phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
subg_inv u	:= Subg (groupVr (subgP u)).	Definition	subg_inv	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "groupVr", "subgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subg_mul u v	:= Subg (groupM (subgP u) (subgP v)).	Definition	subg_mul	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "groupM", "subgP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subg_oneP : left_id subg_one subg_mul.	Proof. by move=> u; apply: val_inj; apply: mul1g. Qed.	Lemma	subg_oneP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "mul1g", "subg_mul", "subg_one", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subg_invP : left_inverse subg_one subg_inv subg_mul.	Proof. by move=> u; apply: val_inj; apply: mulVg. Qed.	Lemma	subg_invP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "mulVg", "subg_inv", "subg_mul", "subg_one", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subg_mulP : associative subg_mul.	Proof. by move=> u v w; apply: val_inj; apply: mulgA. Qed.	Lemma	subg_mulP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "mulgA", "subg_mul", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sgvalM : {in setT &, {morph sgval : x y / x * y}}.	Proof. by []. Qed.	Lemma	sgvalM	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "setT", "sgval" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
valgM : {in setT &, {morph val : x y / (x : subg_of) * y >-> x * y}}.	Proof. by []. Qed.	Lemma	valgM	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "setT", "subg_of", "val" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subg : gT -> subg_of	:= insubd (1 : subg_of).	Definition	subg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "gT", "insubd", "subg_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subgK x : x \in G -> val (subg x) = x.	Proof. by move=> Gx; rewrite insubdK. Qed.	Lemma	subgK	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "insubdK", "subg", "val" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sgvalK : cancel sgval subg.	Proof. by case=> x Gx; apply: val_inj; apply: subgK. Qed.	Lemma	sgvalK	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "sgval", "subg", "subgK", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subg_default x : (x \in G) = false -> val (subg x) = 1.	Proof. by move=> Gx; rewrite val_insubd Gx. Qed.	Lemma	subg_default	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "subg", "val", "val_insubd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subgM : {in G &, {morph subg : x y / x * y}}.	Proof. by move=> x y Gx Gy; apply: val_inj; rewrite /= !subgK ?groupM. Qed.	Lemma	subgM	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "groupM", "subg", "subgK", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupD1_inj G H : G^# = H^# -> G :=: H.	Proof. by move/(congr1 (setU 1)); rewrite !setD1K. Qed.	Lemma	groupD1_inj	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "setD1K", "setU" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invMG G H : (G * H)^-1 = H * G.	Proof. by rewrite invMg !invGid. Qed.	Lemma	invMG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "invGid", "invMg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulSGid G H : H \subset G -> H * G = G.	Proof. exact: mulSgGid (group1 H). Qed.	Lemma	mulSGid	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group1", "mulSgGid" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulGSid G H : H \subset G -> G * H = G.	Proof. exact: mulGSgid (group1 H). Qed.	Lemma	mulGSid	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group1", "mulGSgid" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulGidPl G H : reflect (G * H = G) (H \subset G).	Proof. by apply: (iffP idP) => [\|<-]; [apply: mulGSid \| apply: mulG_subr]. Qed.	Lemma	mulGidPl	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "mulGSid", "mulG_subr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulGidPr G H : reflect (G * H = H) (G \subset H).	Proof. by apply: (iffP idP) => [\|<-]; [apply: mulSGid \| apply: mulG_subl]. Qed.	Lemma	mulGidPr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "mulG_subl", "mulSGid" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
comm_group_setP G H : reflect (commute G H) (group_set (G * H)).	Proof. rewrite /group_set (subsetP (mulG_subl _ _)) ?group1 // andbC. have <-: #\|G * H\| <= #\|H * G\| by rewrite -invMG card_invg. by rewrite -mulgA mulGS mulgA mulSG -eqEcard eq_sym; apply: eqP. Qed.	Lemma	comm_group_setP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "card_invg", "commute", "eqEcard", "eq_sym", "group1", "group_set", "invMG", "mulGS", "mulG_subl", "mulSG", "mulgA", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_lcosets G H : #\|lcosets H G\| = #\|G : H\|.	Proof. by rewrite -card_invg invg_lcosets !invGid. Qed.	Lemma	card_lcosets	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "card_invg", "invGid", "invg_lcosets", "lcosets" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_modl A B G : A \subset G -> A * (B :&: G) = A * B :&: G.	Proof. move=> sAG; apply/eqP; rewrite eqEsubset subsetI mulgS ?subsetIl //. rewrite -{2}mulGid mulgSS ?subsetIr //. apply/subsetP => _ /setIP[/mulsgP[a b Aa Bb ->] Gab]. by rewrite mem_mulg // inE Bb -(groupMl _ (subsetP sAG _ Aa)). Qed.	Lemma	group_modl	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "eqEsubset", "groupMl", "inE", "mem_mulg", "mulGid", "mulgS", "mulgSS", "mulsgP", "sAG", "setIP", "subsetI", "subsetIl", "subsetIr", "subsetP" ]	Group Modularity equations	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_modr A B G : B \subset G -> (G :&: A) * B = G :&: A * B.	Proof. move=> sBG; apply: invg_inj; rewrite !(invMg, invIg) invGid !(setIC G). by rewrite group_modl // -invGid invSg. Qed.	Lemma	group_modr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "group_modl", "invGid", "invIg", "invMg", "invSg", "invg_inj", "setIC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"G :^ x"	:= (conjG_group G x) : Group_scope.	Notation	G :^ x	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "conjG_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'subg' G ]"	:= (subg_of G) : type_scope.	Notation	[ 'subg' G ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "subg_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'subg' G ]"	:= [set: subg_of G] : group_scope.	Notation	[ 'subg' G ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "subg_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'subg' G ]"	:= [set: subg_of G]%G : Group_scope.	Notation	[ 'subg' G ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "subg_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_setI G H : group_set (G :&: H).	Proof. apply/group_setP; split=> [\|x y]; rewrite !inE ?group1 //. by case/andP=> Gx Hx; rewrite !groupMl. Qed.	