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Theorem sylow2alem1 19635
Description: Lemma for sylow2a 19637. 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 (𝜑𝑃 pGrp 𝐺)
sylow2a.f (𝜑𝑋 ∈ Fin)
sylow2a.y (𝜑𝑌 ∈ Fin)
sylow2a.z 𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}
sylow2a.r = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
Assertion
Ref Expression
sylow2alem1 ((𝜑𝐴𝑍) → [𝐴] = {𝐴})
Distinct variable groups:   ,   𝑔,,𝑢,𝑥,𝑦,𝐴   𝑔,𝐺,𝑥,𝑦   ,𝑔,,𝑢,𝑥,𝑦   𝑔,𝑋,,𝑢,𝑥,𝑦   𝜑,   𝑔,𝑌,,𝑢,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑢,𝑔)   𝑃(𝑥,𝑦,𝑢,𝑔,)   (𝑥,𝑦,𝑢,𝑔)   𝐺(𝑢,)   𝑍(𝑥,𝑦,𝑢,𝑔,)

Proof of Theorem sylow2alem1
Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3484 . . . . . 6 𝑤 ∈ V
2 simpr 484 . . . . . 6 ((𝜑𝐴𝑍) → 𝐴𝑍)
3 elecg 8789 . . . . . 6 ((𝑤 ∈ V ∧ 𝐴𝑍) → (𝑤 ∈ [𝐴] 𝐴 𝑤))
41, 2, 3sylancr 587 . . . . 5 ((𝜑𝐴𝑍) → (𝑤 ∈ [𝐴] 𝐴 𝑤))
5 sylow2a.r . . . . . . . 8 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
65gaorb 19325 . . . . . . 7 (𝐴 𝑤 ↔ (𝐴𝑌𝑤𝑌 ∧ ∃𝑘𝑋 (𝑘 𝐴) = 𝑤))
76simp3bi 1148 . . . . . 6 (𝐴 𝑤 → ∃𝑘𝑋 (𝑘 𝐴) = 𝑤)
8 oveq2 7439 . . . . . . . . . . . . . 14 (𝑢 = 𝐴 → ( 𝑢) = ( 𝐴))
9 id 22 . . . . . . . . . . . . . 14 (𝑢 = 𝐴𝑢 = 𝐴)
108, 9eqeq12d 2753 . . . . . . . . . . . . 13 (𝑢 = 𝐴 → (( 𝑢) = 𝑢 ↔ ( 𝐴) = 𝐴))
1110ralbidv 3178 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (∀𝑋 ( 𝑢) = 𝑢 ↔ ∀𝑋 ( 𝐴) = 𝐴))
12 sylow2a.z . . . . . . . . . . . 12 𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}
1311, 12elrab2 3695 . . . . . . . . . . 11 (𝐴𝑍 ↔ (𝐴𝑌 ∧ ∀𝑋 ( 𝐴) = 𝐴))
142, 13sylib 218 . . . . . . . . . 10 ((𝜑𝐴𝑍) → (𝐴𝑌 ∧ ∀𝑋 ( 𝐴) = 𝐴))
1514simprd 495 . . . . . . . . 9 ((𝜑𝐴𝑍) → ∀𝑋 ( 𝐴) = 𝐴)
16 oveq1 7438 . . . . . . . . . . 11 ( = 𝑘 → ( 𝐴) = (𝑘 𝐴))
1716eqeq1d 2739 . . . . . . . . . 10 ( = 𝑘 → (( 𝐴) = 𝐴 ↔ (𝑘 𝐴) = 𝐴))
1817rspccva 3621 . . . . . . . . 9 ((∀𝑋 ( 𝐴) = 𝐴𝑘𝑋) → (𝑘 𝐴) = 𝐴)
1915, 18sylan 580 . . . . . . . 8 (((𝜑𝐴𝑍) ∧ 𝑘𝑋) → (𝑘 𝐴) = 𝐴)
20 eqeq1 2741 . . . . . . . 8 ((𝑘 𝐴) = 𝑤 → ((𝑘 𝐴) = 𝐴𝑤 = 𝐴))
2119, 20syl5ibcom 245 . . . . . . 7 (((𝜑𝐴𝑍) ∧ 𝑘𝑋) → ((𝑘 𝐴) = 𝑤𝑤 = 𝐴))
2221rexlimdva 3155 . . . . . 6 ((𝜑𝐴𝑍) → (∃𝑘𝑋 (𝑘 𝐴) = 𝑤𝑤 = 𝐴))
237, 22syl5 34 . . . . 5 ((𝜑𝐴𝑍) → (𝐴 𝑤𝑤 = 𝐴))
244, 23sylbid 240 . . . 4 ((𝜑𝐴𝑍) → (𝑤 ∈ [𝐴] 𝑤 = 𝐴))
25 velsn 4642 . . . 4 (𝑤 ∈ {𝐴} ↔ 𝑤 = 𝐴)
2624, 25imbitrrdi 252 . . 3 ((𝜑𝐴𝑍) → (𝑤 ∈ [𝐴] 𝑤 ∈ {𝐴}))
2726ssrdv 3989 . 2 ((𝜑𝐴𝑍) → [𝐴] ⊆ {𝐴})
28 sylow2a.m . . . . . . 7 (𝜑 ∈ (𝐺 GrpAct 𝑌))
29 sylow2a.x . . . . . . . 8 𝑋 = (Base‘𝐺)
305, 29gaorber 19326 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)
3128, 30syl 17 . . . . . 6 (𝜑 Er 𝑌)
3231adantr 480 . . . . 5 ((𝜑𝐴𝑍) → Er 𝑌)
3314simpld 494 . . . . 5 ((𝜑𝐴𝑍) → 𝐴𝑌)
3432, 33erref 8765 . . . 4 ((𝜑𝐴𝑍) → 𝐴 𝐴)
35 elecg 8789 . . . . 5 ((𝐴𝑍𝐴𝑍) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
362, 35sylancom 588 . . . 4 ((𝜑𝐴𝑍) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
3734, 36mpbird 257 . . 3 ((𝜑𝐴𝑍) → 𝐴 ∈ [𝐴] )
3837snssd 4809 . 2 ((𝜑𝐴𝑍) → {𝐴} ⊆ [𝐴] )
3927, 38eqssd 4001 1 ((𝜑𝐴𝑍) → [𝐴] = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  {csn 4626  {cpr 4628   class class class wbr 5143  {copab 5205  cfv 6561  (class class class)co 7431   Er wer 8742  [cec 8743  Fincfn 8985  Basecbs 17247   GrpAct cga 19307   pGrp cpgp 19544
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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-ec 8747  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-ga 19308
This theorem is referenced by:  sylow2alem2  19636  sylow2a  19637
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