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 |
|---|---|---|---|---|---|---|---|---|---|---|
"''dim' E" | := (abelem_dim' E).+1
(at level 10, E at level 8, format "''dim' E") : abelem_scope. | Notation | ''dim' E | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_dim'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''rV' ( E )" | := 'rV_('dim E) (format "''rV' ( E )") : abelem_scope. | Notation | ''rV' ( E ) | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"dim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''M' ( E )" | := 'M_('dim E) (format "''M' ( E )") : abelem_scope. | Notation | ''M' ( E ) | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"dim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''rV[' F ] ( E )" | := 'rV[F]_('dim E) (only parsing) : abelem_scope. | Notation | ''rV[' F ] ( E ) | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"dim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''M[' F ] ( E )" | := 'M[F]_('dim E) (only parsing) : abelem_scope. | Notation | ''M[' F ] ( E ) | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"dim"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Mmn | := 'M['F_p]_(m, n). | Notation | Mmn | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_Fp_abelem : prime p -> p.-abelem [set: Mmn]. | Proof. exact: fin_Fp_lmod_abelem. Qed. | Lemma | mx_Fp_abelem | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"Mmn",
"abelem",
"fin_Fp_lmod_abelem",
"prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mx_Fp_stable (L : {group Mmn}) : [acts setT, on L | 'Zm]. | Proof.
apply/subsetP=> a _ /[!inE]; apply/subsetP=> A L_A.
by rewrite inE /= /scale_act -[val _]natr_Zp scaler_nat groupX.
Qed. | Lemma | mx_Fp_stable | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"Mmn",
"apply",
"group",
"groupX",
"inE",
"natr_Zp",
"on",
"scale_act",
"scaler_nat",
"setT",
"subsetP",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVn | := 'rV['F_p]_n. | Notation | rVn | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowg_mxK (L : {group rVn}) : rowg (rowg_mx L) = L. | Proof. by apply: stable_rowg_mxK; apply: mx_Fp_stable. Qed. | Lemma | rowg_mxK | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"group",
"mx_Fp_stable",
"rVn",
"rowg",
"rowg_mx",
"stable_rowg_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rowg_mxSK (L : {set rVn}) (M : {group rVn}) :
(rowg_mx L <= rowg_mx M)%MS = (L \subset M). | Proof.
apply/idP/idP; last exact: rowg_mxS.
by rewrite -rowgS rowg_mxK; apply/subset_trans/sub_rowg_mx.
Qed. | Lemma | rowg_mxSK | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"group",
"last",
"rVn",
"rowgS",
"rowg_mx",
"rowg_mxK",
"rowg_mxS",
"sub_rowg_mx",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_rowg (L : {group rVn}) :
prime p -> \rank (rowg_mx L) = logn p #|L|. | Proof.
by move=> p_pr; rewrite -{2}(rowg_mxK L) card_rowg card_Fp ?pfactorK.
Qed. | Lemma | mxrank_rowg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"card_Fp",
"card_rowg",
"group",
"logn",
"p_pr",
"pfactorK",
"prime",
"rVn",
"rank",
"rowg_mx",
"rowg_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(abelE : p.-abelem E) (ntE : E :!=: 1%g). | Hypotheses | abelE | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
pE : p.-group E | := abelem_pgroup abelE. | Let | pE | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelE",
"abelem_pgroup",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
p_pr : prime p. | Proof. by have [] := pgroup_pdiv pE ntE. Qed. | Let | p_pr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"pE",
"pgroup_pdiv",
"prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n' | := (abelem_dim' (gval E)). | Notation | n' | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_dim'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVn | := 'rV['F_p](gval E). | Notation | rVn | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dim_abelemE : n = logn p #|E|. | Proof.
rewrite /n'; have [_ _ [k ->]] := pgroup_pdiv pE ntE.
by rewrite /pdiv primesX ?primes_prime // pfactorK.
Qed. | Lemma | dim_abelemE | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"logn",
"n'",
"pE",
"pdiv",
"pfactorK",
"pgroup_pdiv",
"primesX",
"primes_prime"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_abelem_rV : #|rVn| = #|E|. | Proof.
by rewrite dim_abelemE card_mx mul1n card_Fp // -p_part part_pnat_id.
