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 |
|---|---|---|---|---|---|---|
cprodC: commutative cprod.
Proof.
rewrite /cprod => A B; case: ifP => cAB; rewrite centsC cAB // /pprod.
by rewrite andbCA normC !cents_norm // 1?centsC //; do 2!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 | cprodC | |
cprodA: associative cprod.
Proof.
move=> A B C; case A1: (A == 1); first by rewrite (eqP A1) !cprod1g.
case B1: (B == 1); first by rewrite (eqP B1) cprod1g cprodg1.
case C1: (C == 1); first by rewrite (eqP C1) !cprodg1.
rewrite !(triv_cprod, cprod_ntriv) ?{}A1 ?{}B1 ?{}C1 //.
case: isgroupP => [[G ->{A}] | _]; last by rewrite group0.
case: (isgroupP B) => [[H ->{B}] | _]; last by rewrite group0.
case: (isgroupP C) => [[K ->{C}] | _]; last by rewrite group0 !andbF.
case cGH: (H \subset 'C(G)); case cHK: (K \subset 'C(H)); last first.
- by rewrite group0.
- by rewrite group0 /= mulG_subG cGH andbF.
- by rewrite group0 /= centM subsetI cHK !andbF.
rewrite /= mulgA mulG_subG centM subsetI cGH cHK andbT -(cent_joinEr cHK).
by rewrite -(cent_joinEr cGH) !groupP.
Qed.
HB.instance Definition _ := Monoid.isComLaw.Build {set gT} 1 cprod
cprodA cprodC cprod1g. | 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 | cprodA | |
cprod_modlA B G H :
A \* B = G -> A \subset H -> A \* (B :&: H) = G :&: H.
Proof.
case/cprodP=> [[U V -> -> {A B}]] defG cUV sUH.
by rewrite cprodE; [rewrite group_modl ?defG | rewrite subIset ?cUV].
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_modl | |
cprod_modrA B G H :
A \* B = G -> B \subset H -> (H :&: A) \* B = H :&: G.
Proof. by rewrite -!(cprodC B) !(setIC H); apply: cprod_modl. 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_modr | |
bigcprodYP(I : finType) (P : pred I) (H : I -> {group gT}) :
reflect (forall i j, P i -> P j -> i != j -> H i \subset 'C(H j))
(\big[cprod/1]_(i | P i) H i == (\prod_(i | P i) H i)%G).
Proof.
apply: (iffP eqP) => [defG i j Pi Pj neq_ij | cHH].
rewrite (bigD1 j) // (bigD1 i) /= ?cprodA in defG; last exact/andP.
by case/cprodP: defG => [[K _ /cprodP[//]]].
set Q := P; have sQP: subpred Q P by []; have [n leQn] := ubnP #|Q|.
elim: n => // n IHn in (Q) leQn sQP *.
have [i Qi | Q0] := pickP Q; last by rewrite !big_pred0.
rewrite (cardD1x Qi) add1n ltnS !(bigD1 i Qi) /= in leQn *.
rewrite {}IHn {n leQn}// => [|j /andP[/sQP //]].
rewrite bigprodGE cprodEY // gen_subG; apply/bigcupsP=> j /andP[neq_ji Qj].
by rewrite cHH ?sQP.
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 | bigcprodYP | |
bigcprodEYI r (P : pred I) (H : I -> {group gT}) G :
abelian G -> (forall i, P i -> H i \subset G) ->
\big[cprod/1]_(i <- r | P i) H i = (\prod_(i <- r | P i) H i)%G.
Proof.
move=> cGG sHG; apply/eqP; rewrite !(big_tnth _ _ r).
by apply/bigcprodYP=> i j Pi Pj _; rewrite (sub_abelian_cent2 cGG) ?sHG.
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 | bigcprodEY | |
perm_bigcprod(I : eqType) r1 r2 (A : I -> {set gT}) G x :
\big[cprod/1]_(i <- r1) A i = G -> {in r1, forall i, x i \in A i} ->
perm_eq r1 r2 ->
\prod_(i <- r1) x i = \prod_(i <- r2) x i.