Lemma	group_setI	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "group1", "groupMl", "group_set", "group_setP", "inE", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setI_group G H	:= group (group_setI G H).	Canonical	setI_group	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group", "group_setI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_set_bigcap : group_set (\bigcap_(i \| P i) F i).	Proof. by elim/big_rec: _ => [\|i G _ gG]; rewrite -1?(insubdK 1%G gG) groupP. Qed.	Lemma	group_set_bigcap	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "big_rec", "groupP", "group_set", "insubdK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bigcap_group	:= group group_set_bigcap.	Canonical	bigcap_group	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group", "group_set_bigcap" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_set_generated (A : {set gT}) : group_set <<A>>.	Proof. by rewrite unlock group_set_bigcap. Qed.	Lemma	group_set_generated	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "gT", "group_set", "group_set_bigcap" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
generated_group A	:= group (group_set_generated A).	Canonical	generated_group	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group", "group_set_generated" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gcore_group G A : {group _}	:= Eval hnf in [group of gcore G A].	Canonical	gcore_group	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "gcore", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commutator_group A B : {group _}	:= Eval hnf in [group of [~: A, B]].	Canonical	commutator_group	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joing_group A B : {group _}	:= Eval hnf in [group of A <*> B].	Canonical	joing_group	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cycle_group x : {group _}	:= Eval hnf in [group of <[x]>].	Canonical	cycle_group	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
joinG G H	:= joing_group G H.	Definition	joinG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joing_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subgroups A	:= [set G : {group gT} \| G \subset A].	Definition	subgroups	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "gT", "group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
order_gt0 (x : gT) : 0 < #[x].	Proof. exact: cardG_gt0. Qed.	Lemma	order_gt0	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cardG_gt0", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"G :&: H"	:= (setI_group G H) : Group_scope.	Notation	G :&: H	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "setI_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<< A >>"	:= (generated_group A) : Group_scope.	Notation	<< A >>	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "generated_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<[ x ] >"	:= (cycle_group x) : Group_scope.	Notation	<[ x ] >	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cycle_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ ~: A1 , A2 , .. , An ]"	:= (commutator_group .. (commutator_group A1 A2) .. An) : Group_scope.	Notation	[ ~: A1 , A2 , .. , An ]	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "commutator_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"A <*> B"	:= (joing_group A B) : Group_scope.	Notation	A <*> B	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joing_group" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"G * H"	:= (joinG G H) : Group_scope.	Notation	G * H	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joinG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i <- r \| P ) F"	:= (\big[joinG/1%G]_(i <- r \| P%B) F%G) : Group_scope.	Notation	\prod_ ( i <- r \| P ) F	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joinG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i <- r ) F"	:= (\big[joinG/1%G]_(i <- r) F%G) : Group_scope.	Notation	\prod_ ( i <- r ) F	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joinG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( m <= i < n \| P ) F"	:= (\big[joinG/1%G]_(m <= i < n \| P%B) F%G) : Group_scope.	Notation	\prod_ ( m <= i < n \| P ) F	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joinG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( m <= i < n ) F"	:= (\big[joinG/1%G]_(m <= i < n) F%G) : Group_scope.	Notation	\prod_ ( m <= i < n ) F	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joinG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i \| P ) F"	:= (\big[joinG/1%G]_(i \| P%B) F%G) : Group_scope.	Notation	\prod_ ( i \| P ) F	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joinG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ i F"	:= (\big[joinG/1%G]_i F%G) : Group_scope.	Notation	\prod_ i F	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joinG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i : t \| P ) F"	:= (\big[joinG/1%G]_(i : t \| P%B) F%G) (only parsing) : Group_scope.	Notation	\prod_ ( i : t \| P ) F	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joinG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i : t ) F"	:= (\big[joinG/1%G]_(i : t) F%G) (only parsing) : Group_scope.	Notation	\prod_ ( i : t ) F	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joinG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i < n \| P ) F"	:= (\big[joinG/1%G]_(i < n \| P%B) F%G) : Group_scope.	Notation	\prod_ ( i < n \| P ) F	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joinG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i < n ) F"	:= (\big[joinG/1%G]_(i < n) F%G) : Group_scope.	Notation	\prod_ ( i < n ) F	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joinG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i 'in' A \| P ) F"	:= (\big[joinG/1%G]_(i in A \| P%B) F%G) : Group_scope.	Notation	\prod_ ( i 'in' A \| P ) F	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joinG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i 'in' A ) F"	:= (\big[joinG/1%G]_(i in A) F%G) : Group_scope.	Notation	\prod_ ( i 'in' A ) F	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "joinG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