Qed. | Lemma | card_abelem_rV | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"card_Fp",
"card_mx",
"dim_abelemE",
"mul1n",
"p_part",
"part_pnat_id",
"rVn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isog_abelem_rV : E \isog [set: rVn]. | Proof.
by rewrite (isog_abelem_card _ abelE) cardsT card_abelem_rV mx_Fp_abelem /=.
Qed. | Lemma | isog_abelem_rV | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelE",
"card_abelem_rV",
"cardsT",
"isog",
"isog_abelem_card",
"mx_Fp_abelem",
"rVn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ab_rV_P | := (existsP isog_abelem_rV). | Notation | ab_rV_P | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"existsP",
"isog_abelem_rV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rV : gT -> rVn | := xchoose ab_rV_P. | Definition | abelem_rV | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ab_rV_P",
"gT",
"rVn",
"xchoose"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ErV | := abelem_rV. | Notation | ErV | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_rV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rV_M : {in E &, {morph ErV : x y / (x * y)%g >-> x + y}}. | Proof. by case/misomP: (xchooseP ab_rV_P) => fM _; move/morphicP: fM. Qed. | Lemma | abelem_rV_M | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"ab_rV_P",
"fM",
"misomP",
"morphicP",
"xchooseP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rV_morphism | := Morphism abelem_rV_M. | Canonical | abelem_rV_morphism | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_rV_M"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rV_isom : isom E setT ErV. | Proof. by case/misomP: (xchooseP ab_rV_P). Qed. | Lemma | abelem_rV_isom | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"ab_rV_P",
"isom",
"misomP",
"setT",
"xchooseP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rV_injm : 'injm ErV. | Proof. by case/isomP: abelem_rV_isom. Qed. | Lemma | abelem_rV_injm | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"abelem_rV_isom",
"isomP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rV_inj : {in E &, injective ErV}. | Proof. by apply/injmP; apply: abelem_rV_injm. Qed. | Lemma | abelem_rV_inj | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"abelem_rV_injm",
"apply",
"injmP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_abelem_rV : ErV @* E = setT. | Proof. by case/isomP: abelem_rV_isom. Qed. | Lemma | im_abelem_rV | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"abelem_rV_isom",
"isomP",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_im_abelem_rV u : u \in ErV @* E. | Proof. by rewrite im_abelem_rV inE. Qed. | Lemma | mem_im_abelem_rV | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"im_abelem_rV",
"inE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_im_abelem_rV mA : subset mA (mem (ErV @* E)). | Proof. by rewrite unlock; apply/pred0P=> v /=; rewrite mem_im_abelem_rV. Qed. | Lemma | sub_im_abelem_rV | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"apply",
"mem_im_abelem_rV",
"pred0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rV_1 : ErV 1 = 0%R. | Proof. by rewrite morph1. Qed. | Lemma | abelem_rV_1 | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"morph1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rV_X x i : x \in E -> ErV (x ^+ i) = i%:R *: ErV x. | Proof. by move=> Ex; rewrite morphX // scaler_nat. Qed. | Lemma | abelem_rV_X | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"morphX",
"scaler_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rV_V x : x \in E -> ErV x^-1 = - ErV x. | Proof. by move=> Ex; rewrite morphV. Qed. | Lemma | abelem_rV_V | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"morphV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelem : rVn -> gT | := invm abelem_rV_injm. | Definition | rVabelem | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_rV_injm",
"gT",
"invm",
"rVn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelem_morphism | := [morphism of rVabelem]. | Canonical | rVabelem_morphism | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"morphism",
"rVabelem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rV_E | := rVabelem. | Notation | rV_E | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"rVabelem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelem0 : rV_E 0 = 1%g. | Proof. exact: morph1. Qed. | Lemma | rVabelem0 | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"morph1",
"rV_E"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelemD : {morph rV_E : u v / u + v >-> (u * v)%g}. | Proof. by move=> u v /=; rewrite -morphM. Qed. | Lemma | rVabelemD | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"morphM",
"rV_E"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelemN : {morph rV_E: u / - u >-> (u^-1)%g}. | Proof. by move=> u /=; rewrite -morphV. Qed. | Lemma | rVabelemN | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"morphV",
"rV_E"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelemZ (m : 'F_p) : {morph rV_E : u / m *: u >-> (u ^+ m)%g}. | Proof. by move=> u; rewrite /= -morphX -?[(u ^+ m)%g]scaler_nat ?natr_Zp. Qed. | Lemma | rVabelemZ | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"morphX",