Proof.
elim: r1 r2 G => [|i r1 IHr] r2 G defG Ax eq_r12.
by rewrite perm_sym in eq_r12; rewrite (perm_small_eq _ eq_r12) ?big_nil.
have /rot_to[n r3 Dr2]: i \in r2 by rewrite -(perm_mem eq_r12) mem_head.
transitivity (\prod_(j <- rot n r2) x j).
rewrite Dr2 !big_cons in defG Ax *; have [[_ G1 _ defG1] _ _] := cprodP defG.
rewrite (IHr r3 G1) //; first by case/allP/andP: Ax => _ /allP.
by rewrite -(perm_cons i) -Dr2 perm_sym perm_rot perm_sym.
rewrite -(cat_take_drop n r2) [in LHS]cat_take_drop in eq_r12 *.
rewrite (perm_big _ eq_r12) !big_cat /= !(big_nth i) !big_mkord in defG *.
have /cprodP[[G1 G2 defG1 defG2] _ /centsP-> //] := defG.
rewrite defG2 -(bigcprodW defG2) mem_prodg // => k _; apply: Ax.
by rewrite (perm_mem eq_r12) mem_cat orbC mem_nth.
rewrite defG1 -(bigcprodW defG1) mem_prodg // => k _; apply: Ax.
by rewrite (perm_mem eq_r12) mem_cat mem_nth.
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 | perm_bigcprod | |
reindex_bigcprod(I J : finType) (h : J -> I) P (A : I -> {set gT}) G x :
{on SimplPred P, bijective h} -> \big[cprod/1]_(i | P i) A i = G ->
{in SimplPred P, forall i, x i \in A i} ->
\prod_(i | P i) x i = \prod_(j | P (h j)) x (h j).
Proof.
case=> h1 hK h1K defG Ax; have [e big_e [Ue mem_e] _] := big_enumP P.
rewrite -!big_e in defG *; rewrite -(big_map h P x) -[RHS]big_filter filter_map.
apply: perm_bigcprod defG _ _ => [i|]; first by rewrite mem_e => /Ax.
have [r _ [Ur /= mem_r] _] := big_enumP; apply: uniq_perm Ue _ _ => [|i].
by rewrite map_inj_in_uniq // => i j; rewrite !mem_r ; apply: (can_in_inj hK).
rewrite mem_e; apply/idP/mapP=> [Pi|[j r_j ->]]; last by rewrite -mem_r.
by exists (h1 i); rewrite ?mem_r h1K.
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 | reindex_bigcprod | |
dprod1g: left_id 1 dprod.
Proof. by move=> A; rewrite /dprod subsetIl cprod1g. 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 | dprod1g | |
dprodg1: right_id 1 dprod.
Proof. by move=> A; rewrite /dprod subsetIr cprodg1. 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 | dprodg1 | |
dprodPA B G :
A \x B = G -> [/\ are_groups A B, A * B = G, B \subset 'C(A) & A :&: B = 1].
Proof.
rewrite /dprod; case: ifP => trAB; last by case/group_not0.
by case/cprodP=> gAB; split=> //; case: gAB trAB => ? ? -> -> /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 | dprodP | |
dprodEG H : H \subset 'C(G) -> G :&: H = 1 -> G \x H = G * H.
Proof. by move=> cGH trGH; rewrite /dprod trGH sub1G cprodE. 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 | dprodE | |
dprodEYG H : H \subset 'C(G) -> G :&: H = 1 -> G \x H = G <*> H.
Proof. by move=> cGH trGH; rewrite /dprod trGH subxx cprodEY. 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 | dprodEY | |
dprodEcpA B : A :&: B = 1 -> A \x B = A \* B.
Proof. by move=> trAB; rewrite /dprod trAB 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 | dprodEcp | |
dprodEsdA B : B \subset 'C(A) -> A \x B = A ><| B.
Proof. by rewrite /dprod /cprod => ->. 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 | dprodEsd | |
dprodWcpA B G : A \x B = G -> A \* B = G.
Proof. by move=> defG; have [_ _ _ /dprodEcp <-] := dprodP 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 | dprodWcp | |
dprodWsdA B G : A \x B = G -> A ><| B = G.
Proof. by move=> defG; have [_ _ /dprodEsd <-] := dprodP 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 | dprodWsd | |
dprodWA B G : A \x B = G -> A * B = G.
Proof. by move/dprodWsd/sdprodW. 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 | dprodW | |
dprodWCA B G : A \x B = G -> B * A = G.
Proof. by move/dprodWsd/sdprodWC. 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 | dprodWC | |
dprodWYA B G : A \x B = G -> A <*> B = G.
