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
extnprod_invg (x : gTn) : gTn	:= [ffun i => (x i)^-1%g].	Definition	extnprod_invg	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "gTn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
extnprod_mul1g : left_id [ffun=> 1%g] extnprod_mulg.	Proof. by move=> x; apply/ffunP => i; rewrite !ffunE mul1g. Qed.	Lemma	extnprod_mul1g	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "extnprod_mulg", "ffunE", "ffunP", "mul1g" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
extnprod_mulVg : left_inverse [ffun=> 1%g] extnprod_invg extnprod_mulg.	Proof. by move=> x; apply/ffunP => i; rewrite !ffunE mulVg. Qed.	Lemma	extnprod_mulVg	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "extnprod_invg", "extnprod_mulg", "ffunE", "ffunP", "mulVg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
extnprod_mulgA : associative extnprod_mulg.	Proof. by move=> x y z; apply/ffunP => i; rewrite !ffunE mulgA. Qed.	Lemma	extnprod_mulgA	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "extnprod_mulg", "ffunE", "ffunP", "mulgA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
oneg_ffun i : (1 : gTn) i = 1.	Proof. by rewrite ffunE. Qed.	Lemma	oneg_ffun	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "ffunE", "gTn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulg_ffun i (x y : gTn) : (x * y) i = x i * y i.	Proof. by rewrite ffunE. Qed.	Lemma	mulg_ffun	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "ffunE", "gTn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invg_ffun i (x : gTn) : x^-1 i = (x i)^-1.	Proof. by rewrite ffunE. Qed.	Lemma	invg_ffun	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "ffunE", "gTn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodg_ffun T (r : seq T) (F : T -> gTn) (P : {pred T}) i : (\prod_(t <- r \| P t) F t) i = \prod_(t <- r \| P t) F t i.	Proof. exact: (big_morph _ (@mulg_ffun i) (@oneg_ffun i)). Qed.	Lemma	prodg_ffun	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "big_morph", "gTn", "mulg_ffun", "oneg_ffun", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_setXn H : group_set (setXn H).	Proof. by apply/group_setP; split=> [\|x y] /[!inE]/= => [\|/forallP xH /forallP yH]; apply/forallP => i; rewrite ?ffunE (group1, groupM)// ?xH ?yH. Qed.	Lemma	group_setXn	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "ffunE", "forallP", "group1", "groupM", "group_set", "group_setP", "inE", "setXn", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setXn_group H	:= Group (group_setXn H).	Canonical	setXn_group	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "group_setXn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dfung1 i (g : gT i) : gTn	:= finfun (dfwith (fun=> 1 : gT _) g).	Definition	dfung1	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "dfwith", "gT", "gTn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dfung1_id i (g : gT i) : dfung1 g i = g.	Proof. by rewrite ffunE dfwith_in. Qed.	Lemma	dfung1_id	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "dfung1", "dfwith_in", "ffunE", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dfung1_dflt i (g : gT i) j : i != j -> dfung1 g j = 1.	Proof. by move=> ij; rewrite ffunE dfwith_out. Qed.	Lemma	dfung1_dflt	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "dfung1", "dfwith_out", "ffunE", "gT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dfung1_morphM i : {morph @dfung1 i : g h / g * h}.	Proof. move=> g h; apply/ffunP=> j; have [{j}<-\|nij] := eqVneq i j. by rewrite !(dfung1_id, ffunE). by rewrite !(dfung1_dflt, ffunE)// mulg1. Qed.	Lemma	dfung1_morphM	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "dfung1", "dfung1_dflt", "dfung1_id", "eqVneq", "ffunE", "ffunP", "mulg1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dfung1_morphism i	:= @Morphism _ _ setT _ (in2W (@dfung1_morphM i)).	Canonical	dfung1_morphism	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "dfung1_morphM", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dffunM i : {morph (fun x => x i) : x y / x * y}.	Proof. by move=> x y; rewrite !ffunE. Qed.	Lemma	dffunM	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "ffunE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dffun_morphism i	:= @Morphism _ _ setT _ (in2W (@dffunM i)).	Canonical	dffun_morphism	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "dffunM", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_dfung1 i : 'injm (@dfung1 i).	Proof. apply/subsetP => x /morphpreP[_ /set1P /ffunP/=/(_ i)]. by rewrite !(ffunE, dfung1_id) => ->; apply: set11. Qed.	Lemma	injm_dfung1	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "dfung1", "dfung1_id", "ffunE", "ffunP", "morphpreP", "set11", "set1P", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_set_dfwith H i (G : {group gT i}) j : group_set (dfwith (H : forall k, {set gT k}) (G : {set _}) j).	Proof. have [<-\|ij] := eqVneq i j; first by rewrite !dfwith_in// groupP. by rewrite !dfwith_out // groupP. Qed.	Lemma	group_set_dfwith	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "dfwith", "dfwith_in", "dfwith_out", "eqVneq", "gT", "group", "groupP", "group_set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_dfwith H i G j	:= Group (@group_set_dfwith H i G j).	Canonical	group_dfwith	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "group_set_dfwith" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
group_dfwithE H i G j : @group_dfwith H i G j = dfwith H G j.	Proof. by apply/val_inj; have [<-\|nij]/= := eqVneq i j; [rewrite !dfwith_in\|rewrite !dfwith_out]. Qed.	Lemma	group_dfwithE	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "dfwith", "dfwith_in", "dfwith_out", "eqVneq", "group_dfwith", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set1gXn_key : unit.	Proof. by []. Qed.	Fact	set1gXn_key	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "unit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set1gXn {i} (H : {set gT i}) : {set {dffun forall i : I, gT i}}	:= locked_with set1gXn_key (setXn (dfwith (fun i0 : I => [1 gT _]%g) H)).	Definition	set1gXn	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "dfwith", "gT", "i0", "set1gXn_key", "setXn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set1gXnE {i} (H : {set gT i}) : set1gXn H = setXn (dfwith (fun i0 : I => [1 gT _]%g) H).	Proof. by rewrite /set1gXn unlock. Qed.	Lemma	set1gXnE	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "dfwith", "gT", "i0", "set1gXn", "setXn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set1gXnP {i} (H : {set gT i}) x : reflect (exists2 h, h \in H & x = dfung1 h) (x \in set1gXn H).	Proof. rewrite set1gXnE/=; apply: (iffP setXnP) => [xP\|[h hH ->] j]; last first. by rewrite ffunE; case: dfwithP => [\|k ?]; rewrite (dfwith_in, dfwith_out). exists (x i); first by have := xP i; rewrite dfwith_in. apply/ffunP => j; have := xP j; rewrite ffunE. case: dfwithP => // [xiH\|k neq_ik]; first by rewrite dfwit...	Lemma	set1gXnP	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "dfung1", "dfwithP", "dfwith_in", "dfwith_out", "ffunE", "ffunP", "gT", "last", "set1gP", "set1gXn", "set1gXnE", "setXnP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_dfung1 i (G : {set gT i}) : @dfung1 i @* G = set1gXn G.	Proof. by rewrite morphimEsub//=; apply/setP=> /= x; apply/imsetP/set1gXnP. Qed.	Lemma	morphim_dfung1	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "dfung1", "gT", "imsetP", "morphimEsub", "set1gXn", "set1gXnP", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_dffunXn i H : dffun_morphism i @* setXn H = H i.	Proof. apply/eqP; rewrite eqEsubset morphimE setTI /=. apply/andP; split; apply/subsetP=> x. by case/imsetP => x0 /[1!inE] /forallP/(_ i)/= ? ->. move=> Hx1; apply/imsetP; exists (dfung1 x); last by rewrite dfung1_id. by rewrite in_setXn; apply/forallP => j /[!ffunE]; case: dfwithP. Qed.	Lemma	morphim_dffunXn	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "dffun_morphism", "dfung1", "dfung1_id", "dfwithP", "eqEsubset", "ffunE", "forallP", "imsetP", "inE", "in_setXn", "last", "morphimE", "setTI", "setXn", "split", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set1gXn_group_set {i} (H : {group gT i}) : group_set (set1gXn H).	Proof. by rewrite set1gXnE; exact: group_setXn. Qed.	Lemma	set1gXn_group_set	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "gT", "group", "group_set", "group_setXn", "set1gXn", "set1gXnE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupXn1 {i} (H : {group gT i})	:= Group (set1gXn_group_set H).	