"natr_Zp",
"rV_E",
"scaler_nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rV_K : {in E, cancel ErV rV_E}. | Proof. exact: invmE. Qed. | Lemma | abelem_rV_K | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"invmE",
"rV_E"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelemK : cancel rV_E ErV. | Proof. by move=> u; rewrite invmK. Qed. | Lemma | rVabelemK | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"invmK",
"rV_E"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelem_inj : injective rV_E. | Proof. exact: can_inj rVabelemK. Qed. | Lemma | rVabelem_inj | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"rV_E",
"rVabelemK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelem_injm : 'injm rV_E. | Proof. exact: injm_invm abelem_rV_injm. Qed. | Lemma | rVabelem_injm | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_rV_injm",
"injm_invm",
"rV_E"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_rVabelem : rV_E @* setT = E. | Proof. by rewrite -im_abelem_rV im_invm. Qed. | Lemma | im_rVabelem | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"im_abelem_rV",
"im_invm",
"rV_E",
"setT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mem_rVabelem u : rV_E u \in E. | Proof. by rewrite -im_rVabelem mem_morphim. Qed. | Lemma | mem_rVabelem | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"im_rVabelem",
"mem_morphim",
"rV_E"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_rVabelem L : rV_E @* L \subset E. | Proof. by rewrite -[_ @* L]morphimIim im_invm subsetIl. Qed. | Lemma | sub_rVabelem | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"im_invm",
"morphimIim",
"rV_E",
"subsetIl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_rVabelem L : #|rV_E @* L| = #|L|. | Proof. by rewrite card_injm ?rVabelem_injm. Qed. | Lemma | card_rVabelem | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"card_injm",
"rV_E",
"rVabelem_injm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rV_mK (H : {set gT}) : H \subset E -> rV_E @* (ErV @* H) = H. | Proof. exact: morphim_invm abelem_rV_injm H. Qed. | Lemma | abelem_rV_mK | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"abelem_rV_injm",
"gT",
"morphim_invm",
"rV_E"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelem_mK L : ErV @* (rV_E @* L) = L. | Proof. by rewrite morphim_invmE morphpreK. Qed. | Lemma | rVabelem_mK | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"morphim_invmE",
"morphpreK",
"rV_E"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelem_minj : injective (morphim (MorPhantom rV_E)). | Proof. exact: can_inj rVabelem_mK. Qed. | Lemma | rVabelem_minj | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"MorPhantom",
"morphim",
"rV_E",
"rVabelem_mK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelemS L M : (rV_E @* L \subset rV_E @* M) = (L \subset M). | Proof. by rewrite injmSK ?rVabelem_injm. Qed. | Lemma | rVabelemS | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"injmSK",
"rV_E",
"rVabelem_injm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rV_S (H K : {set gT}) :
H \subset E -> (ErV @* H \subset ErV @* K) = (H \subset K). | Proof. by move=> sHE; rewrite injmSK ?abelem_rV_injm. Qed. | Lemma | abelem_rV_S | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"abelem_rV_injm",
"gT",
"injmSK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_rVabelem_im L (H : {set gT}) :
(rV_E @* L \subset H) = (L \subset ErV @* H). | Proof. by rewrite sub_morphim_pre ?morphpre_invm. Qed. | Lemma | sub_rVabelem_im | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"gT",
"morphpre_invm",
"rV_E",
"sub_morphim_pre"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_abelem_rV_im (H : {set gT}) (L : {set 'rV['F_p]_n}) :
H \subset E -> (ErV @* H \subset L) = (H \subset rV_E @* L). | Proof. by move=> sHE; rewrite sub_morphim_pre ?morphim_invmE. Qed. | Lemma | sub_abelem_rV_im | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"gT",
"morphim_invmE",
"rV_E",
"sub_morphim_pre"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_mx_fun (g : subg_of G) v | := ErV ((rV_E v) ^ val g). | Definition | abelem_mx_fun | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"rV_E",
"subg_of",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_mx & G \subset 'N(E) | :=
fun x => lin1_mx (abelem_mx_fun (subg G x)). | Definition | abelem_mx | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_mx_fun",
"lin1_mx",
"subg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nEG : G \subset 'N(E). | Hypothesis | nEG | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
r | := (abelem_mx nEG). | Notation | r | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_mx",
"nEG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_mx_linear_proof g : linear (abelem_mx_fun g). | Proof.
rewrite /abelem_mx_fun; case: g => x /= /(subsetP nEG) Nx /= m u v.
rewrite rVabelemD rVabelemZ conjMg conjXg.
by rewrite abelem_rV_M ?abelem_rV_X ?groupX ?memJ_norm // natr_Zp.
