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 |
|---|---|---|---|---|---|---|---|---|---|---|
injgz : 'injm gz. | Proof. by case/isomP: isoZ. Qed. | Let | injgz | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"isoZ",
"isomP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gzZ : gz @* 'Z(H) = 'Z(K). | Proof. by case/isomP: isoZ. Qed. | Let | gzZ | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"isoZ",
"isomP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gzZchar : gz @* 'Z(H) \char 'Z(K). | Proof. by rewrite gzZ. Qed. | Let | gzZchar | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"char",
"gzZ"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgzZZ : gz @* 'Z(H) \subset 'Z(K) | := char_sub gzZchar. | Let | sgzZZ | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"char_sub",
"gzZchar"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sZH | := center_sub H. | Let | sZH | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"center_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sZK | := center_sub K. | Let | sZK | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"center_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sgzZG : gz @* 'Z(H) \subset K | := subset_trans sgzZZ sZK. | Let | sgzZG | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"sZK",
"sgzZZ",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ker_cprod_by_is_group : group_set kerHK. | Proof.
apply/group_setP; rewrite inE /= group1 morph1 invg1 /=.
split=> // [[x1 y1] [x2 y2]].
rewrite inE /= => /andP[Zx1 /eqP->]; have [_ cGx1] := setIP Zx1.
rewrite inE /= => /andP[Zx2 /eqP->]; have [Gx2 _] := setIP Zx2.
by rewrite inE /= groupM //= -invMg (centP cGx1) // morphM.
Qed. | Lemma | ker_cprod_by_is_group | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"apply",
"centP",
"group1",
"groupM",
"group_set",
"group_setP",
"inE",
"invMg",
"invg1",
"kerHK",
"morph1",
"morphM",
"setIP",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ker_cprod_by_group | := Group ker_cprod_by_is_group. | Canonical | ker_cprod_by_group | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"ker_cprod_by_is_group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ker_cprod_by_central : kerHK \subset 'Z(setX H K). | Proof.
rewrite -(center_dprod (setX_dprod H K)) -morphim_pairg1 -morphim_pair1g.
rewrite -!injm_center ?subsetT ?injm_pair1g ?injm_pairg1 //=.
rewrite morphim_pairg1 morphim_pair1g setX_dprod.
apply/subsetP=> [[x y]] /[1!inE] /andP[Zx /eqP->].
by rewrite inE /= Zx groupV (subsetP sgzZZ) ?mem_morphim.
Qed. | Lemma | ker_cprod_by_central | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"apply",
"center_dprod",
"groupV",
"inE",
"injm_center",
"injm_pair1g",
"injm_pairg1",
"kerHK",
"mem_morphim",
"morphim_pair1g",
"morphim_pairg1",
"setX",
"setX_dprod",
"sgzZZ",
"subsetP",
"subsetT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprod_by_key : unit. | Proof. by []. Qed. | Fact | cprod_by_key | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprod_by_def : finGroupType | := subg_of (setX H K / kerHK). | Definition | cprod_by_def | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"kerHK",
"setX",
"subg_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cprod_by | := locked_with cprod_by_key cprod_by_def. | Definition | cprod_by | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cprod_by_def",
"cprod_by_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
C | := [set: FinGroup.sort cprod_by]. | Notation | C | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cprod_by",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_cprod : gTH * gTK -> cprod_by | :=
let: tt as k := cprod_by_key return _ -> locked_with k cprod_by_def in
subg _ \o coset kerHK. | Definition | in_cprod | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"coset",
"cprod_by",
"cprod_by_def",
"cprod_by_key",
"kerHK",
"subg"
] | FIXME : Check if we need arg_sort instead of sort | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
in_cprodM : {in setX H K &, {morph in_cprod : u v / u * v}}. | Proof.
rewrite /in_cprod /cprod_by; case: cprod_by_key => /= u v Gu Gv.
have nkerHKG := normal_norm (sub_center_normal ker_cprod_by_central).
by rewrite -!morphM ?mem_quotient // (subsetP nkerHKG).