Canonical	groupXn1	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "gT", "group", "set1gXn_group_set" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setXn_prod H : \prod_i set1gXn (H i) = setXn H.	Proof. apply/setP => /= x; apply/prodsgP /setXnP => [[/= f fH {x}-> i]\|xH /=]. rewrite prodg_ffun group_prod// => j _. by have /set1gXnP[x xH ->] := fH j isT; rewrite ffunE; case: dfwithP. exists (fun i => dfung1 (x i)) => [i _\|]; first by apply/set1gXnP; exists (x i). apply/ffunP => i; rewrite prodg_ffun (big_only...	Lemma	setXn_prod	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "big_only1", "dfung1", "dfung1_dflt", "dfung1_id", "dfwithP", "fH", "ffunE", "ffunP", "group_prod", "prodg_ffun", "prodsgP", "set1gXn", "set1gXnP", "setP", "setXn", "setXnP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set1gXn_commute (H : forall i, {group gT i}) i j : commute (set1gXn (H i)) (set1gXn (H j)).	Proof. have [-> //\|neqij] := eqVneq j i. apply/centC/centsP => _ /set1gXnP [hi hiH ->] _ /set1gXnP [hj hjH ->]. apply/ffunP => k; rewrite !ffunE. by case: dfwithP => [\|?]; rewrite ?mulg1 ?mul1g// dfwith_out// mulg1 mul1g. Qed.	Lemma	set1gXn_commute	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "centC", "centsP", "commute", "dfwithP", "dfwith_out", "eqVneq", "ffunE", "ffunP", "gT", "group", "mul1g", "mulg1", "set1gXn", "set1gXnP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setXn_dprod H : \big[dprod/1]_i set1gXn (H i) = setXn H.	Proof. rewrite -setXn_prod//=. suff -> : \big[dprod/1]_i groupXn1 (H i) = (\prod_i groupXn1 (H i))%G. by rewrite comm_prodG//=; apply: in2W; apply: set1gXn_commute. apply/eqP; apply/bigdprodYP => i //= _; rewrite subsetD. apply/andP; split. rewrite comm_prodG; first by apply: in2W; apply: set1gXn_commute. apply/c...	Lemma	setXn_dprod	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "big1", "big_morph", "bigdprodYP", "centsP", "comm_prodG", "contra_neqT", "dffunM", "dfwithP", "dfwith_in", "dfwith_out", "dprod", "ffunE", "ffunP", "gH", "groupXn1", "inE", "last", "mul1g", "mulg1", "prodsgP", "set1gXn", "set1gXnP", "set1gXn_commute", "setI_...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isog_setXn i (G : {group gT i}) : G \isog set1gXn G.	Proof. apply/(@isogP _ _ G); exists [morphism of restrm (subsetT G) (@dfung1 i)]. by rewrite injm_restrm ?injm_dfung1. by rewrite morphim_restrm morphim_dfung1 setIid. Qed.	Lemma	isog_setXn	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "dfung1", "gT", "group", "injm_dfung1", "injm_restrm", "isog", "isogP", "morphim_dfung1", "morphim_restrm", "morphism", "restrm", "set1gXn", "setIid", "subsetT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
setXn_gen H : (forall i, 1 \in H i) -> <<setXn H>> = setXn (fun i => <<H i>>).	Proof. move=> H1; apply/eqP; rewrite eqEsubset gen_subG setXnS/=. by move=> ?; rewrite subset_gen. rewrite -[in X in X \subset _]setXn_prod; under eq_bigr do rewrite -morphim_dfung1 morphim_gen ?subsetT// morphim_dfung1. rewrite prod_subG// => i; rewrite genS // set1gXnE setXnS // => j. by case: dfwithP => // k _;...	Lemma	setXn_gen	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "dfwithP", "eqEsubset", "eq_bigr", "genS", "gen_subG", "morphim_dfung1", "morphim_gen", "prod_subG", "set1gXnE", "setXn", "setXnS", "setXn_prod", "sub1set", "subsetT", "subset_gen" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
groupX0 (gT : 'I_0 -> finGroupType) (G : forall i, {group gT i}) : setXn G = 1%g.	Proof. by apply/setP => ?; apply/setXnP/set1P => [_\|_ []//]; apply/ffunP => -[]. Qed.	Lemma	groupX0	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "ffunP", "gT", "group", "set1P", "setP", "setXn", "setXnP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_by (to : groupAction D R) : predArgType	:= SdPair (ax : aT * rT) of ax \in setX D R.	Inductive	sdprod_by	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "aT", "groupAction", "setX", "to" ]	The pair (a, x) denotes the product sdpair2 a * sdpair1 x	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_of_sd to (u : sdprod_by to)	:= let: SdPair ax _ := u in ax.	Coercion	pair_of_sd	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "sdprod_by", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdT	:= (sdprod_by to).	Notation	sdT	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "sdprod_by", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdval	:= (@pair_of_sd to).	Notation	sdval	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "pair_of_sd", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_one	:= SdPair to (group1 _).	Definition	sdprod_one	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "group1", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_inv_proof (u : sdT) : (u.1^-1, to u.2^-1 u.1^-1) \in setX D R.	Proof. by case: u => [[a x]] /= /setXP[Da Rx]; rewrite inE gact_stable !groupV ?Da. Qed.	Lemma	sdprod_inv_proof	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "gact_stable", "groupV", "inE", "sdT", "setX", "setXP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_inv u	:= SdPair to (sdprod_inv_proof u).	Definition	sdprod_inv	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "sdprod_inv_proof", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_mul_proof (u v : sdT) : (u.1 * v.1, to u.2 v.1 * v.2) \in setX D R.	Proof. case: u v => [[a x] /= /setXP[Da Rx]] [[b y] /= /setXP[Db Ry]]. by rewrite inE !groupM //= gact_stable. Qed.	Lemma	sdprod_mul_proof	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "gact_stable", "groupM", "inE", "sdT", "setX", "setXP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_mul u v	:= SdPair to (sdprod_mul_proof u v).	Definition	sdprod_mul	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "sdprod_mul_proof", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_mul1g : left_id sdprod_one sdprod_mul.	Proof. move=> u; apply: val_inj; case: u => [[a x] /=]; case/setXP=> Da _. by rewrite gact1 // !mul1g. Qed.	Lemma	sdprod_mul1g	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "apply", "gact1", "mul1g", "sdprod_mul", "sdprod_one", "setXP", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_mulVg : left_inverse sdprod_one sdprod_inv sdprod_mul.	Proof. move=> u; apply: val_inj; case: u => [[a x] /=]; case/setXP=> Da _. by rewrite actKVin ?mulVg. Qed.	Lemma	sdprod_mulVg	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "actKVin", "apply", "mulVg", "sdprod_inv", "sdprod_mul", "sdprod_one", "setXP", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_mulgA : associative sdprod_mul.	Proof. move=> u v w; apply: val_inj; case: u => [[a x]] /=; case/setXP=> Da Rx. case: v w => [[b y]] /=; case/setXP=> Db Ry [[c z]] /=; case/setXP=> Dc Rz. by rewrite !(actMin to) // gactM ?gact_stable // !mulgA. Qed.	Lemma	sdprod_mulgA	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "actMin", "apply", "gactM", "gact_stable", "mulgA", "sdprod_mul", "setXP", "to", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_groupType : finGroupType	:= sdT.	Definition	sdprod_groupType	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "sdT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdpair1 x	:= insubd sdprod_one (1, x) : sdT.	Definition	sdpair1	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "insubd", "sdT", "sdprod_one" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdpair2 a	:= insubd sdprod_one (a, 1) : sdT.	Definition	sdpair2	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "insubd", "sdT", "sdprod_one" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdpair1_morphM : {in R &, {morph sdpair1 : x y / x * y}}.	Proof. move=> x y Rx Ry; apply: val_inj. by rewrite /= !val_insubd !inE !group1 !groupM ?Rx ?Ry //= mulg1 act1. Qed.	Lemma	sdpair1_morphM	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "act1", "apply", "group1", "groupM", "inE", "mulg1", "sdpair1", "val_inj", "val_insubd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdpair2_morphM : {in D &, {morph sdpair2 : a b / a * b}}.	Proof. move=> a b Da Db; apply: val_inj. by rewrite /= !val_insubd !inE !group1 !groupM ?Da ?Db //= mulg1 gact1. Qed.	Lemma	sdpair2_morphM	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "apply", "gact1", "group1", "groupM", "inE", "mulg1", "sdpair2", "val_inj", "val_insubd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdpair1_morphism	:= Morphism sdpair1_morphM.	Canonical	sdpair1_morphism	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "sdpair1_morphM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdpair2_morphism	:= Morphism sdpair2_morphM.	Canonical	sdpair2_morphism	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "sdpair2_morphM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_sdpair1 : 'injm sdpair1.	