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 |
|---|---|---|---|---|---|---|
sub_abelian_cent2: B \subset A -> C \subset A -> B \subset 'C(C).
Proof. by move=> sBA; move/sub_abelian_cent; apply: subset_trans. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
sub_abelian_cent2
| |
sub_abelian_norm: C \subset A -> A \subset 'N(C).
Proof. by move=> sCA; rewrite cents_norm ?sub_abelian_cent. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
sub_abelian_norm
| |
sub_abelian_normal: (C \subset A) = (C <| A).
Proof.
by rewrite /normal; case sHG: (C \subset A); rewrite // sub_abelian_norm.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
sub_abelian_normal
| |
maxgroupA gP := maxset (fun A => group_set A && gP <<A>>%G) A.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
maxgroup
| |
mingroupA gP := minset (fun A => group_set A && gP <<A>>%G) A.
Variable gP : pred {group gT}.
Arguments gP _%_G.
|
Definition
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
mingroup
| |
ex_maxgroup: (exists G, gP G) -> {G : {group gT} | maxgroup G gP}.
Proof.
move=> exP; have [A maxA]: {A | maxgroup A gP}.
apply: ex_maxset; case: exP => G gPG.
by exists (G : {set gT}); rewrite groupP genGidG.
by exists <<A>>%G; rewrite /= gen_set_id; case/andP: (maxsetp maxA).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
ex_maxgroup
| |
ex_mingroup: (exists G, gP G) -> {G : {group gT} | mingroup G gP}.
Proof.
move=> exP; have [A minA]: {A | mingroup A gP}.
apply: ex_minset; case: exP => G gPG.
by exists (G : {set gT}); rewrite groupP genGidG.
by exists <<A>>%G; rewrite /= gen_set_id; case/andP: (minsetp minA).
Qed.
Variable G : {group gT}.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
ex_mingroup
| |
mingroupP:
reflect (gP G /\ forall H, gP H -> H \subset G -> H :=: G) (mingroup G gP).
Proof.
apply: (iffP minsetP); rewrite /= groupP genGidG /= => [] [-> minG].
by split=> // H gPH sGH; apply: minG; rewrite // groupP genGidG.
by split=> // A; case/andP=> gA gPA; rewrite -(gen_set_id gA); apply: minG.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
mingroupP
| |
maxgroupP:
reflect (gP G /\ forall H, gP H -> G \subset H -> H :=: G) (maxgroup G gP).
Proof.
apply: (iffP maxsetP); rewrite /= groupP genGidG /= => [] [-> maxG].
by split=> // H gPH sGH; apply: maxG; rewrite // groupP genGidG.
by split=> // A; case/andP=> gA gPA; rewrite -(gen_set_id gA); apply: maxG.
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
maxgroupP
| |
maxgroupp: maxgroup G gP -> gP G. Proof. by case/maxgroupP. Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
maxgroupp
| |
mingroupp: mingroup G gP -> gP G. Proof. by case/mingroupP. Qed.
Hypothesis gPG : gP G.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
mingroupp
| |
maxgroup_exists: {H : {group gT} | maxgroup H gP & G \subset H}.
Proof.
have [A maxA sGA]: {A | maxgroup A gP & G \subset A}.
by apply: maxset_exists; rewrite groupP genGidG.
by exists <<A>>%G; rewrite /= gen_set_id; case/andP: (maxsetp maxA).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
maxgroup_exists
| |
mingroup_exists: {H : {group gT} | mingroup H gP & H \subset G}.
Proof.
have [A maxA sGA]: {A | mingroup A gP & A \subset G}.
by apply: minset_exists; rewrite groupP genGidG.
by exists <<A>>%G; rewrite /= gen_set_id; case/andP: (minsetp maxA).