Qed. | Lemma | in_cprodM | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cprod_by",
"cprod_by_key",
"in_cprod",
"ker_cprod_by_central",
"mem_quotient",
"morphM",
"normal_norm",
"setX",
"sub_center_normal",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
in_cprod_morphism | := Morphism in_cprodM. | Canonical | in_cprod_morphism | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"in_cprodM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ker_in_cprod : 'ker in_cprod = kerHK. | Proof.
transitivity ('ker (subg [group of setX H K / kerHK] \o coset kerHK)).
rewrite /ker /morphpre /= /in_cprod /cprod_by; case: cprod_by_key => /=.
by rewrite ['N(_) :&: _]quotientGK ?sub_center_normal ?ker_cprod_by_central.
by rewrite ker_comp ker_subg -kerE ker_coset.
Qed. | Lemma | ker_in_cprod | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"coset",
"cprod_by",
"cprod_by_key",
"group",
"in_cprod",
"ker",
"kerE",
"kerHK",
"ker_comp",
"ker_coset",
"ker_cprod_by_central",
"ker_subg",
"morphpre",
"quotientGK",
"setX",
"sub_center_normal",
"subg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cpairg1_dom : H \subset 'dom (in_cprod \o @pairg1 gTH gTK). | Proof. by rewrite -sub_morphim_pre ?subsetT // morphim_pairg1 setXS ?sub1G. Qed. | Lemma | cpairg1_dom | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"dom",
"in_cprod",
"morphim_pairg1",
"pairg1",
"setXS",
"sub1G",
"sub_morphim_pre",
"subsetT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cpair1g_dom : K \subset 'dom (in_cprod \o @pair1g gTH gTK). | Proof. by rewrite -sub_morphim_pre ?subsetT // morphim_pair1g setXS ?sub1G. Qed. | Lemma | cpair1g_dom | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"dom",
"in_cprod",
"morphim_pair1g",
"pair1g",
"setXS",
"sub1G",
"sub_morphim_pre",
"subsetT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cpairg1 | := tag (restrmP _ cpairg1_dom). | Definition | cpairg1 | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cpairg1_dom",
"restrmP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cpair1g | := tag (restrmP _ cpair1g_dom). | Definition | cpair1g | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cpair1g_dom",
"restrmP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CH | := (mfun cpairg1 @* gval H). | Notation | CH | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cpairg1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
CK | := (mfun cpair1g @* gval K). | Notation | CK | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cpair1g"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_cpairg1 : 'injm cpairg1. | Proof.
rewrite /cpairg1; case: restrmP => _ [_ -> _ _].
rewrite ker_comp ker_in_cprod; apply/subsetP=> x; rewrite !inE /=.
by case/and3P=> _ Zx; rewrite eq_sym (inv_eq invgK) invg1 morph_injm_eq1.
Qed. | Lemma | injm_cpairg1 | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"apply",
"cpairg1",
"eq_sym",
"inE",
"inv_eq",
"invg1",
"invgK",
"ker_comp",
"ker_in_cprod",
"morph_injm_eq1",
"restrmP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injH | := injm_cpairg1. | Let | injH | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"injm_cpairg1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_cpair1g : 'injm cpair1g. | Proof.
rewrite /cpair1g; case: restrmP => _ [_ -> _ _].
rewrite ker_comp ker_in_cprod; apply/subsetP=> y; rewrite !inE /= morph1 invg1.
by case/and3P.
Qed. | Lemma | injm_cpair1g | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"apply",
"cpair1g",
"inE",
"invg1",
"ker_comp",
"ker_in_cprod",
"morph1",
"restrmP",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injK | := injm_cpair1g. | Let | injK | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"injm_cpair1g"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_cpair_cent : CK \subset 'C(CH). | Proof.
rewrite /cpairg1 /cpair1g; do 2!case: restrmP => _ [_ _ _ -> //].
rewrite !morphim_comp morphim_cents // morphim_pair1g morphim_pairg1.
by case/dprodP: (setX_dprod H K).
Qed. | Lemma | im_cpair_cent | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"CH",
"CK",
"cpair1g",
"cpairg1",
"dprodP",
"morphim_cents",
"morphim_comp",
"morphim_pair1g",
"morphim_pairg1",
"restrmP",
"setX_dprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_cpair : CH * CK = C. | Proof.
rewrite /cpairg1 /cpair1g; do 2!case: restrmP => _ [_ _ _ -> //].
rewrite !morphim_comp -morphimMl morphim_pairg1 ?setXS ?sub1G //.
rewrite morphim_pair1g setX_prod morphimEdom /= /in_cprod /cprod_by.
by case: cprod_by_key; rewrite /= imset_comp imset_coset -morphimEdom im_subg.
