Theorem sylow2alem1 18744

Description: Lemma for sylow2a 18746. An equivalence class of fixed points is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)

Hypotheses

Ref	Expression
sylow2a.x	⊢ 𝑋 = (Base‘𝐺)
sylow2a.m	⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑌))
sylow2a.p	⊢ (𝜑 → 𝑃 pGrp 𝐺)
sylow2a.f	⊢ (𝜑 → 𝑋 ∈ Fin)
sylow2a.y	⊢ (𝜑 → 𝑌 ∈ Fin)
sylow2a.z	⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢}
sylow2a.r	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}

Assertion

Ref	Expression
sylow2alem1	⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → [𝐴] ∼ = {𝐴})

Distinct variable groups:   ∼ ,ℎ   𝑔,ℎ,𝑢,𝑥,𝑦,𝐴   𝑔,𝐺,𝑥,𝑦   ⊕ ,𝑔,ℎ,𝑢,𝑥,𝑦   𝑔,𝑋,ℎ,𝑢,𝑥,𝑦   𝜑,ℎ   𝑔,𝑌,ℎ,𝑢,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑢,𝑔)   𝑃(𝑥,𝑦,𝑢,𝑔,ℎ)   ∼ (𝑥,𝑦,𝑢,𝑔)   𝐺(𝑢,ℎ)   𝑍(𝑥,𝑦,𝑢,𝑔,ℎ)

Proof of Theorem sylow2alem1

Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3499	. . . . . 6 ⊢ 𝑤 ∈ V
2		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → 𝐴 ∈ 𝑍)
3		elecg 8334	. . . . . 6 ⊢ ((𝑤 ∈ V ∧ 𝐴 ∈ 𝑍) → (𝑤 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝑤))
4	1, 2, 3	sylancr 589	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (𝑤 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝑤))
5		sylow2a.r	. . . . . . . 8 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
6	5	gaorb 18439	. . . . . . 7 ⊢ (𝐴 ∼ 𝑤 ↔ (𝐴 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝐴) = 𝑤))
7	6	simp3bi 1143	. . . . . 6 ⊢ (𝐴 ∼ 𝑤 → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝐴) = 𝑤)
8		oveq2 7166	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝐴 → (ℎ ⊕ 𝑢) = (ℎ ⊕ 𝐴))
9		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝐴 → 𝑢 = 𝐴)
10	8, 9	eqeq12d 2839	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝐴 → ((ℎ ⊕ 𝑢) = 𝑢 ↔ (ℎ ⊕ 𝐴) = 𝐴))
11	10	ralbidv 3199	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝐴 → (∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢 ↔ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐴))
12		sylow2a.z	. . . . . . . . . . . 12 ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢}
13	11, 12	elrab2 3685	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑍 ↔ (𝐴 ∈ 𝑌 ∧ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐴))
14	2, 13	sylib 220	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (𝐴 ∈ 𝑌 ∧ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐴))
15	14	simprd 498	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐴)
16		oveq1 7165	. . . . . . . . . . 11 ⊢ (ℎ = 𝑘 → (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
17	16	eqeq1d 2825	. . . . . . . . . 10 ⊢ (ℎ = 𝑘 → ((ℎ ⊕ 𝐴) = 𝐴 ↔ (𝑘 ⊕ 𝐴) = 𝐴))
18	17	rspccva 3624	. . . . . . . . 9 ⊢ ((∀ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐴 ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) = 𝐴)
19	15, 18	sylan 582	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) = 𝐴)
20		eqeq1 2827	. . . . . . . 8 ⊢ ((𝑘 ⊕ 𝐴) = 𝑤 → ((𝑘 ⊕ 𝐴) = 𝐴 ↔ 𝑤 = 𝐴))
21	19, 20	syl5ibcom 247	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝑘 ⊕ 𝐴) = 𝑤 → 𝑤 = 𝐴))
22	21	rexlimdva 3286	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝐴) = 𝑤 → 𝑤 = 𝐴))
23	7, 22	syl5 34	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (𝐴 ∼ 𝑤 → 𝑤 = 𝐴))
24	4, 23	sylbid 242	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (𝑤 ∈ [𝐴] ∼ → 𝑤 = 𝐴))
25		velsn 4585	. . . 4 ⊢ (𝑤 ∈ {𝐴} ↔ 𝑤 = 𝐴)
26	24, 25	syl6ibr 254	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (𝑤 ∈ [𝐴] ∼ → 𝑤 ∈ {𝐴}))
27	26	ssrdv 3975	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → [𝐴] ∼ ⊆ {𝐴})
28		sylow2a.m	. . . . . . 7 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑌))
29		sylow2a.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
30	5, 29	gaorber 18440	. . . . . . 7 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
31	28, 30	syl 17	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝑌)
32	31	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → ∼ Er 𝑌)
33	14	simpld 497	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → 𝐴 ∈ 𝑌)
34	32, 33	erref 8311	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → 𝐴 ∼ 𝐴)
35		elecg 8334	. . . . 5 ⊢ ((𝐴 ∈ 𝑍 ∧ 𝐴 ∈ 𝑍) → (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
36	2, 35	sylancom 590	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
37	34, 36	mpbird 259	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → 𝐴 ∈ [𝐴] ∼ )
38	37	snssd 4744	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → {𝐴} ⊆ [𝐴] ∼ )
39	27, 38	eqssd 3986	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → [𝐴] ∼ = {𝐴})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  {crab 3144  Vcvv 3496   ⊆ wss 3938  {csn 4569  {cpr 4571   class class class wbr 5068  {copab 5130  ‘cfv 6357  (class class class)co 7158   Er wer 8288  [cec 8289  Fincfn 8511  Basecbs 16485   GrpAct cga 18421   pGrp cpgp 18656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-er 8291  df-ec 8293  df-map 8410  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-minusg 18109  df-ga 18422

This theorem is referenced by:  sylow2alem2  18745  sylow2a  18746