Qed.
|
Lemma
|
fingroup
|
[
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype div path tuple bigop prime finset",
"From mathcomp Require Export monoid"
] |
fingroup/fingroup.v
|
mingroup_exists
| |
partial_productA B :=
if A == 1 then B else if B == 1 then A else
if [&& group_set A, group_set B & B \subset 'N(A)] then A * B else set0.
|
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
|
partial_product
| |
semidirect_productA B :=
if A :&: B \subset 1%G then partial_product A B else set0.
|
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
|
semidirect_product
| |
central_productA B :=
if B \subset 'C(A) then partial_product A B else set0.
|
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
|
central_product
| |
direct_productA B :=
if A :&: B \subset 1%G then central_product A B else set0.
|
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
|
direct_product
| |
complements_to_inA B :=
[set K : {group gT} | A :&: K == 1 & A * K == B].
|
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
|
complements_to_in
| |
splits_overB A := complements_to_in A B != set0.
|
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
|
splits_over
| |
remgrA B x := repr (A :* x :&: B).
|
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
|
remgr
| |
divgrA B x := x * (remgr A B x)^-1.
|
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
|
divgr
| |
pprod:= (partial_product _).
|
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
|
pprod
| |
sdprod:= (semidirect_product _).
|
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
|
sdprod
| |
cprod:= (central_product _).
|
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
|
cprod
| |
dprod:= (direct_product _).
|
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
|
dprod
| |
pprod1g: left_id 1 pprod.
Proof. by move=> A; rewrite /pprod 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
|
pprod1g
| |
pprodg1: right_id 1 pprod.
Proof. by move=> A; rewrite /pprod eqxx; case: eqP. 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
|
pprodg1
| |
are_groupsA B : Prop := AreGroups K H of A = K & B = H.
|
Variant
|
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
|
are_groups
| |
group_not0G : set0 <> G.
Proof. by move/setP/(_ 1); rewrite inE group1. 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_not0
| |
mulg0: right_zero (@set0 gT) mul.
Proof.
by move=> A; apply/setP=> x; rewrite inE; apply/imset2P=> [[y z]]; rewrite inE.
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
|
mulg0
| |
mul0g: left_zero (@set0 gT) mul.
Proof.
by move=> A; apply/setP=> x; rewrite inE; apply/imset2P=> [[y z]]; rewrite inE.
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
|
mul0g
| |
pprodPA B G :
pprod A B = G -> [/\ are_groups A B, A * B = G & B \subset 'N(A)].
Proof.
have Gnot0 := @group_not0 G; rewrite /pprod; do 2?case: eqP => [-> ->| _].
- by rewrite mul1g norms1; split; first exists 1%G G.
- by rewrite mulg1 sub1G; split; first exists G 1%G.
by case: and3P => // [[gA gB ->]]; split; first exists (Group gA) (Group gB).
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
|
pprodP
| |
pprodEK H : H \subset 'N(K) -> pprod K H = K * H.
Proof.
move=> nKH; rewrite /pprod nKH !groupP /=.
by do 2?case: eqP => [-> | _]; rewrite ?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
|
pprodE
| |
pprodEYK H : H \subset 'N(K) -> pprod K H = K <*> H.
Proof. by move=> nKH; rewrite pprodE ?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
|
pprodEY
| |
pprodWA B G : pprod A B = G -> A * B = G. Proof. by case/pprodP. 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
|
pprodW
| |
pprodWCA B G : pprod A B = G -> B * A = G.
Proof. by case/pprodP=> _ <- /normC. 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
|
pprodWC
| |
pprodWYA B G : pprod A B = G -> A <*> B = G.
Proof. by case/pprodP=> [[K H -> ->] <- /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
|
pprodWY
| |
pprodJA B x : pprod A B :^ x = pprod (A :^ x) (B :^ x).
Proof.
rewrite /pprod !conjsg_eq1 !group_setJ normJ conjSg -conjsMg.
by do 3?case: ifP => // _; apply: conj0g.
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
|
pprodJ
| |
remgrMlK B x y : y \in K -> remgr K B (y * x) = remgr K B x.