Proof. apply/subsetP=> x /setIP[Rx]. by rewrite !inE -val_eqE val_insubd inE Rx group1 /=; case/andP. Qed.	Lemma	injm_sdpair1	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "group1", "inE", "sdpair1", "setIP", "subsetP", "val_eqE", "val_insubd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_sdpair2 : 'injm sdpair2.	Proof. apply/subsetP=> a /setIP[Da]. by rewrite !inE -val_eqE val_insubd inE Da group1 /=; case/andP. Qed.	Lemma	injm_sdpair2	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "apply", "group1", "inE", "sdpair2", "setIP", "subsetP", "val_eqE", "val_insubd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdpairE (u : sdT) : u = sdpair2 u.1 * sdpair1 u.2.	Proof. apply: val_inj; case: u => [[a x] /= /setXP[Da Rx]]. by rewrite !val_insubd !inE Da Rx !(group1, gact1) // mulg1 mul1g. Qed.	Lemma	sdpairE	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "apply", "gact1", "group1", "inE", "mul1g", "mulg1", "sdT", "sdpair1", "sdpair2", "setXP", "val_inj", "val_insubd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdpair_act : {in R & D, forall x a, sdpair1 (to x a) = sdpair1 x ^ sdpair2 a}.	Proof. move=> x a Rx Da; apply: val_inj. rewrite /= !val_insubd !inE !group1 gact_stable ?Da ?Rx //=. by rewrite !mul1g mulVg invg1 mulg1 actKVin ?mul1g. Qed.	Lemma	sdpair_act	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "actKVin", "apply", "gact_stable", "group1", "inE", "invg1", "mul1g", "mulVg", "mulg1", "sdpair1", "sdpair2", "to", "val_inj", "val_insubd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdpair_setact (G : {set rT}) a : G \subset R -> a \in D -> sdpair1 @* (to^~ a @: G) = (sdpair1 @* G) :^ sdpair2 a.	Proof. move=> sGR Da; have GtoR := subsetP sGR; apply/eqP. rewrite eqEcard cardJg !(card_injm injm_sdpair1) //. by apply/subsetP=> _ /imsetP[x Gx ->]; rewrite gact_stable ?GtoR. rewrite (card_imset _ (act_inj _ _)) leqnn andbT. apply/subsetP=> _ /morphimP[xa Rxa /imsetP[x Gx def_xa ->]]. rewrite mem_conjg -morphV // ...	Lemma	sdpair_setact	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "actKin", "act_inj", "apply", "cardJg", "card_imset", "card_injm", "eqEcard", "gact_stable", "groupV", "imsetP", "injm_sdpair1", "leqnn", "mem_conjg", "mem_morphim", "morphV", "morphimP", "sdpair1", "sdpair2", "sdpair_act", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_sdpair_norm : sdpair2 @* D \subset 'N(sdpair1 @* R).	Proof. apply/subsetP=> _ /morphimP[a _ Da ->]. rewrite inE -sdpair_setact // morphimS //. by apply/subsetP=> _ /imsetP[x Rx ->]; rewrite gact_stable. Qed.	Lemma	im_sdpair_norm	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "apply", "gact_stable", "imsetP", "inE", "morphimP", "morphimS", "sdpair1", "sdpair2", "sdpair_setact", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_sdpair_TI : (sdpair1 @* R) :&: (sdpair2 @* D) = 1.	Proof. apply/trivgP; apply/subsetP=> _ /setIP[/morphimP[x _ Rx ->]]. case/morphimP=> a _ Da /eqP; rewrite inE -!val_eqE. by rewrite !val_insubd !inE Da Rx !group1 /eq_op /= eqxx; case/andP. Qed.	Lemma	im_sdpair_TI	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "apply", "eqxx", "group1", "inE", "morphimP", "sdpair1", "sdpair2", "setIP", "subsetP", "trivgP", "val_eqE", "val_insubd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_sdpair : (sdpair1 @* R) * (sdpair2 @* D) = setT.	Proof. apply/eqP; rewrite -subTset -(normC im_sdpair_norm). apply/subsetP=> /= u _; rewrite [u]sdpairE. by case: u => [[a x] /= /setXP[Da Rx]]; rewrite mem_mulg ?mem_morphim. Qed.	Lemma	im_sdpair	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "apply", "im_sdpair_norm", "mem_morphim", "mem_mulg", "normC", "sdpair1", "sdpair2", "sdpairE", "setT", "setXP", "subTset", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprod_sdpair : sdpair1 @* R ><\| sdpair2 @* D = setT.	Proof. by rewrite sdprodE ?(im_sdpair_norm, im_sdpair, im_sdpair_TI). Qed.	Lemma	sdprod_sdpair	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "im_sdpair", "im_sdpair_TI", "im_sdpair_norm", "sdpair1", "sdpair2", "sdprodE", "setT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gacentEsd : 'C_(\|to)(A) = sdpair1 @^-1 'C(sdpair2 @ A).	Proof. apply/setP=> x; apply/idP/idP. case/setIP=> Rx /afixP cDAx; rewrite mem_morphpre //. apply/centP=> _ /morphimP[a Da Aa ->]; red. by rewrite conjgC -sdpair_act // cDAx // inE Da. case/morphpreP=> Rx cAx; rewrite inE Rx; apply/afixP=> a /setIP[Da Aa]. apply: (injmP injm_sdpair1); rewrite ?gact_stable /= ?sdp...	Lemma	gacentEsd	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "afixP", "apply", "centP", "conjg", "conjgC", "gact_stable", "inE", "injmP", "injm_sdpair1", "mem_morphim", "mem_morphpre", "morphimP", "morphpreP", "mulKg", "sdpair1", "sdpair2", "sdpair_act", "setIP", "setP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
(sAD : A \subset D) (sGR : G \subset R).		Hypotheses	sAD	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabEsd : 'C(G \| to) = sdpair2 @^-1 'C(sdpair1 @ G).	Proof. have ssGR := subsetP sGR; apply/setP=> a; apply/idP/idP=> [cGa\|]. rewrite mem_morphpre ?(astab_dom cGa) //. apply/centP=> _ /morphimP[x Rx Gx ->]; symmetry. by rewrite conjgC -sdpair_act ?(astab_act cGa) ?(astab_dom cGa). case/morphpreP=> Da cGa; rewrite !inE Da; apply/subsetP=> x Gx; rewrite inE. apply/e...	Lemma	astabEsd	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "apply", "astab_act", "astab_dom", "centP", "conjg", "conjgC", "gact_stable", "inE", "injmP", "injm_sdpair1", "mem_morphim", "mem_morphpre", "morphimP", "morphpreP", "mulKg", "sdpair1", "sdpair2", "sdpair_act", "setP", "subsetP", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
astabsEsd : 'N(G \| to) = sdpair2 @^-1 'N(sdpair1 @ G).	Proof. apply/setP=> a; apply/idP/idP=> [nGa\|]. have Da := astabs_dom nGa; rewrite mem_morphpre // inE sub_conjg. apply/subsetP=> _ /morphimP[x Rx Gx ->]. by rewrite mem_conjgV -sdpair_act // mem_morphim ?gact_stable ?astabs_act. case/morphpreP=> Da nGa; rewrite !inE Da; apply/subsetP=> x Gx. have Rx := subsetP sG...	Lemma	astabsEsd	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "Da", "apply", "astabs_act", "astabs_dom", "gact_stable", "inE", "injmSK", "injm_sdpair1", "memJ_norm", "mem_conjgV", "mem_morphim", "mem_morphpre", "morphimP", "morphim_set1", "morphpreP", "sdpair1", "sdpair2", "sdpair_act", "setP", "sub1set", "sub_conjg", "subsetP", "to...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actsEsd : [acts A, on G \| to] = (sdpair2 @* A \subset 'N(sdpair1 @* G)).	Proof. by rewrite sub_morphim_pre -?astabsEsd. Qed.	Lemma	actsEsd	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "astabsEsd", "on", "sdpair1", "sdpair2", "sub_morphim_pre", "to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodm & B \subset 'N(A) & {in A & B, morph_act 'J 'J fA fB} & {in A :&: B, fA =1 fB}	:= fun x => fA (divgr A B x) * fB (remgr A B x).	Definition	pprodm	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "divgr", "fA", "morph_act", "remgr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nHK : K \subset 'N(H).		Hypothesis	nHK	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
actf : {in H & K, morph_act 'J 'J fH fK}.		Hypothesis	actf	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "fH", "fK", "morph_act" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqfHK : {in H :&: K, fH =1 fK}.		Hypothesis	eqfHK	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "fH", "fK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