LagrangeI G H : (#\|G :&: H\| * #\|G : H\|)%N = #\|G\|.	Proof. rewrite -[#\|G\|]sum1_card (partition_big_imset (rcoset H)) /=. rewrite mulnC -sum_nat_const; apply: eq_bigr => _ /rcosetsP[x Gx ->]. rewrite -(card_rcoset _ x) -sum1_card; apply: eq_bigl => y. by rewrite rcosetE (sameP eqP rcoset_eqP) group_modr ?sub1set // !inE. Qed.	Lemma	LagrangeI	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "card_rcoset", "eq_bigl", "eq_bigr", "group_modr", "inE", "mulnC", "partition_big_imset", "rcoset", "rcosetE", "rcoset_eqP", "rcosetsP", "sub1set", "sum1_card", "sum_nat_const" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divgI G H : #\|G\| %/ #\|G :&: H\| = #\|G : H\|.	Proof. by rewrite -(LagrangeI G H) mulKn ?cardG_gt0. Qed.	Lemma	divgI	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "LagrangeI", "cardG_gt0", "mulKn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divg_index G H : #\|G\| %/ #\|G : H\| = #\|G :&: H\|.	Proof. by rewrite -(LagrangeI G H) mulnK. Qed.	Lemma	divg_index	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "LagrangeI", "mulnK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dvdn_indexg G H : #\|G : H\| %\| #\|G\|.	Proof. by rewrite -(LagrangeI G H) dvdn_mull. Qed.	Lemma	dvdn_indexg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "LagrangeI", "dvdn_mull" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Lagrange G H : H \subset G -> (#\|H\| * #\|G : H\|)%N = #\|G\|.	Proof. by move/setIidPr=> sHG; rewrite -{1}sHG LagrangeI. Qed.	Theorem	Lagrange	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "LagrangeI", "sHG", "setIidPr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardSg G H : H \subset G -> #\|H\| %\| #\|G\|.	Proof. by move/Lagrange <-; rewrite dvdn_mulr. Qed.	Lemma	cardSg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "Lagrange", "dvdn_mulr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lognSg p G H : G \subset H -> logn p #\|G\| <= logn p #\|H\|.	Proof. by move=> sGH; rewrite dvdn_leq_log ?cardSg. Qed.	Lemma	lognSg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cardSg", "dvdn_leq_log", "logn", "sGH" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
piSg G H : G \subset H -> {subset \pi(gval G) <= \pi(gval H)}.	Proof. move=> sGH p; rewrite !mem_primes !cardG_gt0 => /and3P[-> _ pG]. exact: dvdn_trans (cardSg sGH). Qed.	Lemma	piSg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cardG_gt0", "cardSg", "dvdn_trans", "mem_primes", "pG", "pi", "sGH" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divgS G H : H \subset G -> #\|G\| %/ #\|H\| = #\|G : H\|.	Proof. by move/Lagrange <-; rewrite mulKn. Qed.	Lemma	divgS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "Lagrange", "mulKn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divg_indexS G H : H \subset G -> #\|G\| %/ #\|G : H\| = #\|H\|.	Proof. by move/Lagrange <-; rewrite mulnK. Qed.	Lemma	divg_indexS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "Lagrange", "mulnK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprimeSg G H p : H \subset G -> coprime #\|G\| p -> coprime #\|H\| p.	Proof. by move=> sHG; apply: coprime_dvdl (cardSg sHG). Qed.	Lemma	coprimeSg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cardSg", "coprime", "coprime_dvdl", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprimegS G H p : H \subset G -> coprime p #\|G\| -> coprime p #\|H\|.	Proof. by move=> sHG; apply: coprime_dvdr (cardSg sHG). Qed.	Lemma	coprimegS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cardSg", "coprime", "coprime_dvdr", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
indexJg G H x : #\|G :^ x : H :^ x\| = #\|G : H\|.	Proof. by rewrite -!divgI -conjIg !cardJg. Qed.	Lemma	indexJg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cardJg", "conjIg", "divgI" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
indexgg G : #\|G : G\| = 1%N.	Proof. by rewrite -divgS // divnn cardG_gt0. Qed.	Lemma	indexgg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cardG_gt0", "divgS", "divnn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcosets_id G : rcosets G G = [set G : {set gT}].	Proof. apply/esym/eqP; rewrite eqEcard sub1set [#\|_\|]indexgg cards1 andbT. by apply/rcosetsP; exists 1; rewrite ?mulg1. Qed.	Lemma	rcosets_id	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cards1", "eqEcard", "gT", "indexgg", "mulg1", "rcosets", "rcosetsP", "sub1set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Lagrange_index G H K : H \subset G -> K \subset H -> (#\|G : H\| * #\|H : K\|)%N = #\|G : K\|.	Proof. move=> sHG sKH; apply/eqP; rewrite mulnC -(eqn_pmul2l (cardG_gt0 K)). by rewrite mulnA !Lagrange // (subset_trans sKH). Qed.	Lemma	Lagrange_index	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "Lagrange", "apply", "cardG_gt0", "eqn_pmul2l", "mulnA", "mulnC", "sHG", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
indexgI G H : #\|G : G :&: H\| = #\|G : H\|.	Proof. by rewrite -[RHS]divgI divgS ?subsetIl. Qed.	Lemma	indexgI	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "divgI", "divgS", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
indexgS G H K : H \subset K -> #\|G : K\| %\| #\|G : H\|.	Proof. move=> sHK; rewrite -(@dvdn_pmul2l #\|G :&: K\|) ?cardG_gt0 // LagrangeI. by rewrite -(Lagrange (setIS G sHK)) mulnAC LagrangeI dvdn_mulr. Qed.	Lemma	indexgS	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "Lagrange", "LagrangeI", "cardG_gt0", "dvdn_mulr", "dvdn_pmul2l", "mulnAC", "sHK", "setIS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
indexSg G H K : H \subset K -> K \subset G -> #\|K : H\| %\| #\|G : H\|.	Proof. move=> sHK sKG; rewrite -(@dvdn_pmul2l #\|H\|) ?cardG_gt0 //. by rewrite !Lagrange ?(cardSg, subset_trans sHK). Qed.	Lemma	indexSg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "Lagrange", "cardG_gt0", "cardSg", "dvdn_pmul2l", "sHK", "sKG", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
indexg_eq1 G H : (#\|G : H\| == 1%N) = (G \subset H).	Proof. rewrite eqn_leq -(leq_pmul2l (cardG_gt0 (G :&: H))) LagrangeI muln1. by rewrite indexg_gt0 andbT (sameP setIidPl eqP) eqEcard subsetIl. Qed.	Lemma	indexg_eq1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "LagrangeI", "cardG_gt0", "eqEcard", "eqn_leq", "indexg_gt0", "leq_pmul2l", "muln1", "setIidPl", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
indexg_gt1 G H : (#\|G : H\| > 1) = ~~ (G \subset H).	Proof. by rewrite -indexg_eq1 eqn_leq indexg_gt0 andbT -ltnNge. Qed.	Lemma	indexg_gt1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "eqn_leq", "indexg_eq1", "indexg_gt0", "ltnNge" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
index1g G H : H \subset G -> #\|G : H\| = 1%N -> H :=: G.	Proof. by move=> sHG iHG; apply/eqP; rewrite eqEsubset sHG -indexg_eq1 iHG. Qed.	Lemma	index1g	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "eqEsubset", "indexg_eq1", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
indexg1 G : #\|G : 1\| = #\|G\|.	Proof. by rewrite -divgS ?sub1G // cards1 divn1. Qed.	Lemma	indexg1	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cards1", "divgS", "divn1", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
