Theorem sylow2alem1 19399

Description: Lemma for sylow2a 19401. 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 3449	. . . . . 6 ⊢ 𝑤 ∈ V
2		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → 𝐴 ∈ 𝑍)
3		elecg 8691	. . . . . 6 ⊢ ((𝑤 ∈ V ∧ 𝐴 ∈ 𝑍) → (𝑤 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝑤))
4	1, 2, 3	sylancr 587	. . . . 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 19087	. . . . . . 7 ⊢ (𝐴 ∼ 𝑤 ↔ (𝐴 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝐴) = 𝑤))
7	6	simp3bi 1147	. . . . . 6 ⊢ (𝐴 ∼ 𝑤 → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝐴) = 𝑤)
8		oveq2 7365	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝐴 → (ℎ ⊕ 𝑢) = (ℎ ⊕ 𝐴))
9		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝐴 → 𝑢 = 𝐴)
10	8, 9	eqeq12d 2752	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝐴 → ((ℎ ⊕ 𝑢) = 𝑢 ↔ (ℎ ⊕ 𝐴) = 𝐴))
11	10	ralbidv 3174	. . . . . . . . . . . 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 3648	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑍 ↔ (𝐴 ∈ 𝑌 ∧ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐴))
14	2, 13	sylib 217	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (𝐴 ∈ 𝑌 ∧ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐴))
15	14	simprd 496	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐴)
16		oveq1 7364	. . . . . . . . . . 11 ⊢ (ℎ = 𝑘 → (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
17	16	eqeq1d 2738	. . . . . . . . . 10 ⊢ (ℎ = 𝑘 → ((ℎ ⊕ 𝐴) = 𝐴 ↔ (𝑘 ⊕ 𝐴) = 𝐴))
18	17	rspccva 3580	. . . . . . . . 9 ⊢ ((∀ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐴 ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) = 𝐴)
19	15, 18	sylan 580	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) = 𝐴)
20		eqeq1 2740	. . . . . . . 8 ⊢ ((𝑘 ⊕ 𝐴) = 𝑤 → ((𝑘 ⊕ 𝐴) = 𝐴 ↔ 𝑤 = 𝐴))
21	19, 20	syl5ibcom 244	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝑘 ⊕ 𝐴) = 𝑤 → 𝑤 = 𝐴))
22	21	rexlimdva 3152	. . . . . 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 239	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (𝑤 ∈ [𝐴] ∼ → 𝑤 = 𝐴))
25		velsn 4602	. . . 4 ⊢ (𝑤 ∈ {𝐴} ↔ 𝑤 = 𝐴)
26	24, 25	syl6ibr 251	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (𝑤 ∈ [𝐴] ∼ → 𝑤 ∈ {𝐴}))
27	26	ssrdv 3950	. 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 19088	. . . . . . 7 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
31	28, 30	syl 17	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝑌)
32	31	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → ∼ Er 𝑌)
33	14	simpld 495	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → 𝐴 ∈ 𝑌)
34	32, 33	erref 8668	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → 𝐴 ∼ 𝐴)
35		elecg 8691	. . . . 5 ⊢ ((𝐴 ∈ 𝑍 ∧ 𝐴 ∈ 𝑍) → (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
36	2, 35	sylancom 588	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
37	34, 36	mpbird 256	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → 𝐴 ∈ [𝐴] ∼ )
38	37	snssd 4769	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → {𝐴} ⊆ [𝐴] ∼ )
39	27, 38	eqssd 3961	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → [𝐴] ∼ = {𝐴})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445   ⊆ wss 3910  {csn 4586  {cpr 4588   class class class wbr 5105  {copab 5167  ‘cfv 6496  (class class class)co 7357   Er wer 8645  [cec 8646  Fincfn 8883  Basecbs 17083   GrpAct cga 19069   pGrp cpgp 19308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-er 8648  df-ec 8650  df-map 8767  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-ga 19070

This theorem is referenced by:  sylow2alem2  19400  sylow2a  19401