Qed. | Lemma | im_cpair | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"CH",
"CK",
"cpair1g",
"cpairg1",
"cprod_by",
"cprod_by_key",
"im_subg",
"imset_comp",
"imset_coset",
"in_cprod",
"morphimEdom",
"morphimMl",
"morphim_comp",
"morphim_pair1g",
"morphim_pairg1",
"restrmP",
"setXS",
"setX_prod",
"sub1G"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_cpair_cprod : CH \* CK = C. | Proof. by rewrite cprodE ?im_cpair. Qed. | Lemma | im_cpair_cprod | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"CH",
"CK",
"cprodE",
"im_cpair"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_cpairZ : {in 'Z(H), cpairg1 =1 cpair1g \o gz}. | Proof.
rewrite /cpairg1 /cpair1g => z1 Zz1; set z2 := gz z1.
have Zz2: z2 \in 'Z(K) by rewrite (subsetP sgzZZ) ?mem_morphim.
have [[Gz1 _] [/= Gz2 _]]:= (setIP Zz1, setIP Zz2).
do 2![case: restrmP => f /= [df _ _ _]; rewrite {f}df].
apply/rcoset_kerP; rewrite ?inE ?group1 ?andbT //.
by rewrite ker_in_cprod mem_rcoset i... | Lemma | eq_cpairZ | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"apply",
"cpair1g",
"cpairg1",
"group1",
"inE",
"invg1",
"ker_in_cprod",
"mem_morphim",
"mem_rcoset",
"mul1g",
"mulg1",
"rcoset_kerP",
"restrmP",
"setIP",
"sgzZZ",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
setI_im_cpair : CH :&: CK = 'Z(CH). | Proof.
apply/eqP; rewrite eqEsubset setIS //=.
rewrite subsetI center_sub -injm_center //.
rewrite (eq_in_morphim _ eq_cpairZ); last by rewrite morphim_comp morphimS.
by rewrite !(setIidPr _) // -sub_morphim_pre.
Qed. | Lemma | setI_im_cpair | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"CH",
"CK",
"apply",
"center_sub",
"eqEsubset",
"eq_cpairZ",
"eq_in_morphim",
"injm_center",
"last",
"morphimS",
"morphim_comp",
"setIS",
"setIidPr",
"sub_morphim_pre",
"subsetI"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cpair1g_center : cpair1g @* 'Z(K) = 'Z(C). | Proof.
case/cprodP: (center_cprod im_cpair_cprod) => _ <- _.
by rewrite injm_center // -setI_im_cpair mulSGid //= setIC setIS 1?centsC.
Qed. | Lemma | cpair1g_center | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"center_cprod",
"centsC",
"cpair1g",
"cprodP",
"im_cpair_cprod",
"injm_center",
"mulSGid",
"setIC",
"setIS",
"setI_im_cpair"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cpair_center_id : 'Z(CH) = 'Z(CK). | Proof.
rewrite -!injm_center // -gzZ -morphim_comp; apply: eq_in_morphim eq_cpairZ.
by rewrite !(setIidPr _) // -sub_morphim_pre.
Qed. | Lemma | cpair_center_id | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"CH",
"CK",
"apply",
"eq_cpairZ",
"eq_in_morphim",
"gzZ",
"injm_center",
"morphim_comp",
"setIidPr",
"sub_morphim_pre"
] | Uses gzZ. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
cpairg1_center : cpairg1 @* 'Z(H) = 'Z(C). | Proof. by rewrite -cpair1g_center !injm_center // cpair_center_id. Qed. | Lemma | cpairg1_center | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cpair1g_center",
"cpair_center_id",
"cpairg1",
"injm_center"
] | Uses gzZ. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
eq_fHK : {in 'Z(H), fH =1 fK \o gz}. | Hypothesis | eq_fHK | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"fH",
"fK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
gH | := ifactm fH injm_cpairg1. | Let | gH | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"fH",
"ifactm",
"injm_cpairg1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gK | := ifactm fK injm_cpair1g. | Let | gK | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"fK",
"ifactm",
"injm_cpair1g"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcprodm_cent : gK @* CK \subset 'C(gH @* CH). | Proof. by rewrite !im_ifactm. Qed. | Lemma | xcprodm_cent | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"CH",
"CK",
"gH",
"gK",
"im_ifactm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcprodmI : {in CH :&: CK, gH =1 gK}. | Proof.