f	:= (pprodm nHK actf eqfHK).	Notation	f	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "actf", "eqfHK", "nHK", "pprodm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodmE x a : x \in H -> a \in K -> f (x * a) = fH x * fK a.	Proof. move=> Hx Ka; have: x * a \in H * K by rewrite mem_mulg. rewrite -remgrP inE /f rcoset_sym mem_rcoset /divgr -mulgA groupMl //. case/andP; move: (remgr H K _) => b Hab Kb; rewrite morphM // -mulgA. have Kab: a * b^-1 \in K by rewrite groupM ?groupV. by congr (_ * _); rewrite eqfHK 1?inE ?Hab // -morphM // mulgKV...	Lemma	pprodmE	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "divgr", "eqfHK", "fH", "fK", "groupM", "groupMl", "groupV", "inE", "mem_mulg", "mem_rcoset", "morphM", "mulgA", "mulgKV", "rcoset_sym", "remgr", "remgrP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodmEl : {in H, f =1 fH}.	Proof. by move=> x Hx; rewrite -(mulg1 x) pprodmE // morph1 !mulg1. Qed.	Lemma	pprodmEl	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "fH", "morph1", "mulg1", "pprodmE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodmEr : {in K, f =1 fK}.	Proof. by move=> a Ka; rewrite -(mul1g a) pprodmE // morph1 !mul1g. Qed.	Lemma	pprodmEr	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "fK", "morph1", "mul1g", "pprodmE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodmM : {in H <> K &, {morph f: x y / x y}}.	Proof. move=> xa yb; rewrite norm_joinEr //. move=> /imset2P[x a Ha Ka ->{xa}] /imset2P[y b Hy Kb ->{yb}]. have Hya: y ^ a^-1 \in H by rewrite -mem_conjg (normsP nHK). rewrite mulgA -(mulgA x) (conjgCV a y) (mulgA x) -mulgA !pprodmE 1?groupMl //. by rewrite morphM // actf ?groupV ?morphV // morphM // !mulgA mulgKV invg...	Lemma	pprodmM	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "actf", "conjgCV", "groupMl", "groupV", "imset2P", "invgK", "mem_conjg", "morphM", "morphV", "mulgA", "mulgKV", "nHK", "norm_joinEr", "normsP", "pprodmE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pprodm_morphism	:= Morphism pprodmM.	Canonical	pprodm_morphism	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "pprodmM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_pprodm A B : A \subset H -> B \subset K -> f @* (A * B) = fH @* A * fK @* B.	Proof. move=> sAH sBK; rewrite [f @* _]morphimEsub /=. by rewrite norm_joinEr // mulgSS. apply/setP=> y; apply/imsetP/idP=> [[_ /mulsgP[x a Ax Ba ->] ->{y}] \|]. have Hx := subsetP sAH x Ax; have Ka := subsetP sBK a Ba. by rewrite pprodmE // imset2_f ?mem_morphim. case/mulsgP=> _ _ /morphimP[x Hx Ax ->] /morphimP[...	Lemma	morphim_pprodm	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "fH", "fK", "imset2_f", "imsetP", "mem_morphim", "mem_mulg", "morphimEsub", "morphimP", "mulgSS", "mulsgP", "norm_joinEr", "pprodmE", "setP", "subsetP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_pprodml A : A \subset H -> f @* A = fH @* A.	Proof. by move=> sAH; rewrite -{1}(mulg1 A) morphim_pprodm ?sub1G // morphim1 mulg1. Qed.	Lemma	morphim_pprodml	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "fH", "morphim1", "morphim_pprodm", "mulg1", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_pprodmr B : B \subset K -> f @* B = fK @* B.	Proof. by move=> sBK; rewrite -{1}(mul1g B) morphim_pprodm ?sub1G // morphim1 mul1g. Qed.	Lemma	morphim_pprodmr	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "fK", "morphim1", "morphim_pprodm", "mul1g", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_pprodm : 'ker f = [set x * a^-1 \| x in H, a in K & fH x == fK a].	Proof. apply/setP=> y; rewrite 3!inE {1}norm_joinEr //=. apply/andP/imset2P=> [[/mulsgP[x a Hx Ka ->{y}]]\|[x a Hx]]. rewrite pprodmE // => fxa1. by exists x a^-1; rewrite ?invgK // inE groupVr ?morphV // eq_mulgV1 invgK. case/setIdP=> Kx /eqP fx ->{y}. by rewrite imset2_f ?pprodmE ?groupV ?morphV // fx mulgV. Qed.	Lemma	ker_pprodm	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "eq_mulgV1", "fH", "fK", "groupV", "groupVr", "imset2P", "imset2_f", "inE", "invgK", "ker", "morphV", "mulgV", "mulsgP", "norm_joinEr", "pprodmE", "setIdP", "setP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_pprodm : 'injm f = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == fH @* K].	Proof. apply/idP/and3P=> [injf \| [injfH injfK]]. rewrite eq_sym -{1}morphimIdom -(morphim_pprodml (subsetIl _ _)) injmI //. rewrite morphim_pprodml // morphim_pprodmr //=; split=> //. apply/injmP=> x y Hx Hy /=; rewrite -!pprodmEl //. by apply: (injmP injf); rewrite ?mem_gen ?inE ?Hx ?Hy. apply/injmP=> a ...	Lemma	injm_pprodm	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "eq_sym", "eqfHK", "fH", "fK", "imset2P", "inE", "injf", "injmI", "injmP", "ker_pprodm", "mem_gen", "mem_morphim", "morphimIdom", "morphimP", "morphim_pprodml", "morphim_pprodmr", "mulgV", "pprodmEl", "pprodmEr", "set11", "setIdP", "setUP", "split", "subsetIl...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqHK_G : H ><\| K = G.		Hypothesis	eqHK_G	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodm_norm : K \subset 'N(H).	Proof. by case/sdprodP: eqHK_G. Qed.	Lemma	sdprodm_norm	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "eqHK_G", "sdprodP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodm_sub : G \subset H <*> K.	Proof. by case/sdprodP: eqHK_G => _ <- nHK _; rewrite norm_joinEr. Qed.	Lemma	sdprodm_sub	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "eqHK_G", "nHK", "norm_joinEr", "sdprodP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodm_eqf : {in H :&: K, fH =1 fK}.	Proof. by case/sdprodP: eqHK_G => _ _ _ -> _ /set1P->; rewrite !morph1. Qed.	Lemma	sdprodm_eqf	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "eqHK_G", "fH", "fK", "morph1", "sdprodP", "set1P" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodm	:= restrm sdprodm_sub (pprodm sdprodm_norm actf sdprodm_eqf).	Definition	sdprodm	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "actf", "pprodm", "restrm", "sdprodm_eqf", "sdprodm_norm", "sdprodm_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodm_morphism	:= Eval hnf in [morphism of sdprodm].	Canonical	sdprodm_morphism	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "morphism", "sdprodm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodmE a b : a \in H -> b \in K -> sdprodm (a * b) = fH a * fK b.	Proof. exact: pprodmE. Qed.	Lemma	sdprodmE	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "fH", "fK", "pprodmE", "sdprodm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodmEl a : a \in H -> sdprodm a = fH a.	Proof. exact: pprodmEl. Qed.	Lemma	sdprodmEl	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "fH", "pprodmEl", "sdprodm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdprodmEr b : b \in K -> sdprodm b = fK b.	Proof. exact: pprodmEr. Qed.	Lemma	sdprodmEr	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "fK", "pprodmEr", "sdprodm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_sdprodm A B : A \subset H -> B \subset K -> sdprodm @* (A * B) = fH @* A * fK @* B.	Proof. move=> sAH sBK; rewrite /sdprodm morphim_restrm /= (setIidPr _) ?morphim_pprodm //. by case/sdprodP: eqHK_G => _ <- _ _; apply: mulgSS. Qed.	Lemma	morphim_sdprodm	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "eqHK_G", "fH", "fK", "morphim_pprodm", "morphim_restrm", "mulgSS", "sdprodP", "sdprodm", "setIidPr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
im_sdprodm : sdprodm @* G = fH @* H * fK @* K.	Proof. by rewrite -morphim_sdprodm //; case/sdprodP: eqHK_G => _ ->. Qed.	Lemma	im_sdprodm	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "eqHK_G", "fH", "fK", "morphim_sdprodm", "sdprodP", "sdprodm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_sdprodml A : A \subset H -> sdprodm @* A = fH @* A.	Proof. by move=> sHA; rewrite -{1}(mulg1 A) morphim_sdprodm ?sub1G // morphim1 mulg1. Qed.	Lemma	morphim_sdprodml	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "fH", "morphim1", "morphim_sdprodm", "mulg1", "sdprodm", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
morphim_sdprodmr B : B \subset K -> sdprodm @* B = fK @* B.	Proof. by move=> sBK; rewrite -{1}(mul1g B) morphim_sdprodm ?sub1G // morphim1 mul1g. Qed.	Lemma	morphim_sdprodmr	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "fK", "morphim1", "morphim_sdprodm", "mul1g", "sdprodm", "sub1G" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ker_sdprodm : 'ker sdprodm = [set a * b^-1 \| a in H, b in K & fH a == fK b].	Proof. rewrite ker_restrm (setIidPr _) ?subIset ?ker_pprodm //; apply/orP; left. by case/sdprodP: eqHK_G => _ <- nHK _; rewrite norm_joinEr. Qed.	Lemma	ker_sdprodm	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "apply", "eqHK_G", "fH", "fK", "ker", "ker_pprodm", "ker_restrm", "nHK", "norm_joinEr", "sdprodP", "sdprodm", "setIidPr", "subIset" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
injm_sdprodm : 'injm sdprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == 1].	Proof. rewrite ker_sdprodm -(ker_pprodm sdprodm_norm actf sdprodm_eqf) injm_pprodm. congr [&& _, _ & _ == _]; have [_ _ _ tiHK] := sdprodP eqHK_G. by rewrite -morphimIdom tiHK morphim1. Qed.	Lemma	injm_sdprodm	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "actf", "eqHK_G", "fH", "fK", "injm_pprodm", "ker_pprodm", "ker_sdprodm", "morphim1", "morphimIdom", "sdprodP", "sdprodm", "sdprodm_eqf", "sdprodm_norm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqHK_G : H \* K = G.		Hypothesis	eqHK_G	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cfHK : fK @* K \subset 'C(fH @* H).		Hypothesis	cfHK	finite_group	finite_group/gproduct.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]	[ "fH", "fK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