Qed. | Fact | abelem_mx_linear_proof | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_mx_fun",
"abelem_rV_M",
"abelem_rV_X",
"conjMg",
"conjXg",
"groupX",
"linear",
"memJ_norm",
"nEG",
"natr_Zp",
"rVabelemD",
"rVabelemZ",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelemJmx v x : x \in G -> rV_E (v *m r x) = (rV_E v) ^ x. | Proof.
move=> Gx; rewrite /= mul_rV_lin1 /= /abelem_mx_fun subgK //.
by rewrite abelem_rV_K // memJ_norm // (subsetP nEG).
Qed. | Let | rVabelemJmx | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_mx_fun",
"abelem_rV_K",
"memJ_norm",
"mul_rV_lin1",
"nEG",
"rV_E",
"subgK",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_mx_repr : mx_repr G r. | Proof.
split=> [|x y Gx Gy]; apply/row_matrixP=> i; apply: rVabelem_inj.
by rewrite rowE -row1 rVabelemJmx // conjg1.
by rewrite !rowE mulmxA !rVabelemJmx ?groupM // conjgM.
Qed. | Fact | abelem_mx_repr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"conjg1",
"conjgM",
"groupM",
"mulmxA",
"mx_repr",
"rVabelemJmx",
"rVabelem_inj",
"row1",
"rowE",
"row_matrixP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_repr | := MxRepresentation abelem_mx_repr. | Canonical | abelem_repr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_mx_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rG | := abelem_repr. | Let | rG | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rVabelemJ v x : x \in G -> rV_E (v *m rG x) = (rV_E v) ^ x. | Proof. exact: rVabelemJmx. Qed. | Lemma | rVabelemJ | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"rG",
"rV_E",
"rVabelemJmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rV_J : {in E & G, forall x y, ErV (x ^ y) = ErV x *m rG y}. | Proof.
by move=> x y Ex Gy; rewrite -{1}(abelem_rV_K Ex) -rVabelemJ ?rVabelemK.
Qed. | Lemma | abelem_rV_J | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"abelem_rV_K",
"rG",
"rVabelemJ",
"rVabelemK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_rowgJ m (A : 'M_(m, n)) x :
x \in G -> rV_E @* rowg (A *m rG x) = (rV_E @* rowg A) :^ x. | Proof.
move=> Gx; apply: (canRL (conjsgKV _)); apply/setP=> y.
rewrite mem_conjgV !morphim_invmE !inE memJ_norm ?(subsetP nEG) //=.
apply: andb_id2l => Ey; rewrite abelem_rV_J //.
by rewrite submxMfree // row_free_unit (repr_mx_unit rG).
Qed. | Lemma | abelem_rowgJ | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_rV_J",
"apply",
"conjsgKV",
"inE",
"memJ_norm",
"mem_conjgV",
"morphim_invmE",
"nEG",
"rG",
"rV_E",
"repr_mx_unit",
"row_free_unit",
"rowg",
"setP",
"submxMfree",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rV_abelem_sJ (L : {group gT}) x :
x \in G -> L \subset E -> ErV @* (L :^ x) = rowg (rowg_mx (ErV @* L) *m rG x). | Proof.
move=> Gx sLE; apply: rVabelem_minj; rewrite abelem_rowgJ //.
by rewrite rowg_mxK !morphim_invm // -(normsP nEG x Gx) conjSg.
Qed. | Lemma | rV_abelem_sJ | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"abelem_rowgJ",
"apply",
"conjSg",
"gT",
"group",
"morphim_invm",
"nEG",
"normsP",
"rG",
"rVabelem_minj",
"rowg",
"rowg_mx",
"rowg_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstab_abelem m (A : 'M_(m, n)) : rstab rG A = 'C_G(rV_E @* rowg A). | Proof.
apply/setP=> x /[!inE]/=; apply: andb_id2l => Gx; apply/eqP/centP => cAx.
move=> _ /morphimP[u _ + ->] => /[1!inE] /submxP[{}u ->].