rewrite setI_im_cpair -injm_center // => fHx; case/morphimP=> x Gx Zx ->{fHx}.
by rewrite {2}eq_cpairZ //= ?ifactmE ?eq_fHK //= (subsetP sgzZG) ?mem_morphim.
Qed. | Lemma | xcprodmI | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"CH",
"CK",
"eq_cpairZ",
"eq_fHK",
"gH",
"gK",
"ifactmE",
"injm_center",
"mem_morphim",
"morphimP",
"setI_im_cpair",
"sgzZG",
"subsetP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcprodm | := cprodm im_cpair_cprod xcprodm_cent xcprodmI. | Definition | xcprodm | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cprodm",
"im_cpair_cprod",
"xcprodmI",
"xcprodm_cent"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcprod_morphism | := [morphism of xcprodm]. | Canonical | xcprod_morphism | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"morphism",
"xcprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcprodmEl : {in H, forall x, xcprodm (cpairg1 x) = fH x}. | Proof. by move=> x Hx; rewrite /xcprodm cprodmEl ?mem_morphim ?ifactmE. Qed. | Lemma | xcprodmEl | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cpairg1",
"cprodmEl",
"fH",
"ifactmE",
"mem_morphim",
"xcprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcprodmEr : {in K, forall y, xcprodm (cpair1g y) = fK y}. | Proof. by move=> y Ky; rewrite /xcprodm cprodmEr ?mem_morphim ?ifactmE. Qed. | Lemma | xcprodmEr | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cpair1g",
"cprodmEr",
"fK",
"ifactmE",
"mem_morphim",
"xcprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcprodmE :
{in H & K, forall x y, xcprodm (cpairg1 x * cpair1g y) = fH x * fK y}. | Proof.
by move=> x y Hx Ky; rewrite /xcprodm cprodmE ?mem_morphim ?ifactmE.
Qed. | Lemma | xcprodmE | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cpair1g",
"cpairg1",
"cprodmE",
"fH",
"fK",
"ifactmE",
"mem_morphim",
"xcprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_xcprodm : xcprodm @* C = fH @* H * fK @* K. | Proof. by rewrite -im_cpair morphim_cprodm // !im_ifactm. Qed. | Lemma | im_xcprodm | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"fH",
"fK",
"im_cpair",
"im_ifactm",
"morphim_cprodm",
"xcprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_xcprodml A : xcprodm @* (cpairg1 @* A) = fH @* A. | Proof.
rewrite -!(morphimIdom _ A) morphim_cprodml ?morphimS ?subsetIl //.
by rewrite morphim_ifactm ?subsetIl.
Qed. | Lemma | im_xcprodml | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cpairg1",
"fH",
"morphimIdom",
"morphimS",
"morphim_cprodml",
"morphim_ifactm",
"subsetIl",
"xcprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
im_xcprodmr A : xcprodm @* (cpair1g @* A) = fK @* A. | Proof.
rewrite -!(morphimIdom _ A) morphim_cprodmr ?morphimS ?subsetIl //.
by rewrite morphim_ifactm ?subsetIl.
Qed. | Lemma | im_xcprodmr | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cpair1g",
"fK",
"morphimIdom",
"morphimS",
"morphim_cprodmr",
"morphim_ifactm",
"subsetIl",
"xcprodm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
injm_xcprodm : 'injm xcprodm = 'injm fH && 'injm fK. | Proof.
rewrite injm_cprodm !ker_ifactm !subG1 !morphim_injm_eq1 ?subsetIl // -!subG1.
apply: andb_id2l => /= injfH; apply: andb_idr => _.
rewrite !im_ifactm // -(morphimIdom gH) setI_im_cpair -injm_center //.
rewrite morphim_ifactm // eqEsubset subsetI morphimS //=.
rewrite {1}injm_center // setIS //=.
