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 |
|---|---|---|---|---|---|---|---|---|---|---|
cprodm_norm : K \subset 'N(H). | Proof. by rewrite cents_norm //; case/cprodP: eqHK_G. Qed. | Lemma | cprodm_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"
] | [
"cents_norm",
"cprodP",
"eqHK_G"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodm_sub : G \subset H <*> K. | Proof. by case/cprodP: eqHK_G => _ <- cHK; rewrite cent_joinEr. Qed. | Lemma | cprodm_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"
] | [
"cent_joinEr",
"cprodP",
"eqHK_G"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodm_actf : {in H & K, morph_act 'J 'J fH fK}. | Proof.
case/cprodP: eqHK_G => _ _ cHK a b Ha Kb /=.
by rewrite /conjg -(centsP cHK b) // -(centsP cfHK (fK b)) ?mulKg ?mem_morphim.
Qed. | Lemma | cprodm_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"
] | [
"centsP",
"cfHK",
"conjg",
"cprodP",
"eqHK_G",
"fH",
"fK",
"mem_morphim",
"morph_act",
"mulKg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodm | := restrm cprodm_sub (pprodm cprodm_norm cprodm_actf eqfHK). | Definition | cprodm | 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"
] | [
"cprodm_actf",
"cprodm_norm",
"cprodm_sub",
"eqfHK",
"pprodm",
"restrm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodm_morphism | := Eval hnf in [morphism of cprodm]. | Canonical | cprodm_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"
] | [
"cprodm",
"morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodmE a b : a \in H -> b \in K -> cprodm (a * b) = fH a * fK b. | Proof. exact: pprodmE. Qed. | Lemma | cprodmE | 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"
] | [
"cprodm",
"fH",
"fK",
"pprodmE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodmEl a : a \in H -> cprodm a = fH a. | Proof. exact: pprodmEl. Qed. | Lemma | cprodmEl | 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"
] | [
"cprodm",
"fH",
"pprodmEl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprodmEr b : b \in K -> cprodm b = fK b. | Proof. exact: pprodmEr. Qed. | Lemma | cprodmEr | 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"
] | [
"cprodm",
"fK",
"pprodmEr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_cprodm A B :
A \subset H -> B \subset K -> cprodm @* (A * B) = fH @* A * fK @* B. | Proof.
move=> sAH sBK; rewrite [LHS]morphim_restrm /= (setIidPr _) ?morphim_pprodm //.
by case/cprodP: eqHK_G => _ <- _; apply: mulgSS.
Qed. | Lemma | morphim_cprodm | 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",
"cprodP",
"cprodm",
"eqHK_G",
"fH",
"fK",
"morphim_pprodm",
"morphim_restrm",
"mulgSS",
"setIidPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_cprodm : cprodm @* G = fH @* H * fK @* K. | Proof.
by have [_ defHK _] := cprodP eqHK_G; rewrite -{2}defHK morphim_cprodm.
Qed. | Lemma | im_cprodm | 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"
] | [
"cprodP",
"cprodm",
"eqHK_G",
"fH",
"fK",
"morphim_cprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_cprodml A : A \subset H -> cprodm @* A = fH @* A. | Proof.
by move=> sHA; rewrite -{1}(mulg1 A) morphim_cprodm ?sub1G // morphim1 mulg1.
Qed. | Lemma | morphim_cprodml | 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"
] | [
"cprodm",
"fH",
"morphim1",
"morphim_cprodm",
"mulg1",
"sub1G"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_cprodmr B : B \subset K -> cprodm @* B = fK @* B. | Proof.
by move=> sBK; rewrite -{1}(mul1g B) morphim_cprodm ?sub1G // morphim1 mul1g.
Qed. | Lemma | morphim_cprodmr | 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"
] | [
"cprodm",
"fK",
"morphim1",
"morphim_cprodm",
"mul1g",
"sub1G"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ker_cprodm : 'ker cprodm = [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/cprodP: eqHK_G => _ <- cHK; rewrite cent_joinEr.
Qed. | Lemma | ker_cprodm | 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",
"cent_joinEr",
"cprodP",
"cprodm",
"eqHK_G",
"fH",
"fK",
"ker",
"ker_pprodm",
"ker_restrm",
"setIidPr",
"subIset"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_cprodm :
'injm cprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == fH @* K]. | Proof.
by rewrite ker_cprodm -(ker_pprodm cprodm_norm cprodm_actf eqfHK) injm_pprodm.