indexMg G A : #\|G * A : G\| = #\|A : G\|.	Proof. apply/eq_card/setP/eqP; rewrite eqEsubset andbC imsetS ?mulG_subr //. by apply/subsetP=> _ /rcosetsP[x GAx ->]; rewrite mem_rcosets. Qed.	Lemma	indexMg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "eqEsubset", "eq_card", "imsetS", "mem_rcosets", "mulG_subr", "rcosetsP", "setP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcosets_partition_mul G H : partition (rcosets H G) (H * G).	Proof. set HG := H * G; have sGHG: {subset G <= HG} by apply/subsetP/mulG_subr. have defHx x: x \in HG -> [set y in HG \| rcoset H x == rcoset H y] = H :* x. move=> HGx; apply/setP=> y; rewrite inE !rcosetE (sameP eqP rcoset_eqP). by rewrite rcoset_sym; apply/andb_idl/subsetP; rewrite mulGS sub1set. have:= preim_par...	Lemma	rcosets_partition_mul	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "HG", "apply", "imsetP", "inE", "mem_rcosets", "mulGS", "mulG_subr", "partition", "preim_partitionP", "rcoset", "rcosetE", "rcoset_eqP", "rcoset_sym", "rcosets", "rcosetsP", "setP", "sub1set", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rcosets_partition G H : H \subset G -> partition (rcosets H G) G.	Proof. by move=> sHG; have:= rcosets_partition_mul G H; rewrite mulSGid. Qed.	Lemma	rcosets_partition	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "mulSGid", "partition", "rcosets", "rcosets_partition_mul", "sHG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
LagrangeMl G H : (#\|G\| * #\|H : G\|)%N = #\|G * H\|.	Proof. rewrite mulnC -(card_uniform_partition _ (rcosets_partition_mul H G)) //. by move=> _ /rcosetsP[x Hx ->]; rewrite card_rcoset. Qed.	Lemma	LagrangeMl	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "card_rcoset", "card_uniform_partition", "mulnC", "rcosetsP", "rcosets_partition_mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
LagrangeMr G H : (#\|G : H\| * #\|H\|)%N = #\|G * H\|.	Proof. by rewrite mulnC LagrangeMl -card_invg invMg !invGid. Qed.	Lemma	LagrangeMr	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "LagrangeMl", "card_invg", "invGid", "invMg", "mulnC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_cardG G H : (#\|G\| * #\|H\| = #\|G * H\|%g * #\|G :&: H\|)%N.	Proof. by rewrite -LagrangeMr -(LagrangeI G H) -mulnA mulnC. Qed.	Lemma	mul_cardG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "LagrangeI", "LagrangeMr", "mulnA", "mulnC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dvdn_cardMg G H : #\|G * H\| %\| #\|G\| * #\|H\|.	Proof. by rewrite mul_cardG dvdn_mulr. Qed.	Lemma	dvdn_cardMg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "dvdn_mulr", "mul_cardG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardMg_divn G H : #\|G * H\| = (#\|G\| * #\|H\|) %/ #\|G :&: H\|.	Proof. by rewrite mul_cardG mulnK ?cardG_gt0. Qed.	Lemma	cardMg_divn	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cardG_gt0", "mul_cardG", "mulnK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardIg_divn G H : #\|G :&: H\| = (#\|G\| * #\|H\|) %/ #\|G * H\|.	Proof. by rewrite mul_cardG mulKn // (cardD1 (1 * 1)) mem_mulg. Qed.	Lemma	cardIg_divn	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cardD1", "mem_mulg", "mulKn", "mul_cardG" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
TI_cardMg G H : G :&: H = 1 -> #\|G * H\| = (#\|G\| * #\|H\|)%N.	Proof. by move=> tiGH; rewrite mul_cardG tiGH cards1 muln1. Qed.	Lemma	TI_cardMg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "cards1", "mul_cardG", "muln1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardMg_TI G H : #\|G\| * #\|H\| <= #\|G * H\| -> G :&: H = 1.	Proof. move=> leGH; apply: card_le1_trivg. rewrite -(@leq_pmul2l #\|G * H\|); last by rewrite -mul_cardG muln1. by apply: leq_trans leGH; rewrite muln_gt0 !cardG_gt0. Qed.	Lemma	cardMg_TI	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cardG_gt0", "card_le1_trivg", "last", "leq_pmul2l", "leq_trans", "mul_cardG", "muln1", "muln_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprime_TIg G H : coprime #\|G\| #\|H\| -> G :&: H = 1.	Proof. move=> coGH; apply/eqP; rewrite trivg_card1 -dvdn1 -{}(eqnP coGH). by rewrite dvdn_gcd /= {2}setIC !cardSg ?subsetIl. Qed.	Lemma	coprime_TIg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "cardSg", "coprime", "dvdn1", "dvdn_gcd", "eqnP", "setIC", "subsetIl", "trivg_card1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prime_TIg G H : prime #\|G\| -> ~~ (G \subset H) -> G :&: H = 1.	Proof. case/primeP=> _ /(_ _ (cardSg (subsetIl G H))). rewrite (sameP setIidPl eqP) eqEcard subsetIl => /pred2P[/card1_trivg\|] //= ->. by case/negP. Qed.	Lemma	prime_TIg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "card1_trivg", "cardSg", "eqEcard", "pred2P", "prime", "primeP", "setIidPl", "subsetIl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prime_meetG G H : prime #\|G\| -> G :&: H != 1 -> G \subset H.	Proof. by move=> prG; apply: contraR; move/prime_TIg->. Qed.	Lemma	prime_meetG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "prime", "prime_TIg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprime_cardMg G H : coprime #\|G\| #\|H\| -> #\|G * H\| = (#\|G\| * #\|H\|)%N.	Proof. by move=> coGH; rewrite TI_cardMg ?coprime_TIg. Qed.	Lemma	coprime_cardMg	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "TI_cardMg", "coprime", "coprime_TIg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coprime_index_mulG G H K : H \subset G -> K \subset G -> coprime #\|G : H\| #\|G : K\| -> H * K = G.	Proof. move=> sHG sKG co_iG_HK; apply/eqP; rewrite eqEcard mul_subG //=. rewrite -(@leq_pmul2r #\|H :&: K\|) ?cardG_gt0 // -mul_cardG. rewrite -(Lagrange sHG) -(LagrangeI K H) mulnAC setIC -mulnA. rewrite !leq_pmul2l ?cardG_gt0 // dvdn_leq // -(Gauss_dvdr _ co_iG_HK). by rewrite -(indexgI K) Lagrange_index ?indexgS ?subs...	Lemma	coprime_index_mulG	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "Gauss_dvdr", "Lagrange", "LagrangeI", "Lagrange_index", "apply", "cardG_gt0", "coprime", "dvdn_leq", "eqEcard", "indexgI", "indexgS", "leq_pmul2l", "leq_pmul2r", "mul_cardG", "mul_subG", "mulnA", "mulnAC", "sHG", "sKG", "setIC", "subsetIl", "subsetIr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_gen A : A \subset <<A>>.	Proof. rewrite [@generated]unlock; exact/bigcapsP. Qed.	Lemma	subset_gen	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "bigcapsP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_gen A B : A \subset B -> A \subset <<B>>.	Proof. by move/subset_trans=> -> //; apply: subset_gen. Qed.	Lemma	sub_gen	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "apply", "subset_gen", "subset_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_gen x A : x \in A -> x \in <<A>>.	Proof. exact: subsetP (subset_gen A) x. Qed.	Lemma	mem_gen	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "subsetP", "subset_gen" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
generatedP x A : reflect (forall G, A \subset G -> x \in G) (x \in <<A>>).	Proof. rewrite [@generated]unlock; exact: bigcapP. Qed.	Lemma	generatedP	finite_group	finite_group/fingroup.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]	[ "bigcapP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