Proof. by move=> Ky; rewrite {1}/remgr rcosetM rcoset_id. 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
|
remgrMl
| |
remgrPK B x : (remgr K B x \in K :* x :&: B) = (x \in K * B).
Proof.
set y := _ x; apply/idP/mulsgP=> [|[g b Kg Bb x_gb]].
rewrite inE rcoset_sym mem_rcoset => /andP[Kxy' By].
by exists (x * y^-1) y; rewrite ?mulgKV.
by apply: (mem_repr b); rewrite inE rcoset_sym mem_rcoset x_gb mulgK Kg.
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
|
remgrP
| |
remgr1K H x : x \in K -> remgr K H x = 1.
Proof. by move=> Kx; rewrite /remgr rcoset_id ?repr_group. 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
|
remgr1
| |
divgr_eqA B x : x = divgr A B x * remgr A B x.
Proof. by rewrite 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
|
divgr_eq
| |
divgrMlK B x y : x \in K -> divgr K B (x * y) = x * divgr K B y.
Proof. by move=> Hx; rewrite /divgr remgrMl ?mulgA. 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
|
divgrMl
| |
divgr_idK H x : x \in K -> divgr K H x = x.
Proof. by move=> Kx; rewrite /divgr remgr1 // invg1 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
|
divgr_id
| |
mem_remgrK B x : x \in K * B -> remgr K B x \in B.
Proof. by rewrite -remgrP => /setIP[]. 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
|
mem_remgr
| |
mem_divgrK B x : x \in K * B -> divgr K B x \in K.
Proof. by rewrite -remgrP inE rcoset_sym mem_rcoset => /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
|
mem_divgr
| |
remgr_idx : x \in H -> remgr K H x = x.
Proof.
move=> Hx; apply/eqP; rewrite eq_mulgV1 (sameP eqP set1gP) -tiKH inE.
rewrite -mem_rcoset groupMr ?groupV // -in_setI remgrP.
by apply: subsetP Hx; apply: mulG_subr.
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
|
remgr_id
| |
remgrMidx y : x \in K -> y \in H -> remgr K H (x * y) = y.
Proof. by move=> Kx Hy; rewrite remgrMl ?remgr_id. 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
|
remgrMid
| |
divgrMidx y : x \in K -> y \in H -> divgr K H (x * y) = x.
Proof. by move=> Kx Hy; rewrite /divgr remgrMid ?mulgK. 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
|
divgrMid
| |
subcent_TImulgK H A :
K :&: H = 1 -> A \subset 'N(K) :&: 'N(H) -> 'C_K(A) * 'C_H(A) = 'C_(K * H)(A).
Proof.
move=> tiKH /subsetIP[nKA nHA]; apply/eqP.
rewrite group_modl ?subsetIr // eqEsubset setSI ?mulSg ?subsetIl //=.
apply/subsetP=> _ /setIP[/mulsgP[x y Kx Hy ->] cAxy].
rewrite inE cAxy mem_mulg // inE Kx /=.
apply/centP=> z Az; apply/commgP/conjg_fixP.
move/commgP/conjg_fixP/(congr1 (divgr K H)): (centP cAxy z Az).
by rewrite conjMg !divgrMid ?memJ_norm // (subsetP nKA, subsetP nHA).
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
|
subcent_TImulg
| |
complPH A B :
reflect (A :&: H = 1 /\ A * H = B) (H \in [complements to A in B]).
Proof. by apply: (iffP setIdP); case; split; apply/eqP. 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
|
complP
| |
splitsPB A :
reflect (exists H, H \in [complements to A in B]) [splits B, over A].
Proof. exact: set0Pn. 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
|
splitsP
| |
complgCH K G :
(H \in [complements to K in G]) = (K \in [complements to H in G]).
Proof.
rewrite !inE setIC; congr (_ && _).
by apply/eqP/eqP=> defG; rewrite -(comm_group_setP _) // defG 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
|
complgC
| |
remgrM: K <| G -> {in G &, {morph remgr K H : x y / x * y}}.
