Theorem sylow2alem1 19657

Description: Lemma for sylow2a 19659. 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 3458	. . . . . 6 ⊢ 𝑤 ∈ V
2		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → 𝐴 ∈ 𝑍)
3		elecg 8723	. . . . . 6 ⊢ ((𝑤 ∈ V ∧ 𝐴 ∈ 𝑍) → (𝑤 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝑤))
4	1, 2, 3	sylancr 596	. . . . 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 19347	. . . . . . 7 ⊢ (𝐴 ∼ 𝑤 ↔ (𝐴 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌 ∧ ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝐴) = 𝑤))
7	6	simp3bi 1160	. . . . . 6 ⊢ (𝐴 ∼ 𝑤 → ∃𝑘 ∈ 𝑋 (𝑘 ⊕ 𝐴) = 𝑤)
8		oveq2 7404	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝐴 → (ℎ ⊕ 𝑢) = (ℎ ⊕ 𝐴))
9		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝐴 → 𝑢 = 𝐴)
10	8, 9	eqeq12d 2778	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝐴 → ((ℎ ⊕ 𝑢) = 𝑢 ↔ (ℎ ⊕ 𝐴) = 𝐴))
11	10	ralbidv 3185	. . . . . . . . . . . 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 3654	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑍 ↔ (𝐴 ∈ 𝑌 ∧ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐴))
14	13	bilani 508	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (𝐴 ∈ 𝑌 ∧ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐴))
15	14	simprd 499	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐴)
16		oveq1 7403	. . . . . . . . . . 11 ⊢ (ℎ = 𝑘 → (ℎ ⊕ 𝐴) = (𝑘 ⊕ 𝐴))
17	16	eqeq1d 2764	. . . . . . . . . 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 589	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝑘 ⊕ 𝐴) = 𝐴)
20		eqeq1 2766	. . . . . . . 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 3163	. . . . . 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 4598	. . . 4 ⊢ (𝑤 ∈ {𝐴} ↔ 𝑤 = 𝐴)
26	24, 25	imbitrrdi 254	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (𝑤 ∈ [𝐴] ∼ → 𝑤 ∈ {𝐴}))
27	26	ssrdv 3942	. 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 19348	. . . . . . 7 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
31	28, 30	syl 17	. . . . . 6 ⊢ (𝜑 → ∼ Er 𝑌)
32	31	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → ∼ Er 𝑌)
33	14	simpld 498	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → 𝐴 ∈ 𝑌)
34	32, 33	erref 8699	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → 𝐴 ∼ 𝐴)
35		elecg 8723	. . . . 5 ⊢ ((𝐴 ∈ 𝑍 ∧ 𝐴 ∈ 𝑍) → (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
36	2, 35	sylancom 597	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
37	34, 36	mpbird 259	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → 𝐴 ∈ [𝐴] ∼ )
38	37	snssd 4745	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → {𝐴} ⊆ [𝐴] ∼ )
39	27, 38	eqssd 3953	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑍) → [𝐴] ∼ = {𝐴})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  {crab 3414  Vcvv 3454   ⊆ wss 3904  {csn 4582  {cpr 4584   class class class wbr 5100  {copab 5162  ‘cfv 6521  (class class class)co 7396   Er wer 8675  [cec 8676  Fincfn 8927  Basecbs 17245   GrpAct cga 19329   pGrp cpgp 19566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-er 8678  df-ec 8680  df-map 8810  df-0g 17470  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-grp 18978  df-minusg 18979  df-ga 19330

This theorem is referenced by:  sylow2alem2  19658  sylow2a  19659