Proof. by move/dprodWsd/sdprodWY. 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 | dprodWY | |
cprod_card_dprodG A B :
A \* B = G -> #|A| * #|B| <= #|G| -> A \x B = G.
Proof. by case/cprodP=> [[K H -> ->] <- cKH] /cardMg_TI; apply: dprodE. 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_card_dprod | |
dprodJA B x : (A \x B) :^ x = A :^ x \x B :^ x.
Proof.
rewrite /dprod -conjIg sub_conjg conjs1g -cprodJ.
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 | dprodJ | |
dprod_normal2A B G : A \x B = G -> A <| G /\ B <| G.
Proof. by move/dprodWcp/cprod_normal2. 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 | dprod_normal2 | |
dprodYPK H : reflect (K \x H = K <*> H) (H \subset 'C(K) :\: K^#).
Proof.
rewrite subsetD -setI_eq0 setIDA setD_eq0 setIC subG1 /=.
by apply: (iffP andP) => [[cKH /eqP/dprodEY->] | /dprodP[_ _ -> ->]].
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 | dprodYP | |
dprodC: commutative dprod.
Proof. by move=> A B; rewrite /dprod setIC cprodC. 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 | dprodC | |
dprodWsdCA B G : A \x B = G -> B ><| A = G.
Proof. by rewrite dprodC => /dprodWsd. 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 | dprodWsdC | |
dprodA: associative dprod.
Proof.
move=> A B C; case A1: (A == 1); first by rewrite (eqP A1) !dprod1g.
case B1: (B == 1); first by rewrite (eqP B1) dprod1g dprodg1.
case C1: (C == 1); first by rewrite (eqP C1) !dprodg1.
rewrite /dprod (fun_if (cprod A)) (fun_if (cprod^~ C)) -cprodA.
rewrite -(cprodC set0) !cprod0g cprod_ntriv ?B1 ?{}C1 //.
case: and3P B1 => [[] | _ _]; last by rewrite cprodC cprod0g !if_same.
case/isgroupP=> H ->; case/isgroupP=> K -> {B C}; move/cent_joinEr=> eHK H1.
rewrite cprod_ntriv ?trivMg ?{}A1 ?{}H1 // mulG_subG.
case: and4P => [[] | _]; last by rewrite !if_same.
case/isgroupP=> G ->{A} _ cGH _; rewrite cprodEY // -eHK.
case trGH: (G :&: H \subset _); case trHK: (H :&: K \subset _); last first.
- by rewrite !if_same.
- rewrite if_same; case: ifP => // trG_HK; case/negP: trGH.
by apply: subset_trans trG_HK; rewrite setIS ?joing_subl.
- rewrite if_same; case: ifP => // trGH_K; case/negP: trHK.
by apply: subset_trans trGH_K; rewrite setSI ?joing_subr.
do 2![case: ifP] => // trGH_K trG_HK; [case/negP: trGH_K | case/negP: trG_HK].
apply: subset_trans trHK; rewrite subsetI subsetIr -{2}(mulg1 H) -mulGS.
rewrite setIC group_modl ?joing_subr //= cent_joinEr // -eHK.
by rewrite -group_modr ?joing_subl //= setIC -(normC (sub1G _)) mulSg.
apply: subset_trans trGH; rewrite subsetI subsetIl -{2}(mul1g H) -mulSG.
rewrite setIC group_modr ?joing_subl //= eHK -(cent_joinEr cGH).
by rewrite -group_modl ?joing_subr //= setIC (normC (sub1G _)) mulgS.
Qed.
HB.instanc
... | 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 | dprodA | |
bigdprodWcpI (r : seq I) P F G :
\big[dprod/1]_(i <- r | P i) F i = G -> \big[cprod/1]_(i <- r | P i) F i = G.
Proof.
elim/big_rec2: _ G => // i A B _ IH G /dprodP[[K H -> defB] <- cKH _].
by rewrite (IH H) // cprodE -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 | bigdprodWcp | |
bigdprodWI (r : seq I) P F G :
\big[dprod/1]_(i <- r | P i) F i = G -> \prod_(i <- r | P i) F i = G.
Proof. by move/bigdprodWcp; apply: bigcprodW. 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 | bigdprodW | |
bigdprodWYI (r : seq I) P F G :
\big[dprod/1]_(i <- r | P i) F i = G -> << \bigcup_(i <- r | P i) F i >> = G.
