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Theorem orbsta2 18097

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 18092	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐻 ∈ (SubGrp‘𝐺))
5	4	adantr 474	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → 𝐻 ∈ (SubGrp‘𝐺))
6		simpr 479	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → 𝑋 ∈ Fin)
7	1, 2, 5, 6	lagsubg2 18006	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘(𝑋 / ∼ )) · (♯‘𝐻)))
8		eqid 2825	. . . . . . 7 ⊢ ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩) = ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩)
9		orbsta2.o	. . . . . . 7 ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
10	1, 3, 2, 8, 9	orbsta 18096	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩):(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂)
11	10	adantr 474	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩):(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂)
12	1	fvexi 6447	. . . . . . 7 ⊢ 𝑋 ∈ V
13	12	qsex 8071	. . . . . 6 ⊢ (𝑋 / ∼ ) ∈ V
14	13	f1oen 8243	. . . . 5 ⊢ (ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩):(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂 → (𝑋 / ∼ ) ≈ [𝐴]𝑂)
15	11, 14	syl 17	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (𝑋 / ∼ ) ≈ [𝐴]𝑂)
16		pwfi 8530	. . . . . . 7 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
17	6, 16	sylib 210	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → 𝒫 𝑋 ∈ Fin)
18	1, 2	eqger 17995	. . . . . . . 8 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
19	5, 18	syl 17	. . . . . . 7 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ∼ Er 𝑋)
20	19	qsss 8073	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (𝑋 / ∼ ) ⊆ 𝒫 𝑋)
21		ssfi 8449	. . . . . 6 ⊢ ((𝒫 𝑋 ∈ Fin ∧ (𝑋 / ∼ ) ⊆ 𝒫 𝑋) → (𝑋 / ∼ ) ∈ Fin)
22	17, 20, 21	syl2anc 581	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (𝑋 / ∼ ) ∈ Fin)
23	15	ensymd 8273	. . . . . 6 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → [𝐴]𝑂 ≈ (𝑋 / ∼ ))
24		enfii 8446	. . . . . 6 ⊢ (((𝑋 / ∼ ) ∈ Fin ∧ [𝐴]𝑂 ≈ (𝑋 / ∼ )) → [𝐴]𝑂 ∈ Fin)
25	22, 23, 24	syl2anc 581	. . . . 5 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → [𝐴]𝑂 ∈ Fin)
26		hashen 13427	. . . . 5 ⊢ (((𝑋 / ∼ ) ∈ Fin ∧ [𝐴]𝑂 ∈ Fin) → ((♯‘(𝑋 / ∼ )) = (♯‘[𝐴]𝑂) ↔ (𝑋 / ∼ ) ≈ [𝐴]𝑂))
27	22, 25, 26	syl2anc 581	. . . 4 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ((♯‘(𝑋 / ∼ )) = (♯‘[𝐴]𝑂) ↔ (𝑋 / ∼ ) ≈ [𝐴]𝑂))
28	15, 27	mpbird 249	. . 3 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘(𝑋 / ∼ )) = (♯‘[𝐴]𝑂))
29	28	oveq1d 6920	. 2 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → ((♯‘(𝑋 / ∼ )) · (♯‘𝐻)) = ((♯‘[𝐴]𝑂) · (♯‘𝐻)))
30	7, 29	eqtrd 2861	1 ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝐴]𝑂) · (♯‘𝐻)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∃wrex 3118 {crab 3121 ⊆ wss 3798 𝒫 cpw 4378 {cpr 4399 ⟨cop 4403 class class class wbr 4873 {copab 4935 ↦ cmpt 4952 ran crn 5343 –1-1-onto→wf1o 6122 ‘cfv 6123 (class class class)co 6905 Er wer 8006 [cec 8007 / cqs 8008 ≈ cen 8219 Fincfn 8222 · cmul 10257 ♯chash 13410 Basecbs 16222 SubGrpcsubg 17939 ~_QG cqg 17941 GrpAct cga 18072

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-disj 4842 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-ec 8011 df-qs 8015 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-sup 8617 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-rp 12113 df-fz 12620 df-fzo 12761 df-seq 13096 df-exp 13155 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-sum 14794 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-minusg 17780 df-subg 17942 df-eqg 17944 df-ga 18073

This theorem is referenced by: sylow1lem5 18368 sylow2alem2 18384 sylow3lem3 18395

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