rewrite (eq_in_m... | Lemma | injm_xcprodm | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"apply",
"eqEsubset",
"eq_fHK",
"eq_in_morphim",
"fH",
"fK",
"gH",
"im_ifactm",
"injm_center",
"injm_cprodm",
"ker_ifactm",
"last",
"morphimIdom",
"morphimS",
"morphim_comp",
"morphim_ifactm",
"morphim_injm_eq1",
"setIS",
"setI_im_cpair",
"setIidPr",
"subG1",
"sub_morphim_p... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aut_cprod_by_full :
Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H) ->
Aut_in (Aut K) 'Z(K) \isog Aut 'Z(K) ->
Aut_in (Aut C) 'Z(C) \isog Aut 'Z(C). | Proof.
move=> AutZinH AutZinK.
have Cfull:= Aut_cprod_full im_cpair_cprod cpair_center_id.
by rewrite Cfull // -injm_center // injm_Aut_full ?center_sub.
Qed. | Lemma | Aut_cprod_by_full | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"Aut",
"Aut_cprod_full",
"Aut_in",
"center_sub",
"cpair_center_id",
"im_cpair_cprod",
"injm_Aut_full",
"injm_center",
"isog"
] | Uses gzZchar. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
gzZ_lone (Y : {group gTK}) :
Y \subset 'Z(K) -> gz @* 'Z(H) \isog Y -> gz @* 'Z(H) = Y. | Proof.
move=> sYZ isoY; apply/eqP.
by rewrite eq_sym eqEcard (card_isog isoY) gzZ sYZ /=.
Qed. | Let | gzZ_lone | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"apply",
"card_isog",
"eqEcard",
"eq_sym",
"group",
"gzZ",
"isog"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(defG : GH \* GK = G) (ziGHK : GH :&: GK = 'Z(GH)). | Hypotheses | defG | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
AutZHfull : Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H). | Hypothesis | AutZHfull | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"Aut",
"Aut_in",
"isog"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
(isoGH : GH \isog H) (isoGK : GK \isog K). | Hypotheses | isoGH | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"isog"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
cprod_by_uniq :
exists f : {morphism G >-> cprod_by},
[/\ isom G C f, f @* GH = CH & f @* GK = CK]. | Proof.
have [_ defGHK cGKH] := cprodP defG.
have AutZinH := Aut_sub_fullP sZH AutZHfull.
have [fH injfH defGH]:= isogP (isog_symr isoGH).
have [fK injfK defGK]:= isogP (isog_symr isoGK).
have sfHZfK: fH @* 'Z(H) \subset fK @* K.
by rewrite injm_center //= defGH defGK -ziGHK subsetIr.
have gzZ_id: gz @* 'Z(H) = invm i... | Lemma | cprod_by_uniq | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"AutZHfull",
"Aut_sub_fullP",
"CH",
"CK",
"apply",
"centsC",
"cprodP",
"cprod_by",
"defG",
"dom",
"domP",
"fH",
"fK",
"gH",
"gzZ_lone",
"im_invm",
"im_xcprodm",
"im_xcprodml",
"im_xcprodmr",
"injf",
"injmK",
"injm_center",
"injm_comp",
"injm_invm",
"injm_xcprodm",
"... | Uses gzZ_lone | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
isog_cprod_by : G \isog C. | Proof. by have [f [isoG _ _]] := cprod_by_uniq; apply: isom_isog isoG. Qed. | Lemma | isog_cprod_by | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"apply",
"cprod_by_uniq",
"isoG",
"isog",
"isom_isog"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
gt_ b | := if b then gTK else gTH. | Let | gt_ | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isob | := ('Z(H) \isog 'Z(K)) (only parsing). | Notation | isob | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"isog"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
G_ b | := if b as b' return {group gt_ b'} then K else H. | Let | G_ | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"group",
"gt_"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcprod_subproof :
{gz : {morphism 'Z(H) >-> gt_ isob} | isom 'Z(H) 'Z(G_ isob) gz}. | Proof.
case: (pickP [pred f : {ffun _} | misom 'Z(H) 'Z(K) f]) => [f isoZ | no_f].
rewrite (misom_isog isoZ); case/andP: isoZ => fM isoZ.
by exists [morphism of morphm fM].
move/pred0P: no_f => not_isoZ; rewrite [isob](congr1 negb not_isoZ).
by exists (idm_morphism _); apply/isomP; rewrite injm_idm im_idm.