by apply/esym/commgP/conjg_fixP; rewrite -rVabelemJ -?mulmxA ?cAx.
apply/row_matrixP=> i; apply: rVabelem_inj.
by rewrite row_mul rVabelemJ // /conjg -cAx ?mulKg ?mem_morphim // i... | Lemma | rstab_abelem | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"centP",
"commgP",
"conjg",
"conjg_fixP",
"inE",
"mem_morphim",
"morphimP",
"mulKg",
"mulmxA",
"rG",
"rV_E",
"rVabelemJ",
"rVabelem_inj",
"row_matrixP",
"row_mul",
"row_sub",
"rowg",
"rstab",
"setP",
"submxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_abelem m (A : 'M_(m, n)) : rstabs rG A = 'N_G(rV_E @* rowg A). | Proof.
apply/setP=> x /[!inE]/=; apply: andb_id2l => Gx.
by rewrite -rowgS -rVabelemS abelem_rowgJ.
Qed. | Lemma | rstabs_abelem | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_rowgJ",
"apply",
"inE",
"rG",
"rV_E",
"rVabelemS",
"rowg",
"rowgS",
"rstabs",
"setP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rstabs_abelemG (L : {group gT}) :
L \subset E -> rstabs rG (rowg_mx (ErV @* L)) = 'N_G(L). | Proof. by move=> sLE; rewrite rstabs_abelem rowg_mxK morphim_invm. Qed. | Lemma | rstabs_abelemG | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"gT",
"group",
"morphim_invm",
"rG",
"rowg_mx",
"rowg_mxK",
"rstabs",
"rstabs_abelem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_abelem m (U : 'M['F_p]_(m, n)) :
mxmodule rG U = (G \subset 'N(rV_E @* rowg U)). | Proof. by rewrite -subsetIidl -rstabs_abelem. Qed. | Lemma | mxmodule_abelem | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"mxmodule",
"rG",
"rV_E",
"rowg",
"rstabs_abelem",
"subsetIidl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_abelemG (L : {group gT}) :
L \subset E -> mxmodule rG (rowg_mx (ErV @* L)) = (G \subset 'N(L)). | Proof. by move=> sLE; rewrite -subsetIidl -rstabs_abelemG. Qed. | Lemma | mxmodule_abelemG | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"gT",
"group",
"mxmodule",
"rG",
"rowg_mx",
"rstabs_abelemG",
"subsetIidl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_abelemP (U : 'M['F_p]_n) :
reflect (mxsimple rG U) (minnormal (rV_E @* rowg U) G). | Proof.
apply: (iffP mingroupP) => [[/andP[ntU modU] minU] | [modU ntU minU]].
split=> [||V modV sVU ntV]; first by rewrite mxmodule_abelem.
by apply: contraNneq ntU => ->; rewrite /= rowg0 morphim1.
rewrite -rowgS -rVabelemS [_ @* rowg V]minU //.
rewrite -subG1 sub_rVabelem_im morphim1 subG1 trivg_rowg ntV ... | Lemma | mxsimple_abelemP | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_rV_injm",
"apply",
"contraNneq",
"eqEsubset",
"mingroupP",
"minnormal",
"morphim1",
"morphim_injm_eq1",
"mxmodule_abelem",
"mxmodule_abelemG",
"mxsimple",
"rG",
"rV_E",
"rVabelemS",
"rowg",
"rowg0",
"rowgK",
"rowgS",
"rowg_mxK",
"rowg_mxSK",
"rowg_mx_eq0",
"split",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_abelemGP (L : {group gT}) :
L \subset E -> reflect (mxsimple rG (rowg_mx (ErV @* L))) (minnormal L G). | Proof.
move/abelem_rV_mK=> {2}<-; rewrite -{2}[_ @* L]rowg_mxK.
exact: mxsimple_abelemP.
Qed. | Lemma | mxsimple_abelemGP | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"abelem_rV_mK",
"gT",
"group",
"minnormal",
"mxsimple",
"mxsimple_abelemP",
"rG",
"rowg_mx",
"rowg_mxK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_mx_irrP : reflect (mx_irreducible rG) (minnormal E G). | Proof.
by rewrite -[E in minnormal E G]im_rVabelem -rowg1; apply: mxsimple_abelemP.