Proof.
case/normalP=> _; case/complP: complH_K => tiKH <- nK_KH x y KHx KHy.
rewrite {1}(divgr_eq K H y) mulgA (conjgCV x) {2}(divgr_eq K H x) -mulgA.
rewrite -[X in _ * X]mulgA mulgA remgrMid //; last first.
by rewrite groupMl mem_remgr.
by rewrite groupMl !(=^~ mem_conjg, nK_KH, mem_divgr).
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
|
remgrM
| |
divgrM: H \subset 'C(K) -> {in G &, {morph divgr K H : x y / x * y}}.
Proof.
move=> cKH; have /complP[_ defG] := complH_K.
have nsKG: K <| G by rewrite -defG -cent_joinEr // normalYl cents_norm.
move=> x y Gx Gy; rewrite {1}/divgr remgrM // invMg -!mulgA (mulgA y).
by congr (_ * _); rewrite -(centsP cKH) ?groupV ?(mem_remgr, mem_divgr, defG).
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
|
divgrM
| |
sdprod1g: left_id 1 sdprod.
Proof. by move=> A; rewrite /sdprod subsetIl pprod1g. 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
|
sdprod1g
| |
sdprodg1: right_id 1 sdprod.
Proof. by move=> A; rewrite /sdprod subsetIr pprodg1. 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
|
sdprodg1
| |
sdprodPA B G :
A ><| B = G -> [/\ are_groups A B, A * B = G, B \subset 'N(A) & A :&: B = 1].
Proof.
rewrite /sdprod; case: ifP => [trAB | _ /group_not0[] //].
case/pprodP=> gAB defG nBA; split=> {defG nBA}//.
by case: gAB trAB => H K -> -> /trivgP.
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
|
sdprodP
| |
sdprodEK H : H \subset 'N(K) -> K :&: H = 1 -> K ><| H = K * H.
Proof. by move=> nKH tiKH; rewrite /sdprod tiKH subxx pprodE. 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
|
sdprodE
| |
sdprodEYK H : H \subset 'N(K) -> K :&: H = 1 -> K ><| H = K <*> H.
Proof. by move=> nKH tiKH; rewrite sdprodE ?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
|
sdprodEY
| |
sdprodWppA B G : A ><| B = G -> pprod A B = G.
Proof. by case/sdprodP=> [[K H -> ->] <- /pprodE]. 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
|
sdprodWpp
| |
sdprodWA B G : A ><| B = G -> A * B = G.
Proof. by move/sdprodWpp/pprodW. 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
|
sdprodW
| |
sdprodWCA B G : A ><| B = G -> B * A = G.
Proof. by move/sdprodWpp/pprodWC. 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
|
sdprodWC
| |
sdprodWYA B G : A ><| B = G -> A <*> B = G.
Proof. by move/sdprodWpp/pprodWY. 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
|
sdprodWY
| |
sdprodJA B x : (A ><| B) :^ x = A :^ x ><| B :^ x.
Proof.
rewrite /sdprod -conjIg sub_conjg conjs1g -pprodJ.
by case: ifP => _ //; apply: imset0.
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
|
sdprodJ
| |
sdprod_contextG K H : K ><| H = G ->
[/\ K <| G, H \subset G, K * H = G, H \subset 'N(K) & K :&: H = 1].
Proof.
case/sdprodP=> _ <- nKH tiKH.
by rewrite /normal mulG_subl mulG_subr mulG_subG normG.
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_context
| |
sdprod_complG K H : K ><| H = G -> H \in [complements to K in G].
Proof. by case/sdprodP=> _ mulKH _ tiKH; apply/complP. 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_compl
| |
sdprod_normal_complPG K H :
K <| G -> reflect (K ><| H = G) (K \in [complements to H in G]).
Proof.
case/andP=> _ nKG; rewrite complgC.
apply: (iffP idP); [case/complP=> tiKH mulKH | exact: sdprod_compl].
by rewrite sdprodE ?(subset_trans _ nKG) // -mulKH mulG_subr.