extnprod_invg (x : gTn) : gTn

:= [ffun i => (x i)^-1%g].

Definition

extnprod_invg

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "gTn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

extnprod_mul1g : left_id [ffun=> 1%g] extnprod_mulg.

Proof. by move=> x; apply/ffunP => i; rewrite !ffunE mul1g. Qed.

Lemma

extnprod_mul1g

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "extnprod_mulg", "ffunE", "ffunP", "mul1g" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

extnprod_mulVg : left_inverse [ffun=> 1%g] extnprod_invg extnprod_mulg.

Proof. by move=> x; apply/ffunP => i; rewrite !ffunE mulVg. Qed.

Lemma

extnprod_mulVg

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "extnprod_invg", "extnprod_mulg", "ffunE", "ffunP", "mulVg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

extnprod_mulgA : associative extnprod_mulg.

Proof. by move=> x y z; apply/ffunP => i; rewrite !ffunE mulgA. Qed.

Lemma

extnprod_mulgA

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "extnprod_mulg", "ffunE", "ffunP", "mulgA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

oneg_ffun i : (1 : gTn) i = 1.

Proof. by rewrite ffunE. Qed.

Lemma

oneg_ffun

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "ffunE", "gTn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulg_ffun i (x y : gTn) : (x * y) i = x i * y i.

Proof. by rewrite ffunE. Qed.

Lemma

mulg_ffun

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "ffunE", "gTn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invg_ffun i (x : gTn) : x^-1 i = (x i)^-1.

Proof. by rewrite ffunE. Qed.

Lemma

invg_ffun

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "ffunE", "gTn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodg_ffun T (r : seq T) (F : T -> gTn) (P : {pred T}) i : (\prod_(t <- r | P t) F t) i = \prod_(t <- r | P t) F t i.

Proof. exact: (big_morph _ (@mulg_ffun i) (@oneg_ffun i)). Qed.

Lemma

prodg_ffun

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "big_morph", "gTn", "mulg_ffun", "oneg_ffun", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_setXn H : group_set (setXn H).

Proof. by apply/group_setP; split=> [|x y] /[!inE]/= => [|/forallP xH /forallP yH]; apply/forallP => i; rewrite ?ffunE (group1, groupM)// ?xH ?yH. Qed.

Lemma

group_setXn

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "ffunE", "forallP", "group1", "groupM", "group_set", "group_setP", "inE", "setXn", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setXn_group H

:= Group (group_setXn H).

Canonical

setXn_group

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "group_setXn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dfung1 i (g : gT i) : gTn

:= finfun (dfwith (fun=> 1 : gT _) g).

Definition

dfung1

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "dfwith", "gT", "gTn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dfung1_id i (g : gT i) : dfung1 g i = g.

Proof. by rewrite ffunE dfwith_in. Qed.

Lemma

dfung1_id

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "dfung1", "dfwith_in", "ffunE", "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dfung1_dflt i (g : gT i) j : i != j -> dfung1 g j = 1.

Proof. by move=> ij; rewrite ffunE dfwith_out. Qed.

Lemma

dfung1_dflt

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "dfung1", "dfwith_out", "ffunE", "gT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dfung1_morphM i : {morph @dfung1 i : g h / g * h}.

Proof. move=> g h; apply/ffunP=> j; have [{j}<-|nij] := eqVneq i j. by rewrite !(dfung1_id, ffunE). by rewrite !(dfung1_dflt, ffunE)// mulg1. Qed.

Lemma

dfung1_morphM

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "dfung1", "dfung1_dflt", "dfung1_id", "eqVneq", "ffunE", "ffunP", "mulg1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dfung1_morphism i

:= @Morphism _ _ setT _ (in2W (@dfung1_morphM i)).

Canonical

dfung1_morphism

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "dfung1_morphM", "setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dffunM i : {morph (fun x => x i) : x y / x * y}.

Proof. by move=> x y; rewrite !ffunE. Qed.

Lemma

dffunM

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "ffunE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dffun_morphism i

:= @Morphism _ _ setT _ (in2W (@dffunM i)).

Canonical

dffun_morphism

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "dffunM", "setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_dfung1 i : 'injm (@dfung1 i).

Proof. apply/subsetP => x /morphpreP[_ /set1P /ffunP/=/(_ i)]. by rewrite !(ffunE, dfung1_id) => ->; apply: set11. Qed.

Lemma

injm_dfung1

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "dfung1", "dfung1_id", "ffunE", "ffunP", "morphpreP", "set11", "set1P", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_set_dfwith H i (G : {group gT i}) j : group_set (dfwith (H : forall k, {set gT k}) (G : {set _}) j).

Proof. have [<-|ij] := eqVneq i j; first by rewrite !dfwith_in// groupP. by rewrite !dfwith_out // groupP. Qed.

Lemma

group_set_dfwith

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "dfwith", "dfwith_in", "dfwith_out", "eqVneq", "gT", "group", "groupP", "group_set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_dfwith H i G j

:= Group (@group_set_dfwith H i G j).

Canonical

group_dfwith

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "group_set_dfwith" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

group_dfwithE H i G j : @group_dfwith H i G j = dfwith H G j.

Proof. by apply/val_inj; have [<-|nij]/= := eqVneq i j; [rewrite !dfwith_in|rewrite !dfwith_out]. Qed.

Lemma

group_dfwithE

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "dfwith", "dfwith_in", "dfwith_out", "eqVneq", "group_dfwith", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set1gXn_key : unit.

Proof. by []. Qed.

Fact

set1gXn_key

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set1gXn {i} (H : {set gT i}) : {set {dffun forall i : I, gT i}}

:= locked_with set1gXn_key (setXn (dfwith (fun i0 : I => [1 gT _]%g) H)).

Definition

set1gXn

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "dfwith", "gT", "i0", "set1gXn_key", "setXn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set1gXnE {i} (H : {set gT i}) : set1gXn H = setXn (dfwith (fun i0 : I => [1 gT _]%g) H).

Proof. by rewrite /set1gXn unlock. Qed.

Lemma

set1gXnE

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "dfwith", "gT", "i0", "set1gXn", "setXn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set1gXnP {i} (H : {set gT i}) x : reflect (exists2 h, h \in H & x = dfung1 h) (x \in set1gXn H).

Proof. rewrite set1gXnE/=; apply: (iffP setXnP) => [xP|[h hH ->] j]; last first. by rewrite ffunE; case: dfwithP => [|k ?]; rewrite (dfwith_in, dfwith_out). exists (x i); first by have := xP i; rewrite dfwith_in. apply/ffunP => j; have := xP j; rewrite ffunE. case: dfwithP => // [xiH|k neq_ik]; first by rewrite dfwit...

Lemma

set1gXnP

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "dfung1", "dfwithP", "dfwith_in", "dfwith_out", "ffunE", "ffunP", "gT", "last", "set1gP", "set1gXn", "set1gXnE", "setXnP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_dfung1 i (G : {set gT i}) : @dfung1 i @* G = set1gXn G.

Proof. by rewrite morphimEsub//=; apply/setP=> /= x; apply/imsetP/set1gXnP. Qed.

Lemma

morphim_dfung1

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "dfung1", "gT", "imsetP", "morphimEsub", "set1gXn", "set1gXnP", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_dffunXn i H : dffun_morphism i @* setXn H = H i.

Proof. apply/eqP; rewrite eqEsubset morphimE setTI /=. apply/andP; split; apply/subsetP=> x. by case/imsetP => x0 /[1!inE] /forallP/(_ i)/= ? ->. move=> Hx1; apply/imsetP; exists (dfung1 x); last by rewrite dfung1_id. by rewrite in_setXn; apply/forallP => j /[!ffunE]; case: dfwithP. Qed.

Lemma

morphim_dffunXn

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "dffun_morphism", "dfung1", "dfung1_id", "dfwithP", "eqEsubset", "ffunE", "forallP", "imsetP", "inE", "in_setXn", "last", "morphimE", "setTI", "setXn", "split", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set1gXn_group_set {i} (H : {group gT i}) : group_set (set1gXn H).

Proof. by rewrite set1gXnE; exact: group_setXn. Qed.

Lemma

set1gXn_group_set

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "gT", "group", "group_set", "group_setXn", "set1gXn", "set1gXnE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupXn1 {i} (H : {group gT i})

:= Group (set1gXn_group_set H).

Canonical

groupXn1

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "gT", "group", "set1gXn_group_set" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setXn_prod H : \prod_i set1gXn (H i) = setXn H.

Proof. apply/setP => /= x; apply/prodsgP /setXnP => [[/= f fH {x}-> i]|xH /=]. rewrite prodg_ffun group_prod// => j _. by have /set1gXnP[x xH ->] := fH j isT; rewrite ffunE; case: dfwithP. exists (fun i => dfung1 (x i)) => [i _|]; first by apply/set1gXnP; exists (x i). apply/ffunP => i; rewrite prodg_ffun (big_only...