Qed. | Lemma | xcprod_subproof | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"G_",
"apply",
"fM",
"gt_",
"idm_morphism",
"im_idm",
"injm_idm",
"isoZ",
"isob",
"isom",
"isomP",
"misom",
"misom_isog",
"morphism",
"morphm",
"pickP",
"pred0P"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcprod | := cprod_by (svalP xcprod_subproof). | Definition | xcprod | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cprod_by",
"xcprod_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcprod_spec : finGroupType -> Prop | :=
XcprodSpec gz isoZ : xcprod_spec (@cprod_by gTH gTK H K gz isoZ). | Inductive | xcprod_spec | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"cprod_by",
"isoZ"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xcprodP : 'Z(H) \isog 'Z(K) -> xcprod_spec xcprod. | Proof. by rewrite /xcprod => isoZ; move: xcprod_subproof; rewrite isoZ. Qed. | Lemma | xcprodP | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"isoZ",
"isog",
"xcprod",
"xcprod_spec",
"xcprod_subproof"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isog_xcprod (rT : finGroupType) (GH GK G : {group rT}) :
Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H) ->
GH \isog H -> GK \isog K -> GH \* GK = G -> 'Z(GH) = 'Z(GK) ->
G \isog [set: xcprod]. | Proof.
move=> AutZinH isoGH isoGK defG eqZGHK; have [_ _ cGHK] := cprodP defG.
have [|gz isoZ] := xcprodP.
have [[fH injfH <-] [fK injfK <-]] := (isogP isoGH, isogP isoGK).
rewrite -!injm_center -?(isog_transl _ (sub_isog _ _)) ?center_sub //=.
by rewrite eqZGHK sub_isog ?center_sub.
rewrite (isog_cprod_by _ defG... | Lemma | isog_xcprod | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"Aut",
"Aut_in",
"apply",
"center_sub",
"cprodP",
"defG",
"eqEsubset",
"fH",
"fK",
"group",
"injm_center",
"isoGH",
"isoZ",
"isog",
"isogP",
"isog_cprod_by",
"isog_transl",
"setIS",
"sub_isog",
"subsetI",
"xcprod",
"xcprodP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ncprod_def n : finGroupType | :=
if n is n'.+1 then xcprod G [set: ncprod_def n']
else subg_of 'Z(G). | Fixpoint | ncprod_def | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"n'",
"subg_of",
"xcprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ncprod_key : unit. | Proof. by []. Qed. | Fact | ncprod_key | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ncprod | := locked_with ncprod_key ncprod_def. | Definition | ncprod | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"ncprod_def",
"ncprod_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
G_ n | := [set: gsort (ncprod n)]. | Notation | G_ | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"gsort",
"ncprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ncprod0 : G_ 0 \isog 'Z(G). | Proof. by rewrite [ncprod]unlock isog_sym isog_subg. Qed. | Lemma | ncprod0 | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"G_",
"isog",
"isog_subg",
"isog_sym",
"ncprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
center_ncprod0 : 'Z(G_ 0) = G_ 0. | Proof. by apply: center_idP; rewrite (isog_abelian ncprod0) center_abelian. Qed. | Lemma | center_ncprod0 | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"G_",
"apply",
"center_abelian",
"center_idP",
"isog_abelian",
"ncprod0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
center_ncprod n : 'Z(G_ n) \isog 'Z(G). | Proof.
elim: n => [|n]; first by rewrite center_ncprod0 ncprod0.
rewrite [ncprod]unlock=> /isog_symr/xcprodP[gz isoZ] /=.
by rewrite -cpairg1_center isog_sym sub_isog ?center_sub ?injm_cpairg1.
Qed. | Lemma | center_ncprod | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"G_",
"center_ncprod0",
"center_sub",
"cpairg1_center",
"injm_cpairg1",
"isoZ",
"isog",
"isog_sym",
"isog_symr",
"ncprod",
"ncprod0",
"sub_isog",
"xcprodP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ncprodS n : xcprod_spec G [set: ncprod n] (ncprod n.+1). | Proof.
by have:= xcprodP (isog_symr (center_ncprod n)); rewrite [ncprod]unlock.