Qed. | Lemma | injm_cprodm | 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"
] | [
"cprodm",
"cprodm_actf",
"cprodm_norm",
"eqfHK",
"fH",
"fK",
"injm_pprodm",
"ker_cprodm",
"ker_pprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqHK_G : H \x 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 | ||
dprodm_cprod : H \* K = G. | Proof.
by rewrite -eqHK_G /dprod; case/dprodP: eqHK_G => _ _ _ ->; rewrite subxx.
Qed. | Lemma | dprodm_cprod | 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"
] | [
"dprod",
"dprodP",
"eqHK_G",
"subxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodm_eqf : {in H :&: K, fH =1 fK}. | Proof. by case/dprodP: eqHK_G => _ _ _ -> _ /set1P->; rewrite !morph1. Qed. | Lemma | dprodm_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"
] | [
"dprodP",
"eqHK_G",
"fH",
"fK",
"morph1",
"set1P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodm | := cprodm dprodm_cprod cfHK dprodm_eqf. | Definition | dprodm | 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"
] | [
"cfHK",
"cprodm",
"dprodm_cprod",
"dprodm_eqf"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodm_morphism | := Eval hnf in [morphism of dprodm]. | Canonical | dprodm_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"
] | [
"dprodm",
"morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodmE a b : a \in H -> b \in K -> dprodm (a * b) = fH a * fK b. | Proof. exact: pprodmE. Qed. | Lemma | dprodmE | 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"
] | [
"dprodm",
"fH",
"fK",
"pprodmE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodmEl a : a \in H -> dprodm a = fH a. | Proof. exact: pprodmEl. Qed. | Lemma | dprodmEl | 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"
] | [
"dprodm",
"fH",
"pprodmEl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dprodmEr b : b \in K -> dprodm b = fK b. | Proof. exact: pprodmEr. Qed. | Lemma | dprodmEr | 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"
] | [
"dprodm",
"fK",
"pprodmEr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_dprodm A B :
A \subset H -> B \subset K -> dprodm @* (A * B) = fH @* A * fK @* B. | Proof. exact: morphim_cprodm. Qed. | Lemma | morphim_dprodm | 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"
] | [
"dprodm",
"fH",
"fK",
"morphim_cprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_dprodm : dprodm @* G = fH @* H * fK @* K. | Proof. exact: im_cprodm. Qed. | Lemma | im_dprodm | 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"
] | [
"dprodm",
"fH",
"fK",
"im_cprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_dprodml A : A \subset H -> dprodm @* A = fH @* A. | Proof. exact: morphim_cprodml. Qed. | Lemma | morphim_dprodml | 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"
] | [
"dprodm",
"fH",
"morphim_cprodml"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_dprodmr B : B \subset K -> dprodm @* B = fK @* B. | Proof. exact: morphim_cprodmr. Qed. | Lemma | morphim_dprodmr | 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"
] | [
"dprodm",
"fK",
"morphim_cprodmr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ker_dprodm : 'ker dprodm = [set a * b^-1 | a in H, b in K & fH a == fK b]. | Proof. exact: ker_cprodm. Qed. | Lemma | ker_dprodm | 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"
] | [
"dprodm",
"fH",
"fK",
"ker",
"ker_cprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_dprodm :
'injm dprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == 1]. | Proof.
rewrite injm_cprodm -(morphimIdom fH K).
by case/dprodP: eqHK_G => _ _ _ ->; rewrite morphim1.
Qed. | Lemma | injm_dprodm | 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"
] | [
"dprodP",
"dprodm",
"eqHK_G",
"fH",
"fK",
"injm_cprodm",
"morphim1",
"morphimIdom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isog_dprod A B G C D L :
A \x B = G -> C \x D = L -> isog A C -> isog B D -> isog G L. | Proof.
move=> defG {C D} /dprodP[[C D -> ->] defL cCD trCD].
case/dprodP: defG (defG) => {A B} [[A B -> ->] defG _ _] dG defC defD.
case/isogP: defC defL cCD trCD => fA injfA <-{C}.
case/isogP: defD => fB injfB <-{D} defL cCD trCD.
apply/isogP; exists (dprodm_morphism dG cCD).