Lemma

setXn_prod

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "big_only1", "dfung1", "dfung1_dflt", "dfung1_id", "dfwithP", "fH", "ffunE", "ffunP", "group_prod", "prodg_ffun", "prodsgP", "set1gXn", "set1gXnP", "setP", "setXn", "setXnP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set1gXn_commute (H : forall i, {group gT i}) i j : commute (set1gXn (H i)) (set1gXn (H j)).

Proof. have [-> //|neqij] := eqVneq j i. apply/centC/centsP => _ /set1gXnP [hi hiH ->] _ /set1gXnP [hj hjH ->]. apply/ffunP => k; rewrite !ffunE. by case: dfwithP => [|?]; rewrite ?mulg1 ?mul1g// dfwith_out// mulg1 mul1g. Qed.

Lemma

set1gXn_commute

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "centC", "centsP", "commute", "dfwithP", "dfwith_out", "eqVneq", "ffunE", "ffunP", "gT", "group", "mul1g", "mulg1", "set1gXn", "set1gXnP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setXn_dprod H : \big[dprod/1]_i set1gXn (H i) = setXn H.

Proof. rewrite -setXn_prod//=. suff -> : \big[dprod/1]_i groupXn1 (H i) = (\prod_i groupXn1 (H i))%G. by rewrite comm_prodG//=; apply: in2W; apply: set1gXn_commute. apply/eqP; apply/bigdprodYP => i //= _; rewrite subsetD. apply/andP; split. rewrite comm_prodG; first by apply: in2W; apply: set1gXn_commute. apply/c...

Lemma

setXn_dprod

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "big1", "big_morph", "bigdprodYP", "centsP", "comm_prodG", "contra_neqT", "dffunM", "dfwithP", "dfwith_in", "dfwith_out", "dprod", "ffunE", "ffunP", "gH", "groupXn1", "inE", "last", "mul1g", "mulg1", "prodsgP", "set1gXn", "set1gXnP", "set1gXn_commute", "setI_...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isog_setXn i (G : {group gT i}) : G \isog set1gXn G.

Proof. apply/(@isogP _ _ G); exists [morphism of restrm (subsetT G) (@dfung1 i)]. by rewrite injm_restrm ?injm_dfung1. by rewrite morphim_restrm morphim_dfung1 setIid. Qed.

Lemma

isog_setXn

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "dfung1", "gT", "group", "injm_dfung1", "injm_restrm", "isog", "isogP", "morphim_dfung1", "morphim_restrm", "morphism", "restrm", "set1gXn", "setIid", "subsetT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

setXn_gen H : (forall i, 1 \in H i) -> <<setXn H>> = setXn (fun i => <<H i>>).

Proof. move=> H1; apply/eqP; rewrite eqEsubset gen_subG setXnS/=. by move=> ?; rewrite subset_gen. rewrite -[in X in X \subset _]setXn_prod; under eq_bigr do rewrite -morphim_dfung1 morphim_gen ?subsetT// morphim_dfung1. rewrite prod_subG// => i; rewrite genS // set1gXnE setXnS // => j. by case: dfwithP => // k _;...

Lemma

setXn_gen

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "dfwithP", "eqEsubset", "eq_bigr", "genS", "gen_subG", "morphim_dfung1", "morphim_gen", "prod_subG", "set1gXnE", "setXn", "setXnS", "setXn_prod", "sub1set", "subsetT", "subset_gen" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

groupX0 (gT : 'I_0 -> finGroupType) (G : forall i, {group gT i}) : setXn G = 1%g.

Proof. by apply/setP => ?; apply/setXnP/set1P => [_|_ []//]; apply/ffunP => -[]. Qed.

Lemma

groupX0

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "ffunP", "gT", "group", "set1P", "setP", "setXn", "setXnP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_by (to : groupAction D R) : predArgType

:= SdPair (ax : aT * rT) of ax \in setX D R.

Inductive

sdprod_by

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "aT", "groupAction", "setX", "to" ]

The pair (a, x) denotes the product sdpair2 a * sdpair1 x

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_of_sd to (u : sdprod_by to)

:= let: SdPair ax _ := u in ax.

Coercion

pair_of_sd

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "sdprod_by", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdT

:= (sdprod_by to).

Notation

sdT

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "sdprod_by", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdval

:= (@pair_of_sd to).

Notation

sdval

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "pair_of_sd", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_one

:= SdPair to (group1 _).

Definition

sdprod_one

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "group1", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_inv_proof (u : sdT) : (u.1^-1, to u.2^-1 u.1^-1) \in setX D R.

Proof. by case: u => [[a x]] /= /setXP[Da Rx]; rewrite inE gact_stable !groupV ?Da. Qed.

Lemma

sdprod_inv_proof

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "gact_stable", "groupV", "inE", "sdT", "setX", "setXP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_inv u

:= SdPair to (sdprod_inv_proof u).

Definition

sdprod_inv

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "sdprod_inv_proof", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_mul_proof (u v : sdT) : (u.1 * v.1, to u.2 v.1 * v.2) \in setX D R.

Proof. case: u v => [[a x] /= /setXP[Da Rx]] [[b y] /= /setXP[Db Ry]]. by rewrite inE !groupM //= gact_stable. Qed.

Lemma

sdprod_mul_proof

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "gact_stable", "groupM", "inE", "sdT", "setX", "setXP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_mul u v

:= SdPair to (sdprod_mul_proof u v).

Definition

sdprod_mul

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "sdprod_mul_proof", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_mul1g : left_id sdprod_one sdprod_mul.

Proof. move=> u; apply: val_inj; case: u => [[a x] /=]; case/setXP=> Da _. by rewrite gact1 // !mul1g. Qed.

Lemma

sdprod_mul1g

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "apply", "gact1", "mul1g", "sdprod_mul", "sdprod_one", "setXP", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_mulVg : left_inverse sdprod_one sdprod_inv sdprod_mul.

Proof. move=> u; apply: val_inj; case: u => [[a x] /=]; case/setXP=> Da _. by rewrite actKVin ?mulVg. Qed.

Lemma

sdprod_mulVg

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "actKVin", "apply", "mulVg", "sdprod_inv", "sdprod_mul", "sdprod_one", "setXP", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_mulgA : associative sdprod_mul.

Proof. move=> u v w; apply: val_inj; case: u => [[a x]] /=; case/setXP=> Da Rx. case: v w => [[b y]] /=; case/setXP=> Db Ry [[c z]] /=; case/setXP=> Dc Rz. by rewrite !(actMin to) // gactM ?gact_stable // !mulgA. Qed.

Lemma

sdprod_mulgA

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "actMin", "apply", "gactM", "gact_stable", "mulgA", "sdprod_mul", "setXP", "to", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_groupType : finGroupType

:= sdT.

Definition

sdprod_groupType

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "sdT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdpair1 x

:= insubd sdprod_one (1, x) : sdT.

Definition

sdpair1

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "insubd", "sdT", "sdprod_one" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdpair2 a

:= insubd sdprod_one (a, 1) : sdT.

Definition

sdpair2

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "insubd", "sdT", "sdprod_one" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdpair1_morphM : {in R &, {morph sdpair1 : x y / x * y}}.

Proof. move=> x y Rx Ry; apply: val_inj. by rewrite /= !val_insubd !inE !group1 !groupM ?Rx ?Ry //= mulg1 act1. Qed.

Lemma

sdpair1_morphM

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "act1", "apply", "group1", "groupM", "inE", "mulg1", "sdpair1", "val_inj", "val_insubd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdpair2_morphM : {in D &, {morph sdpair2 : a b / a * b}}.

Proof. move=> a b Da Db; apply: val_inj. by rewrite /= !val_insubd !inE !group1 !groupM ?Da ?Db //= mulg1 gact1. Qed.

Lemma

sdpair2_morphM

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "apply", "gact1", "group1", "groupM", "inE", "mulg1", "sdpair2", "val_inj", "val_insubd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdpair1_morphism

:= Morphism sdpair1_morphM.

Canonical

sdpair1_morphism

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "sdpair1_morphM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdpair2_morphism

:= Morphism sdpair2_morphM.

Canonical

sdpair2_morphism

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "sdpair2_morphM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_sdpair1 : 'injm sdpair1.

Proof. apply/subsetP=> x /setIP[Rx]. by rewrite !inE -val_eqE val_insubd inE Rx group1 /=; case/andP. Qed.

Lemma

injm_sdpair1

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "group1", "inE", "sdpair1", "setIP", "subsetP", "val_eqE", "val_insubd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_sdpair2 : 'injm sdpair2.

Proof. apply/subsetP=> a /setIP[Da]. by rewrite !inE -val_eqE val_insubd inE Da group1 /=; case/andP. Qed.

Lemma

injm_sdpair2

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "apply", "group1", "inE", "sdpair2", "setIP", "subsetP", "val_eqE", "val_insubd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdpairE (u : sdT) : u = sdpair2 u.1 * sdpair1 u.2.

Proof. apply: val_inj; case: u => [[a x] /= /setXP[Da Rx]]. by rewrite !val_insubd !inE Da Rx !(group1, gact1) // mulg1 mul1g. Qed.