Qed. | Lemma | abelem_mx_irrP | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"im_rVabelem",
"minnormal",
"mx_irreducible",
"mxsimple_abelemP",
"rG",
"rowg1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rfix_abelem (H : {set gT}) :
H \subset G -> (rfix_mx rG H :=: rowg_mx (ErV @* 'C_E(H)%g))%MS. | Proof.
move/subsetP=> sHG; apply/eqmxP/andP; split.
rewrite -rowgS rowg_mxK -sub_rVabelem_im // subsetI sub_rVabelem /=.
apply/centsP=> y /morphimP[v _] /[1!inE] cGv ->{y} x Gx.
by apply/commgP/conjg_fixP; rewrite /= -rVabelemJ ?sHG ?(rfix_mxP H _).
rewrite genmxE; apply/rfix_mxP=> x Hx; apply/row_matrixP=> i.
re... | Lemma | rfix_abelem | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"ErV",
"abelem_rV_J",
"apply",
"centP",
"centsP",
"commgP",
"conjgE",
"conjg_fixP",
"enum_valP",
"eqmxP",
"gT",
"genmxE",
"inE",
"morphimP",
"mulKg",
"rG",
"rVabelemJ",
"rfix_mx",
"rfix_mxP",
"rowK",
"row_matrixP",
"row_mul",
"rowgS",
"rowg_mx",
"rowg_mxK",
"sHG",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rker_abelem : rker rG = 'C_G(E). | Proof. by rewrite /rker rstab_abelem rowg1 im_rVabelem. Qed. | Lemma | rker_abelem | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"im_rVabelem",
"rG",
"rker",
"rowg1",
"rstab_abelem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
abelem_mx_faithful : 'C_G(E) = 1%g -> mx_faithful rG. | Proof. by rewrite /mx_faithful rker_abelem => ->. Qed. | Lemma | abelem_mx_faithful | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"mx_faithful",
"rG",
"rker_abelem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(nEG : G \subset 'N(E)) (sHG : H \subset G). | Hypotheses | nEG | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"sHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
nEH | := subset_trans sHG nEG. | Let | nEH | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"nEG",
"sHG",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rG | := (abelem_repr nEG). | Notation | rG | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_repr",
"nEG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rHG | := (subg_repr rG sHG). | Notation | rHG | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"rG",
"sHG",
"subg_repr"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rH | := (abelem_repr nEH). | Notation | rH | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_repr",
"nEH"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_abelem_subg_repr : {in H, rHG =1 rH}. | Proof.
move=> x Hx; apply/row_matrixP=> i; rewrite !rowE !mul_rV_lin1 /=.
by rewrite /abelem_mx_fun !subgK ?(subsetP sHG).
Qed. | Lemma | eq_abelem_subg_repr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"abelem_mx_fun",
"apply",
"mul_rV_lin1",
"rH",
"rHG",
"rowE",
"row_matrixP",
"sHG",
"subgK",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rsim_abelem_subg : mx_rsim rHG rH. | Proof.
exists 1%:M => [//| |x Hx]; first by rewrite row_free_unit unitmx1.
by rewrite mul1mx mulmx1 eq_abelem_subg_repr.
Qed. | Lemma | rsim_abelem_subg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"eq_abelem_subg_repr",
"mul1mx",
"mulmx1",
"mx_rsim",
"rH",
"rHG",
"row_free_unit",
"unitmx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxmodule_abelem_subg m (U : 'M_(m, n)) : mxmodule rHG U = mxmodule rH U. | Proof.
apply: eq_subset_r => x.
rewrite [LHS]inE inE; apply: andb_id2l => Hx.
by rewrite eq_abelem_subg_repr.
Qed. | Lemma | mxmodule_abelem_subg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"eq_abelem_subg_repr",
"eq_subset_r",
"inE",
"mxmodule",
"rH",
"rHG"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxsimple_abelem_subg U : mxsimple rHG U <-> mxsimple rH U. | Proof.
have eq_modH := mxmodule_abelem_subg; rewrite /mxsimple eq_modH.
by split=> [] [-> -> minU]; split=> [//|//|V]; have:= minU V; rewrite eq_modH.