by rewrite injm_dprodm injfA injfB trCD ... | Lemma | isog_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",
"defG",
"dprodP",
"dprodm_morphism",
"eqxx",
"fA",
"injm_dprodm",
"isog",
"isogP",
"morphim_dprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
actf : {in H & K, morph_act to '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",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
fsH | := (fH \o invm (injm_sdpair1 to)). | Notation | fsH | 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",
"injm_sdpair1",
"invm",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fsK | := (fK \o invm (injm_sdpair2 to)). | Notation | fsK | 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",
"injm_sdpair2",
"invm",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
DgH | := sdpair1 to @* H. | Let | DgH | 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",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
DgK | := sdpair2 to @* K. | Let | DgK | 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",
"to"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xsdprodm_dom1 : DgH \subset 'dom fsH. | Proof. by rewrite ['dom _]morphpre_invm. Qed. | Lemma | xsdprodm_dom1 | 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"
] | [
"DgH",
"dom",
"fsH",
"morphpre_invm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gH | := (restrm xsdprodm_dom1 fsH). | Notation | gH | 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"
] | [
"fsH",
"restrm",
"xsdprodm_dom1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xsdprodm_dom2 : DgK \subset 'dom fsK. | Proof. by rewrite ['dom _]morphpre_invm. Qed. | Lemma | xsdprodm_dom2 | 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"
] | [
"DgK",
"dom",
"fsK",
"morphpre_invm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gK | := (restrm xsdprodm_dom2 fsK). | Notation | gK | 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"
] | [
"fsK",
"restrm",
"xsdprodm_dom2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_sdprodm1 : gH @* DgH = fH @* H. | Proof. by rewrite morphim_restrm setIid morphim_comp im_invm. Qed. | Lemma | im_sdprodm1 | 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"
] | [
"DgH",
"fH",
"gH",
"im_invm",
"morphim_comp",
"morphim_restrm",
"setIid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_sdprodm2 : gK @* DgK = fK @* K. | Proof. by rewrite morphim_restrm setIid morphim_comp im_invm. Qed. | Lemma | im_sdprodm2 | 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"
] | [
"DgK",
"fK",
"gK",
"im_invm",
"morphim_comp",
"morphim_restrm",
"setIid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xsdprodm_act : {in DgH & DgK, morph_act 'J 'J gH gK}. | Proof.
move=> fh fk; case/morphimP=> h _ Hh ->{fh}; case/morphimP=> k _ Kk ->{fk}.
by rewrite /= -sdpair_act // /restrm /= !invmE ?actf ?gact_stable.
Qed. | Lemma | xsdprodm_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"
] | [
"DgH",
"DgK",
"Hh",
"actf",
"gH",
"gK",
"gact_stable",
"invmE",
"morph_act",
"morphimP",
"restrm",
"sdpair_act"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xsdprodm | := sdprodm (sdprod_sdpair to) xsdprodm_act. | Definition | xsdprodm | 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_sdpair",
"sdprodm",
"to",
"xsdprodm_act"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xsdprod_morphism | := [morphism of xsdprodm]. | Canonical | xsdprod_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",
"xsdprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_xsdprodm : xsdprodm @* setT = fH @* H * fK @* K. | Proof. by rewrite -im_sdpair morphim_sdprodm // im_sdprodm1 im_sdprodm2. Qed. | Lemma | im_xsdprodm | 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",
"im_sdpair",
"im_sdprodm1",
"im_sdprodm2",
"morphim_sdprodm",
"setT",
"xsdprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_xsdprodm :
'injm xsdprodm = [&& 'injm fH, 'injm fK & fH @* H :&: fK @* K == 1]. | Proof.
rewrite injm_sdprodm im_sdprodm1 im_sdprodm2 !subG1 /=.
rewrite (ker_restrm xsdprodm_dom1) (ker_restrm xsdprodm_dom2) /= !ker_comp.
rewrite !morphpre_invm !morphimIim.
by rewrite !morphim_injm_eq1 ?subsetIl ?injm_sdpair1 ?injm_sdpair2.
Qed. | Lemma | injm_xsdprodm | 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",
"im_sdprodm1",
"im_sdprodm2",
"injm_sdpair1",
"injm_sdpair2",
"injm_sdprodm",
"ker_comp",
"ker_restrm",
"morphimIim",
"morphim_injm_eq1",
"morphpre_invm",
"subG1",
"subsetIl",
"xsdprodm",
"xsdprodm_dom1",
"xsdprodm_dom2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulgm : gT * gT -> _ | := uncurry mul. | Definition | mulgm | 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",
"mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
imset_mulgm (A B : {set gT}) : mulgm @: setX A B = A * B. | Proof. by rewrite -curry_imset2X. Qed. | Lemma | imset_mulgm | 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"
] | [
"curry_imset2X",
"gT",
"mulgm",
"setX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulgmP H1 H2 G : reflect (H1 \x H2 = G) (misom (setX H1 H2) G mulgm). | Proof.