subg_inv u

:= Subg (groupVr (subgP u)).

Definition

subg_inv

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "groupVr", "subgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subg_mul u v

:= Subg (groupM (subgP u) (subgP v)).

Definition

subg_mul

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "groupM", "subgP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subg_oneP : left_id subg_one subg_mul.

Proof. by move=> u; apply: val_inj; apply: mul1g. Qed.

Lemma

subg_oneP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "mul1g", "subg_mul", "subg_one", "val_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subg_invP : left_inverse subg_one subg_inv subg_mul.

Proof. by move=> u; apply: val_inj; apply: mulVg. Qed.

Lemma

subg_invP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "mulVg", "subg_inv", "subg_mul", "subg_one", "val_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subg_mulP : associative subg_mul.

Proof. by move=> u v w; apply: val_inj; apply: mulgA. Qed.

Lemma

subg_mulP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "mulgA", "subg_mul", "val_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sgvalM : {in setT &, {morph sgval : x y / x * y}}.

Proof. by []. Qed.

Lemma

sgvalM

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "setT", "sgval" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

valgM : {in setT &, {morph val : x y / (x : subg_of) * y >-> x * y}}.

Proof. by []. Qed.

Lemma

valgM

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "setT", "subg_of", "val" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subg : gT -> subg_of

:= insubd (1 : subg_of).

Definition

subg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "gT", "insubd", "subg_of" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subgK x : x \in G -> val (subg x) = x.

Proof. by move=> Gx; rewrite insubdK. Qed.

Lemma

subgK

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "insubdK", "subg", "val" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sgvalK : cancel sgval subg.

Proof. by case=> x Gx; apply: val_inj; apply: subgK. Qed.

Lemma

sgvalK

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "sgval", "subg", "subgK", "val_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subg_default x : (x \in G) = false -> val (subg x) = 1.

Proof. by move=> Gx; rewrite val_insubd Gx. Qed.

Lemma

subg_default

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "subg", "val", "val_insubd" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subgM : {in G &, {morph subg : x y / x * y}}.

Proof. by move=> x y Gx Gy; apply: val_inj; rewrite /= !subgK ?groupM. Qed.

Lemma

subgM

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "groupM", "subg", "subgK", "val_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupD1_inj G H : G^# = H^# -> G :=: H.

Proof. by move/(congr1 (setU 1)); rewrite !setD1K. Qed.

Lemma

groupD1_inj

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "setD1K", "setU" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invMG G H : (G * H)^-1 = H * G.

Proof. by rewrite invMg !invGid. Qed.

Lemma

invMG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "invGid", "invMg" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulSGid G H : H \subset G -> H * G = G.

Proof. exact: mulSgGid (group1 H). Qed.

Lemma

mulSGid

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group1", "mulSgGid" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulGSid G H : H \subset G -> G * H = G.

Proof. exact: mulGSgid (group1 H). Qed.

Lemma

mulGSid

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group1", "mulGSgid" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulGidPl G H : reflect (G * H = G) (H \subset G).

Proof. by apply: (iffP idP) => [|<-]; [apply: mulGSid | apply: mulG_subr]. Qed.

Lemma

mulGidPl

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "mulGSid", "mulG_subr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulGidPr G H : reflect (G * H = H) (G \subset H).

Proof. by apply: (iffP idP) => [|<-]; [apply: mulSGid | apply: mulG_subl]. Qed.

Lemma

mulGidPr

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "mulG_subl", "mulSGid" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

comm_group_setP G H : reflect (commute G H) (group_set (G * H)).

Proof. rewrite /group_set (subsetP (mulG_subl _ _)) ?group1 // andbC. have <-: #|G * H| <= #|H * G| by rewrite -invMG card_invg. by rewrite -mulgA mulGS mulgA mulSG -eqEcard eq_sym; apply: eqP. Qed.

Lemma

comm_group_setP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "card_invg", "commute", "eqEcard", "eq_sym", "group1", "group_set", "invMG", "mulGS", "mulG_subl", "mulSG", "mulgA", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_lcosets G H : #|lcosets H G| = #|G : H|.

Proof. by rewrite -card_invg invg_lcosets !invGid. Qed.

Lemma

card_lcosets

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "card_invg", "invGid", "invg_lcosets", "lcosets" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_modl A B G : A \subset G -> A * (B :&: G) = A * B :&: G.

Proof. move=> sAG; apply/eqP; rewrite eqEsubset subsetI mulgS ?subsetIl //. rewrite -{2}mulGid mulgSS ?subsetIr //. apply/subsetP => _ /setIP[/mulsgP[a b Aa Bb ->] Gab]. by rewrite mem_mulg // inE Bb -(groupMl _ (subsetP sAG _ Aa)). Qed.

Lemma

group_modl

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "eqEsubset", "groupMl", "inE", "mem_mulg", "mulGid", "mulgS", "mulgSS", "mulsgP", "sAG", "setIP", "subsetI", "subsetIl", "subsetIr", "subsetP" ]

Group Modularity equations

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_modr A B G : B \subset G -> (G :&: A) * B = G :&: A * B.

Proof. move=> sBG; apply: invg_inj; rewrite !(invMg, invIg) invGid !(setIC G). by rewrite group_modl // -invGid invSg. Qed.

Lemma

group_modr

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "group_modl", "invGid", "invIg", "invMg", "invSg", "invg_inj", "setIC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"G :^ x"

:= (conjG_group G x) : Group_scope.

Notation

G :^ x

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "conjG_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'subg' G ]"

:= (subg_of G) : type_scope.

Notation

[ 'subg' G ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "subg_of" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'subg' G ]"

:= [set: subg_of G] : group_scope.

Notation

[ 'subg' G ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "subg_of" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'subg' G ]"

:= [set: subg_of G]%G : Group_scope.

Notation

[ 'subg' G ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "subg_of" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_setI G H : group_set (G :&: H).

Proof. apply/group_setP; split=> [|x y]; rewrite !inE ?group1 //. by case/andP=> Gx Hx; rewrite !groupMl. Qed.

Lemma

group_setI

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "group1", "groupMl", "group_set", "group_setP", "inE", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setI_group G H

:= group (group_setI G H).

Canonical

setI_group

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group", "group_setI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_set_bigcap : group_set (\bigcap_(i | P i) F i).

