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 ... | 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/... | 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) ... | 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... | 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 cpro... | 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; ha... | 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 => ... | 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|; eli... | 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... | 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... | 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 def... | 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 ... | 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 ap... | 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 ... | 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... | 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 ?... | 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 /= -!a... | 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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