apply: (iffP misomP) => [[pM /isomP[injf /= <-]] | ].
have /dprodP[_ /= defX cH12] := setX_dprod H1 H2.
rewrite -{4}defX {}defX => /(congr1 (fun A => morphm pM @* A)).
move/(morphimS (morphm_morphism pM)): cH12 => /=.
have sH1H: setX H1 1 \subset setX H1 H2 by rewrite setXS ?sub1G.
have sH2H: setX 1 H2... | Lemma | mulgmP | 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",
"centsP",
"defG",
"dprodE",
"dprodP",
"eq_invg_mul",
"eqxx",
"fM",
"groupV",
"imset_mulgm",
"inE",
"in_setI",
"injf",
"injmI",
"injm_cent",
"invg1",
"isomP",
"last",
"misom",
"misomP",
"morphic",
"morphicP",
"morphim1",
"morphimEsub",
"morphimS",
"morphm",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphism (D : {set aT}) : Type | := Morphism {
mfun :> aT -> FinGroup.sort rT;
_ : {in D &, {morph mfun : x y / x * y}}
}. | Structure | morphism | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"aT",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphism_for D & phant rT | := morphism D. | Definition | morphism_for | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"morphism"
] | available (e.g. its domain is a group). | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
clone_morphism D f | :=
let: Morphism _ fM := f
return {type of @Morphism D for f} -> morphism_for D (Phant rT)
in fun k => k fM. | Definition | clone_morphism | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"fM",
"morphism_for",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_spec : Prop | := MorphimSpec z & z \in D & z \in A & y = f z. | Variant | morphim_spec | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphimP : reflect morphim_spec (y \in f @: (D :&: A)). | Proof.
apply: (iffP imsetP) => [] [z]; first by case/setIP; exists z.
by exists z; first apply/setIP.
Qed. | Lemma | morphimP | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"apply",
"imsetP",
"morphim_spec",
"setIP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphpreP : reflect (x \in D /\ f x \in R) (x \in D :&: f @^-1: R). | Proof. by rewrite !inE; apply: andP. Qed. | Lemma | morphpreP | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"apply",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'morphism' D >-> T }" | := (morphism_for D (Phant T))
(format "{ 'morphism' D >-> T }") : type_scope. | Notation | { 'morphism' D >-> T } | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"morphism_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'morphism' D 'of' f ]" | :=
(@clone_morphism _ _ D _ (fun fM => @Morphism _ _ D f fM))
(format "[ 'morphism' D 'of' f ]") : form_scope. | Notation | [ 'morphism' D 'of' f ] | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"clone_morphism",
"fM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'morphism' 'of' f ]" | := (clone_morphism (@Morphism _ _ _ f))
(format "[ 'morphism' 'of' f ]") : form_scope. | Notation | [ 'morphism' 'of' f ] | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"clone_morphism"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphM : {in D &, {morph f : x y / x * y}}. | Proof. by case f. Qed. | Lemma | morphM | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morPhantom | := (phantom (aT -> rT)). | Notation | morPhantom | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"aT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
MorPhantom | := Phantom (aT -> rT). | Definition | MorPhantom | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"aT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dom & morPhantom f | := D. | Definition | dom | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"morPhantom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim & morPhantom f | := fun A => f @: (D :&: A). | Definition | morphim | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"morPhantom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphpre & morPhantom f | := fun R : {set rT} => D :&: f @^-1: R. | Definition | morphpre | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"morPhantom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ker mph | := morphpre mph 1. | Definition | ker | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"morphpre"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''dom' f" | := (dom (MorPhantom f))
(at level 10, f at level 8, format "''dom' f") : group_scope. | Notation | ''dom' f | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"MorPhantom",
"dom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''ker' f" | := (ker (MorPhantom f))
(at level 10, f at level 8, format "''ker' f") : group_scope. | Notation | ''ker' f | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"MorPhantom",
"ker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''ker_' H f" | := (H :&: 'ker f)
(at level 10, H at level 2, f at level 8, format "''ker_' H f")
: group_scope. | Notation | ''ker_' H f | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"ker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"f @* A" | := (morphim (MorPhantom f) A)
(at level 24, format "f @* A") : group_scope. | Notation | f @* A | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"MorPhantom",
"morphim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"f @*^-1 R" | := (morphpre (MorPhantom f) R)
(at level 24, format "f @*^-1 R") : group_scope. | Notation | f @*^-1 R | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"MorPhantom",
"morphpre"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''injm' f" | := (pred_of_set ('ker f) \subset pred_of_set 1)
(at level 10, f at level 8, format "''injm' f") : group_scope. | Notation | ''injm' f | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"ker",
"pred_of_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morph1 : f 1 = 1. | Proof. by apply: (mulgI (f 1)); rewrite -morphM ?mulg1. Qed. | Lemma | morph1 | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"apply",
"morphM",
"mulg1",
"mulgI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morph_prod I r (P : pred I) F :
(forall i, P i -> F i \in D) ->
f (\prod_(i <- r | P i) F i) = \prod_( i <- r | P i) f (F i). | Proof.