Proof. by move/bigdprodWcp; apply: bigcprodWY. 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 | bigdprodWY | |
bigdprodYP(I : finType) (P : pred I) (F : I -> {group gT}) :
reflect (forall i, P i ->
(\prod_(j | P j && (j != i)) F j)%G \subset 'C(F i) :\: (F i)^#)
(\big[dprod/1]_(i | P i) F i == (\prod_(i | P i) F i)%G).
Proof.
apply: (iffP eqP) => [defG i Pi | dxG].
rewrite !(bigD1 i Pi) /= in defG; have [[_ G' _ defG'] _ _ _] := dprodP defG.
by apply/dprodYP; rewrite -defG defG' bigprodGE (bigdprodWY defG').
set Q := P; have sQP: subpred Q P by []; have [n leQn] := ubnP #|Q|.
elim: n => // n IHn in (Q) leQn sQP *.
have [i Qi | Q0] := pickP Q; last by rewrite !big_pred0.
rewrite (cardD1x Qi) add1n ltnS !(bigD1 i Qi) /= in leQn *.
rewrite {}IHn {n leQn}// => [|j /andP[/sQP //]].
apply/dprodYP; apply: subset_trans (dxG i (sQP i Qi)); rewrite !bigprodGE.
by apply: genS; apply/bigcupsP=> j /andP[Qj ne_ji]; rewrite (bigcup_max j) ?sQP.
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 | bigdprodYP | |
dprod_modlA B G H :
A \x B = G -> A \subset H -> A \x (B :&: H) = G :&: H.
Proof.
case/dprodP=> [[U V -> -> {A B}]] defG cUV trUV sUH.
rewrite dprodEcp; first by apply: cprod_modl; rewrite ?cprodE.
by rewrite setIA trUV (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 | dprod_modl | |
dprod_modrA B G H :
A \x B = G -> B \subset H -> (H :&: A) \x B = H :&: G.
Proof. by rewrite -!(dprodC B) !(setIC H); apply: dprod_modl. 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 | dprod_modr | |
subcent_dprodB C G A :
B \x C = G -> A \subset 'N(B) :&: 'N(C) -> 'C_B(A) \x 'C_C(A) = 'C_G(A).
Proof.
move=> defG; have [_ _ cBC _] := dprodP defG; move: defG.
by rewrite !dprodEsd 1?(centSS _ _ cBC) ?subsetIl //; apply: subcent_sdprod.
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_dprod | |
dprod_cardA B G : A \x B = G -> (#|A| * #|B|)%N = #|G|.
Proof. by case/dprodP=> [[H K -> ->] <- _]; move/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 | dprod_card | |
bigdprod_cardI r (P : pred I) E G :
\big[dprod/1]_(i <- r | P i) E i = G ->
(\prod_(i <- r | P i) #|E i|)%N = #|G|.
Proof.
elim/big_rec2: _ G => [G <- | i A B _ IH G defG]; first by rewrite cards1.
have [[_ H _ defH] _ _ _] := dprodP defG.
by rewrite -(dprod_card defG) (IH H) defH.
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 | bigdprod_card | |
bigcprod_card_dprodI r (P : pred I) (A : I -> {set gT}) G :
\big[cprod/1]_(i <- r | P i) A i = G ->
\prod_(i <- r | P i) #|A i| <= #|G| ->
\big[dprod/1]_(i <- r | P i) A i = G.
Proof.
elim: r G => [|i r IHr]; rewrite !(big_nil, big_cons) //; case: ifP => _ // G.
case/cprodP=> [[K H -> defH]]; rewrite defH => <- cKH leKH_G.
have /implyP := leq_trans leKH_G (dvdn_leq _ (dvdn_cardMg K H)).
rewrite muln_gt0 leq_pmul2l !cardG_gt0 //= => /(IHr H defH){}defH.
by rewrite defH dprodE // cardMg_TI // -(bigdprod_card defH).
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 | bigcprod_card_dprod | |
bigcprod_coprime_dprod(I : finType) (P : pred I) (A : I -> {set gT}) G :
\big[cprod/1]_(i | P i) A i = G ->
(forall i j, P i -> P j -> i != j -> coprime #|A i| #|A j|) ->
\big[dprod/1]_(i | P i) A i = G.