Lemma

sdpairE

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "apply", "gact1", "group1", "inE", "mul1g", "mulg1", "sdT", "sdpair1", "sdpair2", "setXP", "val_inj", "val_insubd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdpair_act : {in R & D, forall x a, sdpair1 (to x a) = sdpair1 x ^ sdpair2 a}.

Proof. move=> x a Rx Da; apply: val_inj. rewrite /= !val_insubd !inE !group1 gact_stable ?Da ?Rx //=. by rewrite !mul1g mulVg invg1 mulg1 actKVin ?mul1g. Qed.

Lemma

sdpair_act

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "actKVin", "apply", "gact_stable", "group1", "inE", "invg1", "mul1g", "mulVg", "mulg1", "sdpair1", "sdpair2", "to", "val_inj", "val_insubd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdpair_setact (G : {set rT}) a : G \subset R -> a \in D -> sdpair1 @* (to^~ a @: G) = (sdpair1 @* G) :^ sdpair2 a.

Proof. move=> sGR Da; have GtoR := subsetP sGR; apply/eqP. rewrite eqEcard cardJg !(card_injm injm_sdpair1) //. by apply/subsetP=> _ /imsetP[x Gx ->]; rewrite gact_stable ?GtoR. rewrite (card_imset _ (act_inj _ _)) leqnn andbT. apply/subsetP=> _ /morphimP[xa Rxa /imsetP[x Gx def_xa ->]]. rewrite mem_conjg -morphV // ...

Lemma

sdpair_setact

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "actKin", "act_inj", "apply", "cardJg", "card_imset", "card_injm", "eqEcard", "gact_stable", "groupV", "imsetP", "injm_sdpair1", "leqnn", "mem_conjg", "mem_morphim", "morphV", "morphimP", "sdpair1", "sdpair2", "sdpair_act", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_sdpair_norm : sdpair2 @* D \subset 'N(sdpair1 @* R).

Proof. apply/subsetP=> _ /morphimP[a _ Da ->]. rewrite inE -sdpair_setact // morphimS //. by apply/subsetP=> _ /imsetP[x Rx ->]; rewrite gact_stable. Qed.

Lemma

im_sdpair_norm

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "apply", "gact_stable", "imsetP", "inE", "morphimP", "morphimS", "sdpair1", "sdpair2", "sdpair_setact", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_sdpair_TI : (sdpair1 @* R) :&: (sdpair2 @* D) = 1.

Proof. apply/trivgP; apply/subsetP=> _ /setIP[/morphimP[x _ Rx ->]]. case/morphimP=> a _ Da /eqP; rewrite inE -!val_eqE. by rewrite !val_insubd !inE Da Rx !group1 /eq_op /= eqxx; case/andP. Qed.

Lemma

im_sdpair_TI

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "apply", "eqxx", "group1", "inE", "morphimP", "sdpair1", "sdpair2", "setIP", "subsetP", "trivgP", "val_eqE", "val_insubd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_sdpair : (sdpair1 @* R) * (sdpair2 @* D) = setT.

Proof. apply/eqP; rewrite -subTset -(normC im_sdpair_norm). apply/subsetP=> /= u _; rewrite [u]sdpairE. by case: u => [[a x] /= /setXP[Da Rx]]; rewrite mem_mulg ?mem_morphim. Qed.

Lemma

im_sdpair

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "apply", "im_sdpair_norm", "mem_morphim", "mem_mulg", "normC", "sdpair1", "sdpair2", "sdpairE", "setT", "setXP", "subTset", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprod_sdpair : sdpair1 @* R ><| sdpair2 @* D = setT.

Proof. by rewrite sdprodE ?(im_sdpair_norm, im_sdpair, im_sdpair_TI). Qed.

Lemma

sdprod_sdpair

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "im_sdpair", "im_sdpair_TI", "im_sdpair_norm", "sdpair1", "sdpair2", "sdprodE", "setT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gacentEsd : 'C_(|to)(A) = sdpair1 @*^-1 'C(sdpair2 @* A).

Proof. apply/setP=> x; apply/idP/idP. case/setIP=> Rx /afixP cDAx; rewrite mem_morphpre //. apply/centP=> _ /morphimP[a Da Aa ->]; red. by rewrite conjgC -sdpair_act // cDAx // inE Da. case/morphpreP=> Rx cAx; rewrite inE Rx; apply/afixP=> a /setIP[Da Aa]. apply: (injmP injm_sdpair1); rewrite ?gact_stable /= ?sdp...

Lemma

gacentEsd

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "afixP", "apply", "centP", "conjg", "conjgC", "gact_stable", "inE", "injmP", "injm_sdpair1", "mem_morphim", "mem_morphpre", "morphimP", "morphpreP", "mulKg", "sdpair1", "sdpair2", "sdpair_act", "setIP", "setP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

(sAD : A \subset D) (sGR : G \subset R).

Hypotheses

sAD

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabEsd : 'C(G | to) = sdpair2 @*^-1 'C(sdpair1 @* G).

Proof. have ssGR := subsetP sGR; apply/setP=> a; apply/idP/idP=> [cGa|]. rewrite mem_morphpre ?(astab_dom cGa) //. apply/centP=> _ /morphimP[x Rx Gx ->]; symmetry. by rewrite conjgC -sdpair_act ?(astab_act cGa) ?(astab_dom cGa). case/morphpreP=> Da cGa; rewrite !inE Da; apply/subsetP=> x Gx; rewrite inE. apply/e...

Lemma

astabEsd

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "apply", "astab_act", "astab_dom", "centP", "conjg", "conjgC", "gact_stable", "inE", "injmP", "injm_sdpair1", "mem_morphim", "mem_morphpre", "morphimP", "morphpreP", "mulKg", "sdpair1", "sdpair2", "sdpair_act", "setP", "subsetP", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

astabsEsd : 'N(G | to) = sdpair2 @*^-1 'N(sdpair1 @* G).

Proof. apply/setP=> a; apply/idP/idP=> [nGa|]. have Da := astabs_dom nGa; rewrite mem_morphpre // inE sub_conjg. apply/subsetP=> _ /morphimP[x Rx Gx ->]. by rewrite mem_conjgV -sdpair_act // mem_morphim ?gact_stable ?astabs_act. case/morphpreP=> Da nGa; rewrite !inE Da; apply/subsetP=> x Gx. have Rx := subsetP sG...

Lemma

astabsEsd

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "Da", "apply", "astabs_act", "astabs_dom", "gact_stable", "inE", "injmSK", "injm_sdpair1", "memJ_norm", "mem_conjgV", "mem_morphim", "mem_morphpre", "morphimP", "morphim_set1", "morphpreP", "sdpair1", "sdpair2", "sdpair_act", "setP", "sub1set", "sub_conjg", "subsetP", "to...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actsEsd : [acts A, on G | to] = (sdpair2 @* A \subset 'N(sdpair1 @* G)).

Proof. by rewrite sub_morphim_pre -?astabsEsd. Qed.

Lemma

actsEsd

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "astabsEsd", "on", "sdpair1", "sdpair2", "sub_morphim_pre", "to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodm & B \subset 'N(A) & {in A & B, morph_act 'J 'J fA fB} & {in A :&: B, fA =1 fB}

:= fun x => fA (divgr A B x) * fB (remgr A B x).

Definition

pprodm

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "divgr", "fA", "morph_act", "remgr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nHK : K \subset 'N(H).

Hypothesis

nHK

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

actf : {in H & K, morph_act 'J 'J fH fK}.

Hypothesis

actf

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "fH", "fK", "morph_act" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eqfHK : {in H :&: K, fH =1 fK}.

Hypothesis

eqfHK

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "fH", "fK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

f

:= (pprodm nHK actf eqfHK).

Notation

f

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "actf", "eqfHK", "nHK", "pprodm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodmE x a : x \in H -> a \in K -> f (x * a) = fH x * fK a.

Proof. move=> Hx Ka; have: x * a \in H * K by rewrite mem_mulg. rewrite -remgrP inE /f rcoset_sym mem_rcoset /divgr -mulgA groupMl //. case/andP; move: (remgr H K _) => b Hab Kb; rewrite morphM // -mulgA. have Kab: a * b^-1 \in K by rewrite groupM ?groupV. by congr (_ * _); rewrite eqfHK 1?inE ?Hab // -morphM // mulgKV...

Lemma

pprodmE

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "divgr", "eqfHK", "fH", "fK", "groupM", "groupMl", "groupV", "inE", "mem_mulg", "mem_rcoset", "morphM", "mulgA", "mulgKV", "rcoset_sym", "remgr", "remgrP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodmEl : {in H, f =1 fH}.

Proof. by move=> x Hx; rewrite -(mulg1 x) pprodmE // morph1 !mulg1. Qed.

Lemma

pprodmEl

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "fH", "morph1", "mulg1", "pprodmE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodmEr : {in K, f =1 fK}.

Proof. by move=> a Ka; rewrite -(mul1g a) pprodmE // morph1 !mul1g. Qed.

Lemma

pprodmEr

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "fK", "morph1", "mul1g", "pprodmE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodmM : {in H <*> K &, {morph f: x y / x * y}}.