move=> D_F; elim/(big_load (fun x => x \in D)): _.
elim/big_rec2: _ => [|i _ x Pi [Dx <-]]; first by rewrite morph1.
by rewrite groupM ?morphM // D_F.
Qed. | Lemma | morph_prod | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"Dx",
"big_load",
"big_rec2",
"groupM",
"morph1",
"morphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphV : {in D, {morph f : x / x^-1}}. | Proof.
move=> x Dx; apply: (mulgI (f x)).
by rewrite -morphM ?groupV // !mulgV morph1.
Qed. | Lemma | morphV | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"Dx",
"apply",
"groupV",
"morph1",
"morphM",
"mulgI",
"mulgV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphJ : {in D &, {morph f : x y / x ^ y}}. | Proof. by move=> * /=; rewrite !morphM ?morphV // ?groupM ?groupV. Qed. | Lemma | morphJ | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"groupM",
"groupV",
"morphM",
"morphV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphX n : {in D, {morph f : x / x ^+ n}}. | Proof.
by elim: n => [|n IHn] x Dx; rewrite ?morph1 // !expgS morphM ?(groupX, IHn).
Qed. | Lemma | morphX | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"Dx",
"expgS",
"groupX",
"morph1",
"morphM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphR : {in D &, {morph f : x y / [~ x, y]}}. | Proof. by move=> * /=; rewrite morphM ?(groupV, groupJ) // morphJ ?morphV. Qed. | Lemma | morphR | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"groupJ",
"groupV",
"morphJ",
"morphM",
"morphV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphimE A : f @* A = f @: (D :&: A). | Proof. by []. Qed. | Lemma | morphimE | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [] | Morphic image, preimage properties w.r.t. set-theoretic operations. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
morphpreE R : f @*^-1 R = D :&: f @^-1: R. | Proof. by []. Qed. | Lemma | morphpreE | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
kerE : 'ker f = f @*^-1 1. | Proof. by []. Qed. | Lemma | kerE | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"ker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphimEsub A : A \subset D -> f @* A = f @: A. | Proof. by move=> sAD; rewrite /morphim (setIidPr sAD). Qed. | Lemma | morphimEsub | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"morphim",
"sAD",
"setIidPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphimEdom : f @* D = f @: D. | Proof. exact: morphimEsub. Qed. | Lemma | morphimEdom | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"morphimEsub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphimIdom A : f @* (D :&: A) = f @* A. | Proof. by rewrite /morphim setIA setIid. Qed. | Lemma | morphimIdom | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"morphim",
"setIA",
"setIid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphpreIdom R : D :&: f @*^-1 R = f @*^-1 R. | Proof. by rewrite /morphim setIA setIid. Qed. | Lemma | morphpreIdom | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"morphim",
"setIA",
"setIid"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphpreIim R : f @*^-1 (f @* D :&: R) = f @*^-1 R. | Proof.
apply/setP=> x; rewrite morphimEdom !inE.
by case Dx: (x \in D); rewrite // imset_f.
Qed. | Lemma | morphpreIim | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"Dx",
"apply",
"imset_f",
"inE",
"morphimEdom",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphimIim A : f @* D :&: f @* A = f @* A. | Proof. by apply/setIidPr; rewrite imsetS // setIid subsetIl. Qed. | Lemma | morphimIim | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"apply",
"imsetS",
"setIid",
"setIidPr",
"subsetIl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_morphim A x : x \in D -> x \in A -> f x \in f @* A. | Proof. by move=> Dx Ax; apply/morphimP; exists x. Qed. | Lemma | mem_morphim | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"Dx",
"apply",
"morphimP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_morphpre R x : x \in D -> f x \in R -> x \in f @*^-1 R. | Proof. by move=> Dx Rfx; apply/morphpreP. Qed. | Lemma | mem_morphpre | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"Dx",
"apply",
"morphpreP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphimS A B : A \subset B -> f @* A \subset f @* B. | Proof. by move=> sAB; rewrite imsetS ?setIS. Qed. | Lemma | morphimS | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"imsetS",
"setIS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_sub A : f @* A \subset f @* D. | Proof. by rewrite imsetS // setIid subsetIl. Qed. | Lemma | morphim_sub | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"imsetS",
"setIid",
"subsetIl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
leq_morphim A : #|f @* A| <= #|A|. | Proof.
by apply: (leq_trans (leq_imset_card _ _)); rewrite subset_leq_card ?subsetIr.