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_normal_complP
| |
sdprod_cardG A B : A ><| B = G -> (#|A| * #|B|)%N = #|G|.
Proof. by case/sdprodP=> [[H K -> ->] <- _ /TI_cardMg]. 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_card
| |
sdprod_isomG A B :
A ><| B = G ->
{nAB : B \subset 'N(A) | isom B (G / A) (restrm nAB (coset A))}.
Proof.
case/sdprodP=> [[K H -> ->] <- nKH tiKH].
by exists nKH; rewrite quotientMidl quotient_isom.
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_isom
| |
sdprod_isogG A B : A ><| B = G -> B \isog G / A.
Proof. by case/sdprod_isom=> nAB; apply: isom_isog. 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_isog
| |
sdprod_subrG A B M : A ><| B = G -> M \subset B -> A ><| M = A <*> M.
Proof.
case/sdprodP=> [[K H -> ->] _ nKH tiKH] sMH.
by rewrite sdprodEY ?(subset_trans sMH) //; apply/trivgP; rewrite -tiKH setIS.
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_subr
| |
index_sdprodG A B : A ><| B = G -> #|B| = #|G : A|.
Proof.
case/sdprodP=> [[K H -> ->] <- _ tiHK].
by rewrite indexMg -indexgI setIC tiHK indexg1.
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
|
index_sdprod
| |
index_sdprodrG A B M :
A ><| B = G -> M \subset B -> #|B : M| = #|G : A <*> M|.
Proof.
move=> defG; case/sdprodP: defG (defG) => [[K H -> ->] mulKH nKH _] defG sMH.
rewrite -!divgS //=; last by rewrite -genM_join gen_subG -mulKH mulgS.
by rewrite -(sdprod_card defG) -(sdprod_card (sdprod_subr defG sMH)) divnMl.
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
|
index_sdprodr
| |
quotient_sdprodr_isomG A B M :
A ><| B = G -> M <| B ->
{f : {morphism B / M >-> coset_of (A <*> M)} |
isom (B / M) (G / (A <*> M)) f
& forall L, L \subset B -> f @* (L / M) = A <*> L / (A <*> M)}.
Proof.
move=> defG nsMH; have [defA defB]: A = <<A>>%G /\ B = <<B>>%G.
by have [[K1 H1 -> ->] _ _ _] := sdprodP defG; rewrite /= !genGid.
do [rewrite {}defA {}defB; move: {A}<<A>>%G {B}<<B>>%G => K H] in defG nsMH *.
have [[nKH /isomP[injKH imKH]] sMH] := (sdprod_isom defG, normal_sub nsMH).
have [[nsKG sHG mulKH _ _] nKM] := (sdprod_context defG, subset_trans sMH nKH).
have nsKMG: K <*> M <| G.
by rewrite -quotientYK // -mulKH -quotientK ?cosetpre_normal ?quotient_normal.
have [/= f inj_f im_f] := third_isom (joing_subl K M) nsKG nsKMG.
rewrite quotientYidl //= -imKH -(restrm_quotientE nKH sMH) in f inj_f im_f.
have /domP[h [_ ker_h _ im_h]]: 'dom (f \o quotm _ nsMH) = H / M.
by rewrite ['dom _]morphpre_quotm injmK.
have{} im_h L: L \subset H -> h @* (L / M) = K <*> L / (K <*> M).
move=> sLH; have [sLG sKKM] := (subset_trans sLH sHG, joing_subl K M).
rewrite im_h morphim_comp morphim_quotm [_ @* L]restrm_quotientE ?im_f //.
rewrite quotientY ?(normsG sKKM) ?(subset_trans sLG) ?normal_norm //.
by rewrite (quotientS1 sKKM) joing1G.
exists h => //; apply/isomP; split; last by rewrite im_h //= (sdprodWY defG).
by rewrite ker_h injm_comp ?injm_quotm.