Proof. by elim/big_rec: _ => [|i G _ gG]; rewrite -1?(insubdK 1%G gG) groupP. Qed.

Lemma

group_set_bigcap

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "big_rec", "groupP", "group_set", "insubdK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bigcap_group

:= group group_set_bigcap.

Canonical

bigcap_group

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group", "group_set_bigcap" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_set_generated (A : {set gT}) : group_set <<A>>.

Proof. by rewrite unlock group_set_bigcap. Qed.

Lemma

group_set_generated

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "gT", "group_set", "group_set_bigcap" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

generated_group A

:= group (group_set_generated A).

Canonical

generated_group

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group", "group_set_generated" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gcore_group G A : {group _}

:= Eval hnf in [group of gcore G A].

Canonical

gcore_group

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "gcore", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commutator_group A B : {group _}

:= Eval hnf in [group of [~: A, B]].

Canonical

commutator_group

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joing_group A B : {group _}

:= Eval hnf in [group of A <*> B].

Canonical

joing_group

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cycle_group x : {group _}

:= Eval hnf in [group of <[x]>].

Canonical

cycle_group

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

joinG G H

:= joing_group G H.

Definition

joinG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joing_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subgroups A

:= [set G : {group gT} | G \subset A].

Definition

subgroups

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

order_gt0 (x : gT) : 0 < #[x].

Proof. exact: cardG_gt0. Qed.

Lemma

order_gt0

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cardG_gt0", "gT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"G :&: H"

:= (setI_group G H) : Group_scope.

Notation

G :&: H

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "setI_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"<< A >>"

:= (generated_group A) : Group_scope.

Notation

<< A >>

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "generated_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"<[ x ] >"

:= (cycle_group x) : Group_scope.

Notation

<[ x ] >

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cycle_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ ~: A1 , A2 , .. , An ]"

:= (commutator_group .. (commutator_group A1 A2) .. An) : Group_scope.

Notation

[ ~: A1 , A2 , .. , An ]

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "commutator_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"A <*> B"

:= (joing_group A B) : Group_scope.

Notation

A <*> B

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joing_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"G * H"

:= (joinG G H) : Group_scope.

Notation

G * H

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joinG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i <- r | P ) F"

:= (\big[joinG/1%G]_(i <- r | P%B) F%G) : Group_scope.

Notation

\prod_ ( i <- r | P ) F

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joinG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i <- r ) F"

:= (\big[joinG/1%G]_(i <- r) F%G) : Group_scope.

Notation

\prod_ ( i <- r ) F

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joinG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( m <= i < n | P ) F"

:= (\big[joinG/1%G]_(m <= i < n | P%B) F%G) : Group_scope.

Notation

\prod_ ( m <= i < n | P ) F

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joinG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( m <= i < n ) F"

:= (\big[joinG/1%G]_(m <= i < n) F%G) : Group_scope.

Notation

\prod_ ( m <= i < n ) F

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joinG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i | P ) F"

:= (\big[joinG/1%G]_(i | P%B) F%G) : Group_scope.

Notation

\prod_ ( i | P ) F

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joinG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ i F"

:= (\big[joinG/1%G]_i F%G) : Group_scope.

Notation

\prod_ i F

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joinG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i : t | P ) F"

:= (\big[joinG/1%G]_(i : t | P%B) F%G) (only parsing) : Group_scope.

Notation

\prod_ ( i : t | P ) F

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joinG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i : t ) F"

:= (\big[joinG/1%G]_(i : t) F%G) (only parsing) : Group_scope.

Notation

\prod_ ( i : t ) F

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joinG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i < n | P ) F"

:= (\big[joinG/1%G]_(i < n | P%B) F%G) : Group_scope.

Notation

\prod_ ( i < n | P ) F

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joinG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i < n ) F"

:= (\big[joinG/1%G]_(i < n) F%G) : Group_scope.

Notation

\prod_ ( i < n ) F

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joinG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i 'in' A | P ) F"

:= (\big[joinG/1%G]_(i in A | P%B) F%G) : Group_scope.

Notation

\prod_ ( i 'in' A | P ) F

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joinG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i 'in' A ) F"

:= (\big[joinG/1%G]_(i in A) F%G) : Group_scope.

Notation

\prod_ ( i 'in' A ) F

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "joinG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

LagrangeI G H : (#|G :&: H| * #|G : H|)%N = #|G|.

Proof. rewrite -[#|G|]sum1_card (partition_big_imset (rcoset H)) /=. rewrite mulnC -sum_nat_const; apply: eq_bigr => _ /rcosetsP[x Gx ->]. rewrite -(card_rcoset _ x) -sum1_card; apply: eq_bigl => y. by rewrite rcosetE (sameP eqP rcoset_eqP) group_modr ?sub1set // !inE. Qed.

Lemma

LagrangeI

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "card_rcoset", "eq_bigl", "eq_bigr", "group_modr", "inE", "mulnC", "partition_big_imset", "rcoset", "rcosetE", "rcoset_eqP", "rcosetsP", "sub1set", "sum1_card", "sum_nat_const" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divgI G H : #|G| %/ #|G :&: H| = #|G : H|.

Proof. by rewrite -(LagrangeI G H) mulKn ?cardG_gt0. Qed.

Lemma

divgI

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "LagrangeI", "cardG_gt0", "mulKn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divg_index G H : #|G| %/ #|G : H| = #|G :&: H|.

Proof. by rewrite -(LagrangeI G H) mulnK. Qed.

Lemma

divg_index

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "LagrangeI", "mulnK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dvdn_indexg G H : #|G : H| %| #|G|.

Proof. by rewrite -(LagrangeI G H) dvdn_mull. Qed.

Lemma

dvdn_indexg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "LagrangeI", "dvdn_mull" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Lagrange G H : H \subset G -> (#|H| * #|G : H|)%N = #|G|.

Proof. by move/setIidPr=> sHG; rewrite -{1}sHG LagrangeI. Qed.

Theorem

Lagrange

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "LagrangeI", "sHG", "setIidPr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardSg G H : H \subset G -> #|H| %| #|G|.

Proof. by move/Lagrange <-; rewrite dvdn_mulr. Qed.

Lemma

cardSg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "Lagrange", "dvdn_mulr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lognSg p G H : G \subset H -> logn p #|G| <= logn p #|H|.

Proof. by move=> sGH; rewrite dvdn_leq_log ?cardSg. Qed.

Lemma

lognSg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cardSg", "dvdn_leq_log", "logn", "sGH" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

piSg G H : G \subset H -> {subset \pi(gval G) <= \pi(gval H)}.