Qed. | Lemma | leq_morphim | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"apply",
"leq_imset_card",
"leq_trans",
"subsetIr",
"subset_leq_card"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphpreS R S : R \subset S -> f @*^-1 R \subset f @*^-1 S. | Proof. by move=> sRS; rewrite setIS ?preimsetS. Qed. | Lemma | morphpreS | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"preimsetS",
"setIS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphpre_sub R : f @*^-1 R \subset D. | Proof. exact: subsetIl. Qed. | Lemma | morphpre_sub | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"subsetIl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_setIpre A R : f @* (A :&: f @*^-1 R) = f @* A :&: R. | Proof.
apply/setP=> fa; apply/morphimP/setIP=> [[a Da] | [/morphimP[a Da Aa ->] Rfa]].
by rewrite !inE Da /= => /andP[Aa Rfa] ->; rewrite mem_morphim.
by exists a; rewrite // !inE Aa Da.
Qed. | Lemma | morphim_setIpre | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"Da",
"apply",
"inE",
"mem_morphim",
"morphimP",
"setIP",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim0 : f @* set0 = set0. | Proof. by rewrite morphimE setI0 imset0. Qed. | Lemma | morphim0 | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"imset0",
"morphimE",
"set0",
"setI0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_eq0 A : A \subset D -> (f @* A == set0) = (A == set0). | Proof. by rewrite imset_eq0 => /setIidPr->. Qed. | Lemma | morphim_eq0 | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"imset_eq0",
"set0",
"setIidPr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim_set1 x : x \in D -> f @* [set x] = [set f x]. | Proof. by rewrite /morphim -sub1set => /setIidPr->; apply: imset_set1. Qed. | Lemma | morphim_set1 | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"apply",
"imset_set1",
"morphim",
"setIidPr",
"sub1set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphim1 : f @* 1 = 1. | Proof. by rewrite morphim_set1 ?morph1. Qed. | Lemma | morphim1 | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"morph1",
"morphim_set1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphimV A : f @* A^-1 = (f @* A)^-1. | Proof.
wlog suffices: A / f @* A^-1 \subset (f @* A)^-1.
by move=> IH; apply/eqP; rewrite eqEsubset IH -invSg invgK -{1}(invgK A) IH.
apply/subsetP=> _ /morphimP[x Dx Ax' ->]; rewrite !inE in Ax' *.
by rewrite -morphV // imset_f // inE groupV Dx.
Qed. | Lemma | morphimV | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"Dx",
"apply",
"eqEsubset",
"groupV",
"imset_f",
"inE",
"invSg",
"invgK",
"morphV",
"morphimP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphpreV R : f @*^-1 R^-1 = (f @*^-1 R)^-1. | Proof.
apply/setP=> x; rewrite !inE groupV; case Dx: (x \in D) => //=.
by rewrite morphV.
Qed. | Lemma | morphpreV | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"Dx",
"apply",
"groupV",
"inE",
"morphV",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
morphimMl A B : A \subset D -> f @* (A * B) = f @* A * f @* B. | Proof.
move=> sAD; rewrite /morphim setIC -group_modl // (setIidPr sAD).
apply/setP=> fxy; apply/idP/idP.
case/imsetP=> _ /imset2P[x y Ax /setIP[Dy By] ->] ->{fxy}.
by rewrite morphM // (subsetP sAD, imset2_f) // imset_f // inE By.
case/imset2P=> _ _ /imsetP[x Ax ->] /morphimP[y Dy By ->] ->{fxy}.
by rewrite -morph... | Lemma | morphimMl | finite_group | finite_group/morphism.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"finfun",
"bigop",
"finset",
"fingroup"
] | [
"apply",
"group_modl",
"imset2P",
"imset2_f",
"imsetP",
"imset_f",
"inE",
"mem_mulg",
"morphM",
"morphim",
"morphimP",
"sAD",
"setIC",
"setIP",
"setIidPr",
"setP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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