
dprodm_morphism
| |
dprodmEa b : a \in H -> b \in K -> dprodm (a * b) = fH a * fK b.
Proof. exact: pprodmE. 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
|
dprodmE
| |
dprodmEla : a \in H -> dprodm a = fH a.
Proof. exact: pprodmEl. 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
|
dprodmEl
| |
dprodmErb : b \in K -> dprodm b = fK b.
Proof. exact: pprodmEr. 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
|
dprodmEr
| |
morphim_dprodmA B :
A \subset H -> B \subset K -> dprodm @* (A * B) = fH @* A * fK @* B.
Proof. exact: morphim_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
|
morphim_dprodm
| |
im_dprodm: dprodm @* G = fH @* H * fK @* K.
Proof. exact: im_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
|
im_dprodm
| |
morphim_dprodmlA : A \subset H -> dprodm @* A = fH @* A.
Proof. exact: morphim_cprodml. 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
|
morphim_dprodml
| |
morphim_dprodmrB : B \subset K -> dprodm @* B = fK @* B.
Proof. exact: morphim_cprodmr. 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
|
morphim_dprodmr
|
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