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
|
quotient_sdprodr_isom
| |
quotient_sdprodr_isogG A B M :
A ><| B = G -> M <| B -> B / M \isog G / (A <*> M).
Proof.
move=> defG; case/sdprodP: defG (defG) => [[K H -> ->] _ _ _] => defG nsMH.
by have [h /isom_isog->] := quotient_sdprodr_isom defG nsMH.
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
|
quotient_sdprodr_isog
| |
sdprod_modlA B G H :
A ><| B = G -> A \subset H -> A ><| (B :&: H) = G :&: H.
Proof.
case/sdprodP=> {A B} [[A B -> ->]] <- nAB tiAB sAH.
rewrite -group_modl ?sdprodE ?subIset ?nAB //.
by rewrite setIA tiAB (setIidPl _) ?sub1G.
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_modl
| |
sdprod_modrA B G H :
A ><| B = G -> B \subset H -> (H :&: A) ><| B = H :&: G.
Proof.
case/sdprodP=> {A B}[[A B -> ->]] <- nAB tiAB sAH.
rewrite -group_modr ?sdprodE ?normsI // ?normsG //.
by rewrite -setIA tiAB (setIidPr _) ?sub1G.
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_modr
| |
subcent_sdprodB C G A :
B ><| C = G -> A \subset 'N(B) :&: 'N(C) -> 'C_B(A) ><| 'C_C(A) = 'C_G(A).
Proof.
case/sdprodP=> [[H K -> ->] <- nHK tiHK] nHKA {B C G}.
rewrite sdprodE ?subcent_TImulg ?normsIG //.
by rewrite -setIIl tiHK (setIidPl (sub1G _)).
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
|
subcent_sdprod
| |
sdprod_recln G K H K1 :
#|G| <= n -> K ><| H = G -> K1 \proper K -> H \subset 'N(K1) ->
exists G1 : {group gT}, [/\ #|G1| < n, G1 \subset G & K1 ><| H = G1].
Proof.
move=> leGn; case/sdprodP=> _ defG nKH tiKH ltK1K nK1H.
have tiK1H: K1 :&: H = 1 by apply/trivgP; rewrite -tiKH setSI ?proper_sub.
exists (K1 <*> H)%G; rewrite /= -defG sdprodE // norm_joinEr //.
rewrite ?mulSg ?proper_sub ?(leq_trans _ leGn) //=.
by rewrite -defG ?TI_cardMg // ltn_pmul2r ?proper_card.
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_recl
| |
sdprod_recrn G K H H1 :
#|G| <= n -> K ><| H = G -> H1 \proper H ->
exists G1 : {group gT}, [/\ #|G1| < n, G1 \subset G & K ><| H1 = G1].
Proof.
move=> leGn; case/sdprodP=> _ defG nKH tiKH ltH1H.
have [sH1H _] := andP ltH1H; have nKH1 := subset_trans sH1H nKH.
have tiKH1: K :&: H1 = 1 by apply/trivgP; rewrite -tiKH setIS.
exists (K <*> H1)%G; rewrite /= -defG sdprodE // norm_joinEr //.
rewrite ?mulgS // ?(leq_trans _ leGn) //=.
by rewrite -defG ?TI_cardMg // ltn_pmul2l ?proper_card.
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_recr
| |
mem_sdprodG A B x : A ><| B = G -> x \in G ->
exists y, exists z,
[/\ y \in A, z \in B, x = y * z &
{in A & B, forall u t, x = u * t -> u = y /\ t = z}].
Proof.
case/sdprodP=> [[K H -> ->{A B}] <- _ tiKH] /mulsgP[y z Ky Hz ->{x}].
exists y; exists z; split=> // u t Ku Ht eqyzut.
move: (congr1 (divgr K H) eqyzut) (congr1 (remgr K H) eqyzut).
by rewrite !remgrMid // !divgrMid.
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
|
mem_sdprod
| |
cprod1g: left_id 1 cprod.
