orbsta2 - Metamath Proof Explorer

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Theorem orbsta2 19297

Description: Relation between the size of the orbit and the size of the stabilizer of a point in a finite group action. (Contributed by Mario Carneiro, 16-Jan-2015.)

**Hypotheses**
Ref	Expression
orbsta2.x	⊢ 𝑋 = (Base‘𝐺)
orbsta2.h	⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴}
orbsta2.r	⊢ ∼ = (𝐺 ~_QG 𝐻)
orbsta2.o	⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}

**Assertion**
Ref	Expression
orbsta2	⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝐴]𝑂) · (♯‘𝐻)))

Distinct variable groups: 𝑢,𝑔,𝑥,𝑦, ⊕ 𝐴,𝑔,𝑢,𝑥,𝑦 𝑔,𝐺,𝑢,𝑥,𝑦 𝑔,𝑌,𝑥,𝑦 ∼ ,𝑔,𝑥,𝑦 𝑥,𝐻,𝑦 𝑔,𝑋,𝑢,𝑥,𝑦 Allowed substitution hints: ∼ (𝑢) 𝐻(𝑢,𝑔) 𝑂(𝑥,𝑦,𝑢,𝑔) 𝑌(𝑢)

Proof of Theorem orbsta2 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orbsta2.x	. . 3 ⊢ 𝑋 = (Base‘𝐺)
2		orbsta2.r	. . 3 ⊢ ∼ = (𝐺 ~_QG 𝐻)
3		orbsta2.h	. . . . 5 ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴}
4	1, 3	gastacl 19292	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐻 ∈ (SubGrp‘𝐺))
5	4	adantr 480	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → 𝐻 ∈ (SubGrp‘𝐺))
6		simpr 484	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → 𝑋 ∈ Fin)
7	1, 2, 5, 6	lagsubg2 19177	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘(𝑋 / ∼ )) · (♯‘𝐻)))
8		pwfi 9329	. . . . . 6 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
9	6, 8	sylib 218	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → 𝒫 𝑋 ∈ Fin)
10	1, 2	eqger 19161	. . . . . . 7 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
11	5, 10	syl 17	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ∼ Er 𝑋)
12	11	qsss 8792	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (𝑋 / ∼ ) ⊆ 𝒫 𝑋)
13	9, 12	ssfid 9273	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (𝑋 / ∼ ) ∈ Fin)
14		eqid 2735	. . . . . 6 ⊢ ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩) = ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩)
15		orbsta2.o	. . . . . 6 ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
16	1, 3, 2, 14, 15	orbsta 19296	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩):(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂)
17	16	adantr 480	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩):(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂)
18	13, 17	hasheqf1od 14371	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘(𝑋 / ∼ )) = (♯‘[𝐴]𝑂))
19	18	oveq1d 7420	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ((♯‘(𝑋 / ∼ )) · (♯‘𝐻)) = ((♯‘[𝐴]𝑂) · (♯‘𝐻)))
20	7, 19	eqtrd 2770	1 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝐴]𝑂) · (♯‘𝐻)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3060 {crab 3415 ⊆ wss 3926 𝒫 cpw 4575 {cpr 4603 ⟨cop 4607 {copab 5181 ↦ cmpt 5201 ran crn 5655 –1-1-onto→wf1o 6530 ‘cfv 6531 (class class class)co 7405 Er wer 8716 [cec 8717 / cqs 8718 Fincfn 8959 · cmul 11134 ♯chash 14348 Basecbs 17228 SubGrpcsubg 19103 ~_QG cqg 19105 GrpAct cga 19272

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-disj 5087 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-ec 8721 df-qs 8725 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-n0 12502 df-z 12589 df-uz 12853 df-rp 13009 df-fz 13525 df-fzo 13672 df-seq 14020 df-exp 14080 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-clim 15504 df-sum 15703 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-0g 17455 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-grp 18919 df-minusg 18920 df-subg 19106 df-eqg 19108 df-ga 19273

This theorem is referenced by: sylow1lem5 19583 sylow2alem2 19599 sylow3lem3 19610

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