Proof.
move=> defG coA; set Q := P in defG *; have sQP: subpred Q P by [].
have [m leQm] := ubnP #|Q|; elim: m => // m IHm in (Q) leQm G defG sQP *.
have [i Qi | Q0] := pickP Q; last by rewrite !big_pred0 in defG *.
move: defG; rewrite !(bigD1 i Qi) /= => /cprodP[[Hi Gi defAi defGi] <-].
rewrite defAi defGi => cHGi.
have{} defGi: \big[dprod/1]_(j | Q j && (j != i)) A j = Gi.
by apply: IHm => [||j /andP[/sQP]] //; rewrite (cardD1x Qi) in leQm.
rewrite defGi dprodE // coprime_TIg // -defAi -(bigdprod_card defGi).
elim/big_rec: _ => [|j n /andP[neq_ji Qj] IHn]; first exact: coprimen1.
by rewrite coprimeMr coprime_sym coA ?sQP.
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 | bigcprod_coprime_dprod | |
mem_dprodG A B x : A \x 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.
move=> defG; have [_ _ cBA _] := dprodP defG.
by apply: mem_sdprod; rewrite -dprodEsd.
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_dprod | |
mem_bigdprod(I : finType) (P : pred I) F G x :
\big[dprod/1]_(i | P i) F i = G -> x \in G ->
exists c, [/\ forall i, P i -> c i \in F i, x = \prod_(i | P i) c i
& forall e, (forall i, P i -> e i \in F i) ->
x = \prod_(i | P i) e i ->
forall i, P i -> e i = c i].
Proof.
move=> defG; rewrite -(bigdprodW defG) => /prodsgP[c Fc ->].
have [r big_r [_ mem_r] _] := big_enumP P.
exists c; split=> // e Fe eq_ce i Pi; rewrite -!{}big_r in defG eq_ce.
have{Pi}: i \in r by rewrite mem_r.
have{mem_r}: all P r by apply/allP=> j; rewrite mem_r.
elim: r G defG eq_ce => // j r IHr G.
rewrite !big_cons inE /= => /dprodP[[K H defK defH] _ _].
rewrite defK defH => tiFjH eq_ce /andP[Pj Pr].
suffices{i IHr} eq_cej: c j = e j.
case/predU1P=> [-> //|]; apply: IHr defH _ Pr.
by apply: (mulgI (c j)); rewrite eq_ce eq_cej.
rewrite !(big_nth j) !big_mkord in defH eq_ce.
move/(congr1 (divgr K H)): eq_ce; move/bigdprodW: defH => defH.
move/(all_nthP j) in Pr.
by rewrite !divgrMid // -?defK -?defH ?mem_prodg // => *; rewrite ?Fc ?Fe ?Pr.
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_bigdprod | |
comm_prodGI r (G : I -> {group gT}) (P : {pred I}) :
{in P &, forall i j, commute (G i) (G j)} ->
(\prod_(i <- r | P i) G i)%G = \prod_(i <- r | P i) G i :> {set gT}.
Proof.
elim: r => /= [|i {}r IHr]; rewrite !(big_nil, big_cons)//=.
case: ifP => //= Pi Gcomm; rewrite comm_joingE {}IHr// /commute.
elim: r => [|j r IHr]; first by rewrite big_nil mulg1 mul1g.
by rewrite big_cons; case: ifP => //= Pj; rewrite mulgA Gcomm// -!mulgA IHr.
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 | comm_prodG | |
morphim_pprod: pprod K H = G -> pprod (f @* K) (f @* H) = f @* G.
Proof.
case/pprodP=> _ defG mKH; rewrite pprodE ?morphim_norms //.
by rewrite -morphimMl ?(subset_trans _ sGD) -?defG // mulG_subl.
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_pprod | |
morphim_coprime_sdprod:
K ><| H = G -> coprime #|K| #|H| -> f @* K ><| f @* H = f @* G.
Proof.
rewrite /sdprod => defG coHK; move: defG.
by rewrite !coprime_TIg ?coprime_morph // !subxx; apply: morphim_pprod.
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_coprime_sdprod | |
injm_sdprod: 'injm f -> K ><| H = G -> f @* K ><| f @* H = f @* G.
Proof.
move=> inj_f; case/sdprodP=> _ defG nKH tiKH.
by rewrite /sdprod -injmI // tiKH morphim1 subxx morphim_pprod // 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 | injm_sdprod | |
morphim_cprod: K \* H = G -> f @* K \* f @* H = f @* G.
Proof.
case/cprodP=> _ defG cKH; rewrite /cprod morphim_cents // morphim_pprod //.
by rewrite pprodE // 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 | morphim_cprod | |
injm_dprod: 'injm f -> K \x H = G -> f @* K \x f @* H = f @* G.