Proof. move=> sGH p; rewrite !mem_primes !cardG_gt0 => /and3P[-> _ pG]. exact: dvdn_trans (cardSg sGH). Qed.

Lemma

piSg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cardG_gt0", "cardSg", "dvdn_trans", "mem_primes", "pG", "pi", "sGH" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divgS G H : H \subset G -> #|G| %/ #|H| = #|G : H|.

Proof. by move/Lagrange <-; rewrite mulKn. Qed.

Lemma

divgS

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "Lagrange", "mulKn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divg_indexS G H : H \subset G -> #|G| %/ #|G : H| = #|H|.

Proof. by move/Lagrange <-; rewrite mulnK. Qed.

Lemma

divg_indexS

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "Lagrange", "mulnK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coprimeSg G H p : H \subset G -> coprime #|G| p -> coprime #|H| p.

Proof. by move=> sHG; apply: coprime_dvdl (cardSg sHG). Qed.

Lemma

coprimeSg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cardSg", "coprime", "coprime_dvdl", "sHG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coprimegS G H p : H \subset G -> coprime p #|G| -> coprime p #|H|.

Proof. by move=> sHG; apply: coprime_dvdr (cardSg sHG). Qed.

Lemma

coprimegS

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cardSg", "coprime", "coprime_dvdr", "sHG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

indexJg G H x : #|G :^ x : H :^ x| = #|G : H|.

Proof. by rewrite -!divgI -conjIg !cardJg. Qed.

Lemma

indexJg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cardJg", "conjIg", "divgI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

indexgg G : #|G : G| = 1%N.

Proof. by rewrite -divgS // divnn cardG_gt0. Qed.

Lemma

indexgg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cardG_gt0", "divgS", "divnn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcosets_id G : rcosets G G = [set G : {set gT}].

Proof. apply/esym/eqP; rewrite eqEcard sub1set [#|_|]indexgg cards1 andbT. by apply/rcosetsP; exists 1; rewrite ?mulg1. Qed.

Lemma

rcosets_id

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cards1", "eqEcard", "gT", "indexgg", "mulg1", "rcosets", "rcosetsP", "sub1set" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Lagrange_index G H K : H \subset G -> K \subset H -> (#|G : H| * #|H : K|)%N = #|G : K|.

Proof. move=> sHG sKH; apply/eqP; rewrite mulnC -(eqn_pmul2l (cardG_gt0 K)). by rewrite mulnA !Lagrange // (subset_trans sKH). Qed.

Lemma

Lagrange_index

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "Lagrange", "apply", "cardG_gt0", "eqn_pmul2l", "mulnA", "mulnC", "sHG", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

indexgI G H : #|G : G :&: H| = #|G : H|.

Proof. by rewrite -[RHS]divgI divgS ?subsetIl. Qed.

Lemma

indexgI

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "divgI", "divgS", "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

indexgS G H K : H \subset K -> #|G : K| %| #|G : H|.

Proof. move=> sHK; rewrite -(@dvdn_pmul2l #|G :&: K|) ?cardG_gt0 // LagrangeI. by rewrite -(Lagrange (setIS G sHK)) mulnAC LagrangeI dvdn_mulr. Qed.

Lemma

indexgS

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "Lagrange", "LagrangeI", "cardG_gt0", "dvdn_mulr", "dvdn_pmul2l", "mulnAC", "sHK", "setIS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

indexSg G H K : H \subset K -> K \subset G -> #|K : H| %| #|G : H|.

Proof. move=> sHK sKG; rewrite -(@dvdn_pmul2l #|H|) ?cardG_gt0 //. by rewrite !Lagrange ?(cardSg, subset_trans sHK). Qed.

Lemma

indexSg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "Lagrange", "cardG_gt0", "cardSg", "dvdn_pmul2l", "sHK", "sKG", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

indexg_eq1 G H : (#|G : H| == 1%N) = (G \subset H).

Proof. rewrite eqn_leq -(leq_pmul2l (cardG_gt0 (G :&: H))) LagrangeI muln1. by rewrite indexg_gt0 andbT (sameP setIidPl eqP) eqEcard subsetIl. Qed.

Lemma

indexg_eq1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "LagrangeI", "cardG_gt0", "eqEcard", "eqn_leq", "indexg_gt0", "leq_pmul2l", "muln1", "setIidPl", "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

indexg_gt1 G H : (#|G : H| > 1) = ~~ (G \subset H).

Proof. by rewrite -indexg_eq1 eqn_leq indexg_gt0 andbT -ltnNge. Qed.

Lemma

indexg_gt1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "eqn_leq", "indexg_eq1", "indexg_gt0", "ltnNge" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

index1g G H : H \subset G -> #|G : H| = 1%N -> H :=: G.

Proof. by move=> sHG iHG; apply/eqP; rewrite eqEsubset sHG -indexg_eq1 iHG. Qed.

Lemma

index1g

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "eqEsubset", "indexg_eq1", "sHG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

indexg1 G : #|G : 1| = #|G|.

Proof. by rewrite -divgS ?sub1G // cards1 divn1. Qed.

Lemma

indexg1

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cards1", "divgS", "divn1", "sub1G" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

indexMg G A : #|G * A : G| = #|A : G|.

Proof. apply/eq_card/setP/eqP; rewrite eqEsubset andbC imsetS ?mulG_subr //. by apply/subsetP=> _ /rcosetsP[x GAx ->]; rewrite mem_rcosets. Qed.

Lemma

indexMg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "eqEsubset", "eq_card", "imsetS", "mem_rcosets", "mulG_subr", "rcosetsP", "setP", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcosets_partition_mul G H : partition (rcosets H G) (H * G).

Proof. set HG := H * G; have sGHG: {subset G <= HG} by apply/subsetP/mulG_subr. have defHx x: x \in HG -> [set y in HG | rcoset H x == rcoset H y] = H :* x. move=> HGx; apply/setP=> y; rewrite inE !rcosetE (sameP eqP rcoset_eqP). by rewrite rcoset_sym; apply/andb_idl/subsetP; rewrite mulGS sub1set. have:= preim_par...

Lemma

rcosets_partition_mul

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "HG", "apply", "imsetP", "inE", "mem_rcosets", "mulGS", "mulG_subr", "partition", "preim_partitionP", "rcoset", "rcosetE", "rcoset_eqP", "rcoset_sym", "rcosets", "rcosetsP", "setP", "sub1set", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rcosets_partition G H : H \subset G -> partition (rcosets H G) G.