Proof. by move=> A; rewrite /cprod cents1 pprod1g. 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
|
cprod1g
| |
cprodg1: right_id 1 cprod.
Proof. by move=> A; rewrite /cprod sub1G pprodg1. 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
|
cprodg1
| |
cprodPA B G :
A \* B = G -> [/\ are_groups A B, A * B = G & B \subset 'C(A)].
Proof. by rewrite /cprod; case: ifP => [cAB /pprodP[] | _ /group_not0[]]. 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
|
cprodP
| |
cprodEG H : H \subset 'C(G) -> G \* H = G * H.
Proof. by move=> cGH; rewrite /cprod cGH pprodE ?cents_norm. 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
|
cprodE
| |
cprodEYG H : H \subset 'C(G) -> G \* H = G <*> H.
Proof. by move=> cGH; rewrite cprodE ?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
|
cprodEY
| |
cprodWppA B G : A \* B = G -> pprod A B = G.
Proof. by case/cprodP=> [[K H -> ->] <- /cents_norm/pprodE]. 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
|
cprodWpp
| |
cprodWA B G : A \* B = G -> A * B = G.
Proof. by move/cprodWpp/pprodW. 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
|
cprodW
| |
cprodWCA B G : A \* B = G -> B * A = G.
Proof. by move/cprodWpp/pprodWC. 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
|
cprodWC
| |
cprodWYA B G : A \* B = G -> A <*> B = G.
Proof. by move/cprodWpp/pprodWY. 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
|
cprodWY
| |
cprodJA B x : (A \* B) :^ x = A :^ x \* B :^ x.
Proof.
by rewrite /cprod centJ conjSg -pprodJ; case: ifP => _ //; apply: imset0.
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
|
cprodJ
| |
cprod_normal2A B G : A \* B = G -> A <| G /\ B <| G.
Proof.
case/cprodP=> [[K H -> ->] <- cKH]; rewrite -cent_joinEr //.
by rewrite normalYl normalYr !cents_norm // centsC.
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
|
cprod_normal2
| |
bigcprodWI (r : seq I) P F G :
\big[cprod/1]_(i <- r | P i) F i = G -> \prod_(i <- r | P i) F i = G.
Proof.
elim/big_rec2: _ G => // i A B _ IH G /cprodP[[_ H _ defB] <- _].
by rewrite (IH H) defB.
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
|
bigcprodW
| |
bigcprodWYI (r : seq I) P F G :
\big[cprod/1]_(i <- r | P i) F i = G -> << \bigcup_(i <- r | P i) F i >> = G.
Proof.
elim/big_rec2: _ G => [|i A B _ IH G]; first by rewrite gen0.
case/cprodP => [[K H -> defB] <- cKH].
by rewrite -[<<_>>]joing_idr (IH H) ?cent_joinEr -?defB.
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
|
bigcprodWY
| |
triv_cprodA B : (A \* B == 1) = (A == 1) && (B == 1).
Proof.
case A1: (A == 1); first by rewrite (eqP A1) cprod1g.
apply/eqP=> /cprodP[[G H defA ->]] /eqP.
by rewrite defA trivMg -defA A1.
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
|
triv_cprod
| |
cprod_ntrivA B : A != 1 -> B != 1 ->
A \* B =
if [&& group_set A, group_set B & B \subset 'C(A)] then A * B else set0.
Proof.
move=> A1 B1; rewrite /cprod; case: ifP => cAB; rewrite ?cAB ?andbF //=.
by rewrite /pprod -if_neg A1 -if_neg B1 cents_norm.
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
|
cprod_ntriv
| |
trivg0: (@set0 gT == 1) = false.
Proof. by rewrite eqEcard cards0 cards1 andbF. 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
|
trivg0
| |
group0: group_set (@set0 gT) = false.
Proof. by rewrite /group_set inE. 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
|
group0
| |
cprod0gA : set0 \* A = set0.
Proof. by rewrite /cprod centsC sub0set /pprod group0 trivg0 !if_same. 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
|
cprod0g
|
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