Proof.
move=> inj_f; case/dprodP=> _ defG cHK tiKH.
by rewrite /dprod -injmI // tiKH morphim1 subxx morphim_cprod // cprodE.
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_dprod | |
morphim_coprime_dprod:
K \x H = G -> coprime #|K| #|H| -> f @* K \x f @* H = f @* G.
Proof.
rewrite /dprod => defG coHK; move: defG.
by rewrite !coprime_TIg ?coprime_morph // !subxx; apply: morphim_cprod.
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_coprime_dprod | |
morphim_bigcprodI r (P : pred I) (H : I -> {group gT}) G :
G \subset D -> \big[cprod/1]_(i <- r | P i) H i = G ->
\big[cprod/1]_(i <- r | P i) f @* H i = f @* G.
Proof.
elim/big_rec2: _ G => [|i fB B Pi def_fB] G sGD defG.
by rewrite -defG morphim1.
case/cprodP: defG (defG) => [[Hi Gi -> defB] _ _]; rewrite defB => defG.
rewrite (def_fB Gi) //; first exact: morphim_cprod.
by apply: subset_trans sGD; case/cprod_normal2: defG => _ /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 | morphim_bigcprod | |
injm_bigdprodI r (P : pred I) (H : I -> {group gT}) G :
G \subset D -> 'injm f -> \big[dprod/1]_(i <- r | P i) H i = G ->
\big[dprod/1]_(i <- r | P i) f @* H i = f @* G.
Proof.
move=> sGD injf; elim/big_rec2: _ G sGD => [|i fB B Pi def_fB] G sGD defG.
by rewrite -defG morphim1.
case/dprodP: defG (defG) => [[Hi Gi -> defB] _ _ _]; rewrite defB => defG.
rewrite (def_fB Gi) //; first exact: injm_dprod.
by apply: subset_trans sGD; case/dprod_normal2: defG => _ /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_bigdprod | |
morphim_coprime_bigdprod(I : finType) P (H : I -> {group gT}) G :
G \subset D -> \big[dprod/1]_(i | P i) H i = G ->
(forall i j, P i -> P j -> i != j -> coprime #|H i| #|H j|) ->
\big[dprod/1]_(i | P i) f @* H i = f @* G.
Proof.
move=> sGD /bigdprodWcp defG coH; have def_fG := morphim_bigcprod sGD defG.
by apply: bigcprod_coprime_dprod => // i j *; rewrite coprime_morph ?coH.
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_coprime_bigdprod | |
quotient_pprod: pprod K H = G -> pprod (K / M) (H / M) = G / M.
Proof. exact: morphim_pprod. 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_pprod | |
quotient_coprime_sdprod:
K ><| H = G -> coprime #|K| #|H| -> (K / M) ><| (H / M) = G / M.
Proof. exact: morphim_coprime_sdprod. 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_coprime_sdprod | |
quotient_cprod: K \* H = G -> (K / M) \* (H / M) = G / M.
Proof. exact: morphim_cprod. 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_cprod | |
quotient_coprime_dprod:
K \x H = G -> coprime #|K| #|H| -> (K / M) \x (H / M) = G / M.
Proof. exact: morphim_coprime_dprod. 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_coprime_dprod | |
extprod_mulg(x y : gT1 * gT2) := (x.1 * y.1, x.2 * y.2). | 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 | extprod_mulg | |
extprod_invg(x : gT1 * gT2) := (x.1^-1, x.2^-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 | extprod_invg | |
extprod_mul1g: left_id (1, 1) extprod_mulg.
Proof. by case=> x1 x2; congr (_, _); apply: 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 | extprod_mul1g | |
extprod_mulVg: left_inverse (1, 1) extprod_invg extprod_mulg.
Proof. by move=> x; congr (_, _); apply: 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 | extprod_mulVg | |
extprod_mulgA: associative extprod_mulg.
Proof. by move=> x y z; congr (_, _); apply: mulgA. Qed.
HB.instance Definition _ := Finite_isGroup.Build (gT1 * gT2)%type
extprod_mulgA extprod_mul1g extprod_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 | extprod_mulgA | |
group_setX(H1 : {group gT1}) (H2 : {group gT2}) : group_set (setX H1 H2).