Proof. by move=> sHG; have:= rcosets_partition_mul G H; rewrite mulSGid. Qed.

Lemma

rcosets_partition

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "mulSGid", "partition", "rcosets", "rcosets_partition_mul", "sHG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

LagrangeMl G H : (#|G| * #|H : G|)%N = #|G * H|.

Proof. rewrite mulnC -(card_uniform_partition _ (rcosets_partition_mul H G)) //. by move=> _ /rcosetsP[x Hx ->]; rewrite card_rcoset. Qed.

Lemma

LagrangeMl

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "card_rcoset", "card_uniform_partition", "mulnC", "rcosetsP", "rcosets_partition_mul" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

LagrangeMr G H : (#|G : H| * #|H|)%N = #|G * H|.

Proof. by rewrite mulnC LagrangeMl -card_invg invMg !invGid. Qed.

Lemma

LagrangeMr

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "LagrangeMl", "card_invg", "invGid", "invMg", "mulnC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul_cardG G H : (#|G| * #|H| = #|G * H|%g * #|G :&: H|)%N.

Proof. by rewrite -LagrangeMr -(LagrangeI G H) -mulnA mulnC. Qed.

Lemma

mul_cardG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "LagrangeI", "LagrangeMr", "mulnA", "mulnC" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dvdn_cardMg G H : #|G * H| %| #|G| * #|H|.

Proof. by rewrite mul_cardG dvdn_mulr. Qed.

Lemma

dvdn_cardMg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "dvdn_mulr", "mul_cardG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardMg_divn G H : #|G * H| = (#|G| * #|H|) %/ #|G :&: H|.

Proof. by rewrite mul_cardG mulnK ?cardG_gt0. Qed.

Lemma

cardMg_divn

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cardG_gt0", "mul_cardG", "mulnK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardIg_divn G H : #|G :&: H| = (#|G| * #|H|) %/ #|G * H|.

Proof. by rewrite mul_cardG mulKn // (cardD1 (1 * 1)) mem_mulg. Qed.

Lemma

cardIg_divn

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cardD1", "mem_mulg", "mulKn", "mul_cardG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

TI_cardMg G H : G :&: H = 1 -> #|G * H| = (#|G| * #|H|)%N.

Proof. by move=> tiGH; rewrite mul_cardG tiGH cards1 muln1. Qed.

Lemma

TI_cardMg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "cards1", "mul_cardG", "muln1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardMg_TI G H : #|G| * #|H| <= #|G * H| -> G :&: H = 1.

Proof. move=> leGH; apply: card_le1_trivg. rewrite -(@leq_pmul2l #|G * H|); last by rewrite -mul_cardG muln1. by apply: leq_trans leGH; rewrite muln_gt0 !cardG_gt0. Qed.

Lemma

cardMg_TI

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cardG_gt0", "card_le1_trivg", "last", "leq_pmul2l", "leq_trans", "mul_cardG", "muln1", "muln_gt0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coprime_TIg G H : coprime #|G| #|H| -> G :&: H = 1.

Proof. move=> coGH; apply/eqP; rewrite trivg_card1 -dvdn1 -{}(eqnP coGH). by rewrite dvdn_gcd /= {2}setIC !cardSg ?subsetIl. Qed.

Lemma

coprime_TIg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "cardSg", "coprime", "dvdn1", "dvdn_gcd", "eqnP", "setIC", "subsetIl", "trivg_card1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prime_TIg G H : prime #|G| -> ~~ (G \subset H) -> G :&: H = 1.

Proof. case/primeP=> _ /(_ _ (cardSg (subsetIl G H))). rewrite (sameP setIidPl eqP) eqEcard subsetIl => /pred2P[/card1_trivg|] //= ->. by case/negP. Qed.

Lemma

prime_TIg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "card1_trivg", "cardSg", "eqEcard", "pred2P", "prime", "primeP", "setIidPl", "subsetIl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prime_meetG G H : prime #|G| -> G :&: H != 1 -> G \subset H.

Proof. by move=> prG; apply: contraR; move/prime_TIg->. Qed.

Lemma

prime_meetG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "prime", "prime_TIg" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coprime_cardMg G H : coprime #|G| #|H| -> #|G * H| = (#|G| * #|H|)%N.

Proof. by move=> coGH; rewrite TI_cardMg ?coprime_TIg. Qed.

Lemma

coprime_cardMg

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "TI_cardMg", "coprime", "coprime_TIg" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coprime_index_mulG G H K : H \subset G -> K \subset G -> coprime #|G : H| #|G : K| -> H * K = G.

Proof. move=> sHG sKG co_iG_HK; apply/eqP; rewrite eqEcard mul_subG //=. rewrite -(@leq_pmul2r #|H :&: K|) ?cardG_gt0 // -mul_cardG. rewrite -(Lagrange sHG) -(LagrangeI K H) mulnAC setIC -mulnA. rewrite !leq_pmul2l ?cardG_gt0 // dvdn_leq // -(Gauss_dvdr _ co_iG_HK). by rewrite -(indexgI K) Lagrange_index ?indexgS ?subs...

Lemma

coprime_index_mulG

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "Gauss_dvdr", "Lagrange", "LagrangeI", "Lagrange_index", "apply", "cardG_gt0", "coprime", "dvdn_leq", "eqEcard", "indexgI", "indexgS", "leq_pmul2l", "leq_pmul2r", "mul_cardG", "mul_subG", "mulnA", "mulnAC", "sHG", "sKG", "setIC", "subsetIl", "subsetIr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_gen A : A \subset <<A>>.

Proof. rewrite [@generated]unlock; exact/bigcapsP. Qed.

Lemma

subset_gen

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "bigcapsP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_gen A B : A \subset B -> A \subset <<B>>.

Proof. by move/subset_trans=> -> //; apply: subset_gen. Qed.

Lemma

sub_gen

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "apply", "subset_gen", "subset_trans" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_gen x A : x \in A -> x \in <<A>>.

Proof. exact: subsetP (subset_gen A) x. Qed.

Lemma

mem_gen

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "subsetP", "subset_gen" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

generatedP x A : reflect (forall G, A \subset G -> x \in G) (x \in <<A>>).

Proof. rewrite [@generated]unlock; exact: bigcapP. Qed.

Lemma

generatedP

finite_group

finite_group/fingroup.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "choice", "fintype", "div", "path", "tuple", "bigop", "prime", "finset", "monoid", "Monoid.Theory" ]

[ "bigcapP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d