Qed. | Lemma | ncprodS | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"center_ncprod",
"isog_symr",
"ncprod",
"xcprodP",
"xcprod_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ncprod1 : G_ 1 \isog G. | Proof.
case: ncprodS => gz isoZ; rewrite isog_sym /= -im_cpair.
rewrite mulGSid /=; last by rewrite sub_isog ?injm_cpairg1.
rewrite -{3}center_ncprod0 injm_center ?injm_cpair1g //.
by rewrite -cpair_center_id center_sub.
Qed. | Lemma | ncprod1 | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"G_",
"center_ncprod0",
"center_sub",
"cpair_center_id",
"im_cpair",
"injm_center",
"injm_cpair1g",
"injm_cpairg1",
"isoZ",
"isog",
"isog_sym",
"last",
"mulGSid",
"ncprodS",
"sub_isog"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Aut_ncprod_full n :
Aut_in (Aut G) 'Z(G) \isog Aut 'Z(G) ->
Aut_in (Aut (G_ n)) 'Z(G_ n) \isog Aut 'Z(G_ n). | Proof.
move=> AutZinG; elim: n => [|n IHn].
by rewrite center_ncprod0; apply/Aut_sub_fullP=> // g injg gG0; exists g.
by case: ncprodS => gz isoZ; apply: Aut_cprod_by_full.
Qed. | Lemma | Aut_ncprod_full | solvable | solvable/center.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"div",
"fintype",
"bigop",
"finset",
"fingroup",
"morphism",
"perm",
"automorphism",
"quotient",
"action",
"gproduct",
"gfunctor",
"cyclic",
"finfun"
] | [
"Aut",
"Aut_cprod_by_full",
"Aut_in",
"Aut_sub_fullP",
"G_",
"apply",
"center_ncprod0",
"isoZ",
"isog",
"ncprodS"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
derived_at n (gT : finGroupType) (A : {set gT}) | :=
iter n (fun B => [~: B, B]) A. | Definition | derived_at | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"gT",
"iter"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"G ^` ( n )" | := (derived_at n G) : group_scope. | Notation | G ^` ( n ) | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"derived_at"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
derg0 A : A^`(0) = A. | Proof. by []. Qed. | Lemma | derg0 | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
derg1 A : A^`(1) = [~: A, A]. | Proof. by []. Qed. | Lemma | derg1 | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dergSn n A : A^`(n.+1) = [~: A^`(n), A^`(n)]. | Proof. by []. Qed. | Lemma | dergSn | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
der_group_set G n : group_set G^`(n). | Proof. by case: n => [|n]; apply: groupP. Qed. | Lemma | der_group_set | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"apply",
"groupP",
"group_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
derived_at_group G n | := Group (der_group_set G n). | Canonical | derived_at_group | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"der_group_set"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"G ^` ( n )" | := (derived_at_group G n) : Group_scope. | Notation | G ^` ( n ) | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"derived_at_group"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjg_mulR x y : x ^ y = x * [~ x, y]. | Proof. by rewrite mulKVg. Qed. | Lemma | conjg_mulR | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"mulKVg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjg_Rmul x y : x ^ y = [~ y, x^-1] * x. | Proof. by rewrite commgEr invgK mulgKV. Qed. | Lemma | conjg_Rmul | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commgEr",
"invgK",
"mulgKV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commMgJ x y z : [~ x * y, z] = [~ x, z] ^ y * [~ y, z]. | Proof. by rewrite !commgEr conjgM mulgA -conjMg mulgK. Qed. | Lemma | commMgJ | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commgEr",
"conjMg",
"conjgM",
"mulgA",
"mulgK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commgMJ x y z : [~ x, y * z] = [~ x, z] * [~ x, y] ^ z. | Proof. by rewrite !commgEl conjgM -mulgA -conjMg mulKVg. Qed. | Lemma | commgMJ | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commgEl",
"conjMg",
"conjgM",
"mulKVg",
"mulgA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commMgR x y z : [~ x * y, z] = [~ x, z] * [~ x, z, y] * [~ y, z]. | Proof. by rewrite commMgJ conjg_mulR. Qed. | Lemma | commMgR | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commMgJ",
"conjg_mulR"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commgMR x y z : [~ x, y * z] = [~ x, z] * [~ x, y] * [~ x, y, z]. | Proof. by rewrite commgMJ conjg_mulR mulgA. Qed. | Lemma | commgMR | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commgMJ",
"conjg_mulR",
"mulgA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Hall_Witt_identity x y z :
[~ x, y^-1, z] ^ y * [~ y, z^-1, x] ^ z * [~ z, x^-1, y] ^ x = 1. | Proof. (* gsimpl *)
pose a x y z : gT := x * z * y ^ x.