Qed. | Lemma | mxsimple_abelem_subg | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"mxmodule_abelem_subg",
"mxsimple",
"rH",
"rHG",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcharFp : p \in [pchar F]. | Hypothesis | pcharFp | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
rfix_pgroup_pchar G H n (rG : mx_representation F G n) :
n > 0 -> p.-group H -> H \subset G -> rfix_mx rG H != 0. | Proof.
move=> n_gt0 pH sHG; rewrite -(rfix_subg rG sHG).
move: {2}_.+1 (ltnSn (n + #|H|)) {rG G sHG}(subg_repr _ _) => m.
elim: m gT H pH => // m IHm gT' G pG in n n_gt0 *; rewrite ltnS => le_nG_m rG.
apply/eqP=> Gregular; have irrG: mx_irreducible rG.
apply/mx_irrP; split=> // U modU; rewrite -mxrank_eq0 -lt0n => Un... | Lemma | rfix_pgroup_pchar | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"Cauchy",
"G1",
"apply",
"capmx0",
"card1_trivg",
"centP",
"center_sub",
"centgmx",
"centgmxP",
"commr1",
"eqn_leq",
"expg_order",
"gT",
"group",
"inE",
"irrG",
"kquo_repr",
"last",
"leqNgt",
"leq_trans",
"linear0",
"lt0n",
"ltnS",
"ltnSn",
"ltn_add2l",
"ltn_add2r",... | This is Gorenstein, Lemma 2.6.3. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
pcore_sub_rstab_mxsimple_pchar M :
mxsimple rG M -> 'O_p(G) \subset rstab rG M. | Proof.
case=> modM nzM simM; have sGpG := pcore_sub p G.
rewrite rfix_mx_rstabC //; set U := rfix_mx _ _.
have:= simM (M :&: U)%MS; rewrite sub_capmx submx_refl.
apply; rewrite ?capmxSl //.
by rewrite capmx_module // normal_rfix_mx_module ?pcore_normal.
rewrite -(in_submodK (capmxSl _ _)) val_submod_eq0 -submx0.
rewr... | Lemma | pcore_sub_rstab_mxsimple_pchar | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"capmxSl",
"capmx_module",
"in_submodK",
"lt0n",
"mxrank_eq0",
"mxsimple",
"normal_rfix_mx_module",
"pcore_normal",
"pcore_pgroup",
"pcore_sub",
"rG",
"rfix_mx",
"rfix_mx_rstabC",
"rfix_pgroup_pchar",
"rfix_submod",
"rstab",
"sub_capmx",
"submx0",
"submx_refl",
"val_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcore_sub_rker_mx_irr_pchar :
mx_irreducible rG -> 'O_p(G) \subset rker rG. | Proof. exact: pcore_sub_rstab_mxsimple_pchar. Qed. | Lemma | pcore_sub_rker_mx_irr_pchar | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"mx_irreducible",
"pcore_sub_rstab_mxsimple_pchar",
"rG",
"rker"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcore_faithful_mx_irr_pchar :
mx_irreducible rG -> mx_faithful rG -> 'O_p(G) = 1%g. | Proof.
move=> irrG ffulG; apply/trivgP; apply: subset_trans ffulG.
exact: pcore_sub_rstab_mxsimple_pchar.
Qed. | Lemma | pcore_faithful_mx_irr_pchar | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"apply",
"irrG",
"mx_faithful",
"mx_irreducible",
"pcore_sub_rstab_mxsimple_pchar",
"rG",
"subset_trans",
"trivgP"
] | This is Gorenstein, Lemma 3.1.3. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
rfix_pgroup_char | := (rfix_pgroup_pchar) (only parsing). | Notation | rfix_pgroup_char | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"rfix_pgroup_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcore_sub_rstab_mxsimple | := (pcore_sub_rstab_mxsimple_pchar) (only parsing). | Notation | pcore_sub_rstab_mxsimple | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"pcore_sub_rstab_mxsimple_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcore_sub_rker_mx_irr | := (pcore_sub_rker_mx_irr_pchar) (only parsing). | Notation | pcore_sub_rker_mx_irr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"pcore_sub_rker_mx_irr_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcore_faithful_mx_irr | := (pcore_faithful_mx_irr_pchar) (only parsing). | Notation | pcore_faithful_mx_irr | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"pcore_faithful_mx_irr_pchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(pS : p.-group S) (esS : extraspecial S). | Hypotheses | pS | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [
"extraspecial",
"group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
oSpn : #|S| = (p ^ n.*2.+1)%N. | Hypothesis | oSpn | group_representation | group_representation/mxabelem.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"div",
"choice",
"fintype",
"tuple",
"finfun",
"bigop",
"prime",
"ssralg",
"poly",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"gproduct",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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