Proof.
apply/group_setP; split; first by rewrite !inE !group1.
by case=> [x1 x2] [y1 y2] /[!inE] /andP[Hx1 Hx2] /andP[Hy1 Hy2] /[!groupM].
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_setX | |
setX_groupH1 H2 := Group (group_setX H1 H2). | 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 | setX_group | |
pairg1x : gT1 * gT2 := (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 | pairg1 | |
pair1gx : gT1 * gT2 := (1, 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 | pair1g | |
pairg1_morphM: {morph pairg1 : x y / x * y}.
Proof. by move=> x y /=; rewrite {2}/mul /= /mul_pair/= 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 | pairg1_morphM | |
pairg1_morphism:= @Morphism _ _ setT _ (in2W pairg1_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 | pairg1_morphism | |
pair1g_morphM: {morph pair1g : x y / x * y}.
Proof. by move=> x y /=; rewrite {2}/mul /= /mul_pair/= 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 | pair1g_morphM | |
pair1g_morphism:= @Morphism _ _ setT _ (in2W pair1g_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 | pair1g_morphism | |
fst_morphM: {morph (@fst gT1 gT2) : x y / x * y}.
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 | fst_morphM | |
snd_morphM: {morph (@snd gT1 gT2) : x y / x * y}.
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 | snd_morphM | |
fst_morphism:= @Morphism _ _ setT _ (in2W fst_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 | fst_morphism | |
snd_morphism:= @Morphism _ _ setT _ (in2W snd_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 | snd_morphism | |
injm_pair1g: 'injm pair1g.
Proof. by apply/subsetP=> x /morphpreP[_ /set1P[->]]; 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_pair1g | |
injm_pairg1: 'injm pairg1.
Proof. by apply/subsetP=> x /morphpreP[_ /set1P[->]]; 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_pairg1 | |
morphim_pairg1(H1 : {set gT1}) : pairg1 @* H1 = setX H1 1.
Proof. by rewrite -imset2_pair imset2_set1r morphimEsub ?subsetT. 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_pairg1 | |
morphim_pair1g(H2 : {set gT2}) : pair1g @* H2 = setX 1 H2.
Proof. by rewrite -imset2_pair imset2_set1l morphimEsub ?subsetT. 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_pair1g | |
morphim_fstX(H1: {set gT1}) (H2 : {group gT2}) :
[morphism of fun x => x.1] @* setX H1 H2 = H1.
Proof.
apply/eqP; rewrite eqEsubset morphimE setTI /=.
apply/andP; split; apply/subsetP=> x.
by case/imsetP=> x0 /[1!inE] /andP[Hx1 _] ->.
move=> Hx1; apply/imsetP; exists (x, 1); last by trivial.
by rewrite in_setX Hx1 /=.
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_fstX | |
morphim_sndX(H1: {group gT1}) (H2 : {set gT2}) :
[morphism of fun x => x.2] @* setX H1 H2 = H2.
Proof.
apply/eqP; rewrite eqEsubset morphimE setTI /=.
apply/andP; split; apply/subsetP=> x.
by case/imsetP=> x0 /[1!inE] /andP[_ Hx2] ->.
move=> Hx2; apply/imsetP; exists (1, x); last by [].
by rewrite in_setX Hx2 andbT.
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_sndX | |
setX_prod(H1 : {set gT1}) (H2 : {set gT2}) :
setX H1 1 * setX 1 H2 = setX H1 H2.
Proof.
apply/setP=> [[x y]]; rewrite !inE /=.
apply/imset2P/andP=> [[[x1 u1] [v1 y1]] | [Hx Hy]].
rewrite !inE /= => /andP[Hx1 /eqP->] /andP[/eqP-> Hx] [-> ->].
by rewrite mulg1 mul1g.
exists (x, 1 : gT2) (1 : gT1, y); rewrite ?inE ?Hx ?eqxx //.
by rewrite /mul /= /mul_pair /= 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 | setX_prod | |
setX_dprod(H1 : {group gT1}) (H2 : {group gT2}) :
setX H1 1 \x setX 1 H2 = setX H1 H2.
Proof.
rewrite dprodE ?setX_prod //.
apply/centsP=> [[x u]] /[!inE]/= /andP[/eqP-> _] [v y].
by rewrite !inE /= => /andP[_ /eqP->]; congr (_, _); rewrite ?mul1g ?mulg1.
apply/trivgP; apply/subsetP=> [[x y]]; rewrite !inE /= -!andbA.
by case/and4P=> _ /eqP-> /eqP->; rewrite 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 | setX_dprod | |
isog_setX1(H1 : {group gT1}) : isog H1 (setX H1 1).