suffices{x y z} hw_aux x y z: [~ x, y^-1, z] ^ y = (a x y z)^-1 * (a y z x).
by rewrite !hw_aux; move: a {hw_aux} => a; rewrite 2!mulgA !mulgK mulVg.
by rewrite commgEr conjMg -conjgM -conjg_Rmul conjgE !invMg !mulgA.
Qed. | Lemma | Hall_Witt_identity | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commgEr",
"conjMg",
"conjgE",
"conjgM",
"conjg_Rmul",
"gT",
"invMg",
"mulVg",
"mulgA",
"mulgK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cxz : commute x [~ x, y]. | Hypothesis | cxz | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commute"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
commVg : [~ x^-1, y] = [~ x, y]^-1. | Proof.
apply/eqP; rewrite commgEl eq_sym eq_invg_mul invgK mulgA -cxz.
by rewrite -conjg_mulR -conjMg mulgV conj1g.
Qed. | Lemma | commVg | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"apply",
"commgEl",
"conj1g",
"conjMg",
"conjg_mulR",
"cxz",
"eq_invg_mul",
"eq_sym",
"invgK",
"mulgA",
"mulgV"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commXg : [~ x ^+ i, y] = [~ x, y] ^+ i. | Proof.
elim: i => [|i' IHi]; first exact: comm1g.
by rewrite !expgS commMgJ /conjg commuteX // mulKg IHi.
Qed. | Lemma | commXg | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"comm1g",
"commMgJ",
"commuteX",
"conjg",
"expgS",
"mulKg"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cyz : commute y [~ x, y]. | Hypothesis | cyz | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commute"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
cyz' | := commuteV cyz. | Let | cyz' | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commuteV",
"cyz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commgV : [~ x, y^-1] = [~ x, y]^-1. | Proof. by rewrite -invg_comm commVg -(invg_comm x y) ?invgK. Qed. | Lemma | commgV | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commVg",
"invgK",
"invg_comm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
commgX : [~ x, y ^+ i] = [~ x, y] ^+ i. | Proof. by rewrite -invg_comm commXg -(invg_comm x y) ?expgVn ?invgK. Qed. | Lemma | commgX | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commXg",
"expgVn",
"invgK",
"invg_comm"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
(cxz : commute x [~ x, y]) (cyz : commute y [~ x, y]). | Hypotheses | cxz | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"commute",
"cyz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
commXXg : [~ x ^+ i, y ^+ j] = [~ x, y] ^+ (i * j). | Proof. by rewrite expgM commgX commXg //; apply: commuteX. Qed. | Lemma | commXXg | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"apply",
"commXg",
"commgX",
"commuteX",
"expgM"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
expMg_Rmul : (y * x) ^+ i = y ^+ i * x ^+ i * [~ x, y] ^+ 'C(i, 2). | Proof.
rewrite -bin2_sum; symmetry.
elim: i => [|k IHk] /=; first by rewrite big_geq ?mulg1.
rewrite big_nat_recr //= addnC expgD !expgS -{}IHk !mulgA; congr (_ * _).
by rewrite -!mulgA commuteX2 // -commgX // [mul y]lock 3!mulgA -commgC.
Qed. | Lemma | expMg_Rmul | solvable | solvable/commutator.v | [
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"fintype",
"bigop",
"finset",
"binomial",
"fingroup",
"morphism",
"automorphism",
"quotient",
"gfunctor"
] | [
"addnC",
"big_geq",
"big_nat_recr",
"bin2_sum",
"commgC",
"commgX",
"commuteX2",
"expgD",
"expgS",
"mul",
"mulg1",
"mulgA"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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