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Theorem sylow2alem1 18078
Description: Lemma for sylow2a 18080. 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 3234 . . . . . 6 𝑤 ∈ V
2 simpr 476 . . . . . 6 ((𝜑𝐴𝑍) → 𝐴𝑍)
3 elecg 7828 . . . . . 6 ((𝑤 ∈ V ∧ 𝐴𝑍) → (𝑤 ∈ [𝐴] 𝐴 𝑤))
41, 2, 3sylancr 696 . . . . 5 ((𝜑𝐴𝑍) → (𝑤 ∈ [𝐴] 𝐴 𝑤))
5 sylow2a.r . . . . . . . 8 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
65gaorb 17786 . . . . . . 7 (𝐴 𝑤 ↔ (𝐴𝑌𝑤𝑌 ∧ ∃𝑘𝑋 (𝑘 𝐴) = 𝑤))
76simp3bi 1098 . . . . . 6 (𝐴 𝑤 → ∃𝑘𝑋 (𝑘 𝐴) = 𝑤)
8 oveq2 6698 . . . . . . . . . . . . . 14 (𝑢 = 𝐴 → ( 𝑢) = ( 𝐴))
9 id 22 . . . . . . . . . . . . . 14 (𝑢 = 𝐴𝑢 = 𝐴)
108, 9eqeq12d 2666 . . . . . . . . . . . . 13 (𝑢 = 𝐴 → (( 𝑢) = 𝑢 ↔ ( 𝐴) = 𝐴))
1110ralbidv 3015 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (∀𝑋 ( 𝑢) = 𝑢 ↔ ∀𝑋 ( 𝐴) = 𝐴))
12 sylow2a.z . . . . . . . . . . . 12 𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}
1311, 12elrab2 3399 . . . . . . . . . . 11 (𝐴𝑍 ↔ (𝐴𝑌 ∧ ∀𝑋 ( 𝐴) = 𝐴))
142, 13sylib 208 . . . . . . . . . 10 ((𝜑𝐴𝑍) → (𝐴𝑌 ∧ ∀𝑋 ( 𝐴) = 𝐴))
1514simprd 478 . . . . . . . . 9 ((𝜑𝐴𝑍) → ∀𝑋 ( 𝐴) = 𝐴)
16 oveq1 6697 . . . . . . . . . . 11 ( = 𝑘 → ( 𝐴) = (𝑘 𝐴))
1716eqeq1d 2653 . . . . . . . . . 10 ( = 𝑘 → (( 𝐴) = 𝐴 ↔ (𝑘 𝐴) = 𝐴))
1817rspccva 3339 . . . . . . . . 9 ((∀𝑋 ( 𝐴) = 𝐴𝑘𝑋) → (𝑘 𝐴) = 𝐴)
1915, 18sylan 487 . . . . . . . 8 (((𝜑𝐴𝑍) ∧ 𝑘𝑋) → (𝑘 𝐴) = 𝐴)
20 eqeq1 2655 . . . . . . . 8 ((𝑘 𝐴) = 𝑤 → ((𝑘 𝐴) = 𝐴𝑤 = 𝐴))
2119, 20syl5ibcom 235 . . . . . . 7 (((𝜑𝐴𝑍) ∧ 𝑘𝑋) → ((𝑘 𝐴) = 𝑤𝑤 = 𝐴))
2221rexlimdva 3060 . . . . . 6 ((𝜑𝐴𝑍) → (∃𝑘𝑋 (𝑘 𝐴) = 𝑤𝑤 = 𝐴))
237, 22syl5 34 . . . . 5 ((𝜑𝐴𝑍) → (𝐴 𝑤𝑤 = 𝐴))
244, 23sylbid 230 . . . 4 ((𝜑𝐴𝑍) → (𝑤 ∈ [𝐴] 𝑤 = 𝐴))
25 velsn 4226 . . . 4 (𝑤 ∈ {𝐴} ↔ 𝑤 = 𝐴)
2624, 25syl6ibr 242 . . 3 ((𝜑𝐴𝑍) → (𝑤 ∈ [𝐴] 𝑤 ∈ {𝐴}))
2726ssrdv 3642 . 2 ((𝜑𝐴𝑍) → [𝐴] ⊆ {𝐴})
28 sylow2a.m . . . . . . 7 (𝜑 ∈ (𝐺 GrpAct 𝑌))
29 sylow2a.x . . . . . . . 8 𝑋 = (Base‘𝐺)
305, 29gaorber 17787 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)
3128, 30syl 17 . . . . . 6 (𝜑 Er 𝑌)
3231adantr 480 . . . . 5 ((𝜑𝐴𝑍) → Er 𝑌)
3314simpld 474 . . . . 5 ((𝜑𝐴𝑍) → 𝐴𝑌)
3432, 33erref 7807 . . . 4 ((𝜑𝐴𝑍) → 𝐴 𝐴)
35 elecg 7828 . . . . 5 ((𝐴𝑍𝐴𝑍) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
362, 35sylancom 702 . . . 4 ((𝜑𝐴𝑍) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
3734, 36mpbird 247 . . 3 ((𝜑𝐴𝑍) → 𝐴 ∈ [𝐴] )
3837snssd 4372 . 2 ((𝜑𝐴𝑍) → {𝐴} ⊆ [𝐴] )
3927, 38eqssd 3653 1 ((𝜑𝐴𝑍) → [𝐴] = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  {csn 4210  {cpr 4212   class class class wbr 4685  {copab 4745  cfv 5926  (class class class)co 6690   Er wer 7784  [cec 7785  Fincfn 7997  Basecbs 15904   GrpAct cga 17768   pGrp cpgp 17992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-ec 7789  df-map 7901  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-ga 17769
This theorem is referenced by:  sylow2alem2  18079  sylow2a  18080
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