Proof.
apply/isogP; exists [morphism of restrm (subsetT H1) pairg1].
by rewrite injm_restrm ?injm_pairg1.
by rewrite morphim_restrm morphim_pairg1 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_setX1 | |
isog_set1X(H2 : {group gT2}) : isog H2 (setX 1 H2).
Proof.
apply/isogP; exists [morphism of restrm (subsetT H2) pair1g].
by rewrite injm_restrm ?injm_pair1g.
by rewrite morphim_restrm morphim_pair1g 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_set1X | |
setX_gen(H1 : {set gT1}) (H2 : {set gT2}) :
1 \in H1 -> 1 \in H2 -> <<setX H1 H2>> = setX <<H1>> <<H2>>.
Proof.
move=> H1_1 H2_1; apply/eqP.
rewrite eqEsubset gen_subG setXS ?subset_gen //. | 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 | setX_gen | |
gTn:= {dffun forall i, gT i}.
Implicit Types (H : forall i, {group gT i}) (x y : {dffun forall i, gT i}). | 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 | gTn | |
extnprod_mulg(x y : gTn) : gTn := [ffun i => (x i * y i)%g]. | 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 | extnprod_mulg | |
extnprod_invg(x : gTn) : gTn := [ffun i => (x i)^-1%g]. | 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 | extnprod_invg | |
extnprod_mul1g: left_id [ffun=> 1%g] extnprod_mulg.
Proof. by move=> x; apply/ffunP => i; rewrite !ffunE 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 | extnprod_mul1g | |
extnprod_mulVg: left_inverse [ffun=> 1%g] extnprod_invg extnprod_mulg.
Proof. by move=> x; apply/ffunP => i; rewrite !ffunE 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 | extnprod_mulVg | |
extnprod_mulgA: associative extnprod_mulg.
Proof. by move=> x y z; apply/ffunP => i; rewrite !ffunE mulgA. Qed.
HB.instance Definition _ := Finite_isGroup.Build {dffun forall i : I, gT i}
extnprod_mulgA extnprod_mul1g extnprod_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 | extnprod_mulgA | |
oneg_ffuni : (1 : gTn) i = 1. Proof. by rewrite ffunE. 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 | oneg_ffun | |
mulg_ffuni (x y : gTn) : (x * y) i = x i * y i.
Proof. by rewrite ffunE. 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 | mulg_ffun | |
invg_ffuni (x : gTn) : x^-1 i = (x i)^-1.
Proof. by rewrite ffunE. 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 | invg_ffun | |
prodg_ffunT (r : seq T) (F : T -> gTn) (P : {pred T}) i :
(\prod_(t <- r | P t) F t) i = \prod_(t <- r | P t) F t i.
Proof. exact: (big_morph _ (@mulg_ffun i) (@oneg_ffun i)). 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 | prodg_ffun | |
group_setXnH : group_set (setXn H).
Proof.
by apply/group_setP; split=> [|x y] /[!inE]/= => [|/forallP xH /forallP yH];
apply/forallP => i; rewrite ?ffunE (group1, groupM)// ?xH ?yH.
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_setXn | |
setXn_groupH := Group (group_setXn 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 | setXn_group | |
dfung1i (g : gT i) : gTn := finfun (dfwith (fun=> 1 : gT _) g). | 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 | dfung1 | |
dfung1_idi (g : gT i) : dfung1 g i = g.
Proof. by rewrite ffunE dfwith_in. 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 | dfung1_id | |
dfung1_dflti (g : gT i) j : i != j -> dfung1 g j = 1.
Proof. by move=> ij; 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 | dfung1_dflt | |
dfung1_morphMi : {morph @dfung1 i : g h / g * h}.
Proof.
move=> g h; apply/ffunP=> j; have [{j}<-|nij] := eqVneq i j.
by rewrite !(dfung1_id, ffunE).
by rewrite !(dfung1_dflt, ffunE)// 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 | dfung1_morphM | |
dfung1_morphismi := @Morphism _ _ setT _ (in2W (@dfung1_morphM 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 | dfung1_morphism | |
dffunMi : {morph (fun x => x i) : x y / x * y}.
Proof. by move=> x y; rewrite !ffunE. 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 | dffunM |
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