Proof. move=> xa yb; rewrite norm_joinEr //. move=> /imset2P[x a Ha Ka ->{xa}] /imset2P[y b Hy Kb ->{yb}]. have Hya: y ^ a^-1 \in H by rewrite -mem_conjg (normsP nHK). rewrite mulgA -(mulgA x) (conjgCV a y) (mulgA x) -mulgA !pprodmE 1?groupMl //. by rewrite morphM // actf ?groupV ?morphV // morphM // !mulgA mulgKV invg...

Lemma

pprodmM

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "actf", "conjgCV", "groupMl", "groupV", "imset2P", "invgK", "mem_conjg", "morphM", "morphV", "mulgA", "mulgKV", "nHK", "norm_joinEr", "normsP", "pprodmE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pprodm_morphism

:= Morphism pprodmM.

Canonical

pprodm_morphism

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "pprodmM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_pprodm A B : A \subset H -> B \subset K -> f @* (A * B) = fH @* A * fK @* B.

Proof. move=> sAH sBK; rewrite [f @* _]morphimEsub /=. by rewrite norm_joinEr // mulgSS. apply/setP=> y; apply/imsetP/idP=> [[_ /mulsgP[x a Ax Ba ->] ->{y}] |]. have Hx := subsetP sAH x Ax; have Ka := subsetP sBK a Ba. by rewrite pprodmE // imset2_f ?mem_morphim. case/mulsgP=> _ _ /morphimP[x Hx Ax ->] /morphimP[...

Lemma

morphim_pprodm

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "fH", "fK", "imset2_f", "imsetP", "mem_morphim", "mem_mulg", "morphimEsub", "morphimP", "mulgSS", "mulsgP", "norm_joinEr", "pprodmE", "setP", "subsetP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_pprodml A : A \subset H -> f @* A = fH @* A.

Proof. by move=> sAH; rewrite -{1}(mulg1 A) morphim_pprodm ?sub1G // morphim1 mulg1. Qed.

Lemma

morphim_pprodml

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "fH", "morphim1", "morphim_pprodm", "mulg1", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_pprodmr B : B \subset K -> f @* B = fK @* B.

Proof. by move=> sBK; rewrite -{1}(mul1g B) morphim_pprodm ?sub1G // morphim1 mul1g. Qed.

Lemma

morphim_pprodmr

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "fK", "morphim1", "morphim_pprodm", "mul1g", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_pprodm : 'ker f = [set x * a^-1 | x in H, a in K & fH x == fK a].

Proof. apply/setP=> y; rewrite 3!inE {1}norm_joinEr //=. apply/andP/imset2P=> [[/mulsgP[x a Hx Ka ->{y}]]|[x a Hx]]. rewrite pprodmE // => fxa1. by exists x a^-1; rewrite ?invgK // inE groupVr ?morphV // eq_mulgV1 invgK. case/setIdP=> Kx /eqP fx ->{y}. by rewrite imset2_f ?pprodmE ?groupV ?morphV // fx mulgV. Qed.

Lemma

ker_pprodm

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "eq_mulgV1", "fH", "fK", "groupV", "groupVr", "imset2P", "imset2_f", "inE", "invgK", "ker", "morphV", "mulgV", "mulsgP", "norm_joinEr", "pprodmE", "setIdP", "setP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_pprodm : 'injm f = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == fH @* K].

Proof. apply/idP/and3P=> [injf | [injfH injfK]]. rewrite eq_sym -{1}morphimIdom -(morphim_pprodml (subsetIl _ _)) injmI //. rewrite morphim_pprodml // morphim_pprodmr //=; split=> //. apply/injmP=> x y Hx Hy /=; rewrite -!pprodmEl //. by apply: (injmP injf); rewrite ?mem_gen ?inE ?Hx ?Hy. apply/injmP=> a ...

Lemma

injm_pprodm

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "eq_sym", "eqfHK", "fH", "fK", "imset2P", "inE", "injf", "injmI", "injmP", "ker_pprodm", "mem_gen", "mem_morphim", "morphimIdom", "morphimP", "morphim_pprodml", "morphim_pprodmr", "mulgV", "pprodmEl", "pprodmEr", "set11", "setIdP", "setUP", "split", "subsetIl...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eqHK_G : H ><| K = G.

Hypothesis

eqHK_G

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodm_norm : K \subset 'N(H).

Proof. by case/sdprodP: eqHK_G. Qed.

Lemma

sdprodm_norm

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "eqHK_G", "sdprodP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodm_sub : G \subset H <*> K.

Proof. by case/sdprodP: eqHK_G => _ <- nHK _; rewrite norm_joinEr. Qed.

Lemma

sdprodm_sub

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "eqHK_G", "nHK", "norm_joinEr", "sdprodP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodm_eqf : {in H :&: K, fH =1 fK}.

Proof. by case/sdprodP: eqHK_G => _ _ _ -> _ /set1P->; rewrite !morph1. Qed.

Lemma

sdprodm_eqf

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "eqHK_G", "fH", "fK", "morph1", "sdprodP", "set1P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodm

:= restrm sdprodm_sub (pprodm sdprodm_norm actf sdprodm_eqf).

Definition

sdprodm

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "actf", "pprodm", "restrm", "sdprodm_eqf", "sdprodm_norm", "sdprodm_sub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodm_morphism

:= Eval hnf in [morphism of sdprodm].

Canonical

sdprodm_morphism

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "morphism", "sdprodm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodmE a b : a \in H -> b \in K -> sdprodm (a * b) = fH a * fK b.

Proof. exact: pprodmE. Qed.

Lemma

sdprodmE

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "fH", "fK", "pprodmE", "sdprodm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodmEl a : a \in H -> sdprodm a = fH a.

Proof. exact: pprodmEl. Qed.

Lemma

sdprodmEl

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "fH", "pprodmEl", "sdprodm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdprodmEr b : b \in K -> sdprodm b = fK b.

Proof. exact: pprodmEr. Qed.

Lemma

sdprodmEr

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "fK", "pprodmEr", "sdprodm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_sdprodm A B : A \subset H -> B \subset K -> sdprodm @* (A * B) = fH @* A * fK @* B.

Proof. move=> sAH sBK; rewrite /sdprodm morphim_restrm /= (setIidPr _) ?morphim_pprodm //. by case/sdprodP: eqHK_G => _ <- _ _; apply: mulgSS. Qed.

Lemma

morphim_sdprodm

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "eqHK_G", "fH", "fK", "morphim_pprodm", "morphim_restrm", "mulgSS", "sdprodP", "sdprodm", "setIidPr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

im_sdprodm : sdprodm @* G = fH @* H * fK @* K.

Proof. by rewrite -morphim_sdprodm //; case/sdprodP: eqHK_G => _ ->. Qed.

Lemma

im_sdprodm

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "eqHK_G", "fH", "fK", "morphim_sdprodm", "sdprodP", "sdprodm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_sdprodml A : A \subset H -> sdprodm @* A = fH @* A.

Proof. by move=> sHA; rewrite -{1}(mulg1 A) morphim_sdprodm ?sub1G // morphim1 mulg1. Qed.

Lemma

morphim_sdprodml

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "fH", "morphim1", "morphim_sdprodm", "mulg1", "sdprodm", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

morphim_sdprodmr B : B \subset K -> sdprodm @* B = fK @* B.

Proof. by move=> sBK; rewrite -{1}(mul1g B) morphim_sdprodm ?sub1G // morphim1 mul1g. Qed.

Lemma

morphim_sdprodmr

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "fK", "morphim1", "morphim_sdprodm", "mul1g", "sdprodm", "sub1G" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ker_sdprodm : 'ker sdprodm = [set a * b^-1 | a in H, b in K & fH a == fK b].

Proof. rewrite ker_restrm (setIidPr _) ?subIset ?ker_pprodm //; apply/orP; left. by case/sdprodP: eqHK_G => _ <- nHK _; rewrite norm_joinEr. Qed.

Lemma

ker_sdprodm

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "apply", "eqHK_G", "fH", "fK", "ker", "ker_pprodm", "ker_restrm", "nHK", "norm_joinEr", "sdprodP", "sdprodm", "setIidPr", "subIset" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

injm_sdprodm : 'injm sdprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == 1].

Proof. rewrite ker_sdprodm -(ker_pprodm sdprodm_norm actf sdprodm_eqf) injm_pprodm. congr [&& _, _ & _ == _]; have [_ _ _ tiHK] := sdprodP eqHK_G. by rewrite -morphimIdom tiHK morphim1. Qed.

Lemma

injm_sdprodm

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "actf", "eqHK_G", "fH", "fK", "injm_pprodm", "ker_pprodm", "ker_sdprodm", "morphim1", "morphimIdom", "sdprodP", "sdprodm", "sdprodm_eqf", "sdprodm_norm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eqHK_G : H \* K = G.

Hypothesis

eqHK_G

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cfHK : fK @* K \subset 'C(fH @* H).

Hypothesis

cfHK

finite_group

finite_group/gproduct.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "finset", "fingroup", "morphism", "quotient", "action", "finfun" ]

[ "fH", "fK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d