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Theorem sylow2alem1 19232
Description: Lemma for sylow2a 19234. 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 3433 . . . . . 6 𝑤 ∈ V
2 simpr 485 . . . . . 6 ((𝜑𝐴𝑍) → 𝐴𝑍)
3 elecg 8528 . . . . . 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 18923 . . . . . . 7 (𝐴 𝑤 ↔ (𝐴𝑌𝑤𝑌 ∧ ∃𝑘𝑋 (𝑘 𝐴) = 𝑤))
76simp3bi 1146 . . . . . 6 (𝐴 𝑤 → ∃𝑘𝑋 (𝑘 𝐴) = 𝑤)
8 oveq2 7275 . . . . . . . . . . . . . 14 (𝑢 = 𝐴 → ( 𝑢) = ( 𝐴))
9 id 22 . . . . . . . . . . . . . 14 (𝑢 = 𝐴𝑢 = 𝐴)
108, 9eqeq12d 2754 . . . . . . . . . . . . 13 (𝑢 = 𝐴 → (( 𝑢) = 𝑢 ↔ ( 𝐴) = 𝐴))
1110ralbidv 3121 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (∀𝑋 ( 𝑢) = 𝑢 ↔ ∀𝑋 ( 𝐴) = 𝐴))
12 sylow2a.z . . . . . . . . . . . 12 𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}
1311, 12elrab2 3626 . . . . . . . . . . 11 (𝐴𝑍 ↔ (𝐴𝑌 ∧ ∀𝑋 ( 𝐴) = 𝐴))
142, 13sylib 217 . . . . . . . . . 10 ((𝜑𝐴𝑍) → (𝐴𝑌 ∧ ∀𝑋 ( 𝐴) = 𝐴))
1514simprd 496 . . . . . . . . 9 ((𝜑𝐴𝑍) → ∀𝑋 ( 𝐴) = 𝐴)
16 oveq1 7274 . . . . . . . . . . 11 ( = 𝑘 → ( 𝐴) = (𝑘 𝐴))
1716eqeq1d 2740 . . . . . . . . . 10 ( = 𝑘 → (( 𝐴) = 𝐴 ↔ (𝑘 𝐴) = 𝐴))
1817rspccva 3558 . . . . . . . . 9 ((∀𝑋 ( 𝐴) = 𝐴𝑘𝑋) → (𝑘 𝐴) = 𝐴)
1915, 18sylan 580 . . . . . . . 8 (((𝜑𝐴𝑍) ∧ 𝑘𝑋) → (𝑘 𝐴) = 𝐴)
20 eqeq1 2742 . . . . . . . 8 ((𝑘 𝐴) = 𝑤 → ((𝑘 𝐴) = 𝐴𝑤 = 𝐴))
2119, 20syl5ibcom 244 . . . . . . 7 (((𝜑𝐴𝑍) ∧ 𝑘𝑋) → ((𝑘 𝐴) = 𝑤𝑤 = 𝐴))
2221rexlimdva 3211 . . . . . 6 ((𝜑𝐴𝑍) → (∃𝑘𝑋 (𝑘 𝐴) = 𝑤𝑤 = 𝐴))
237, 22syl5 34 . . . . 5 ((𝜑𝐴𝑍) → (𝐴 𝑤𝑤 = 𝐴))
244, 23sylbid 239 . . . 4 ((𝜑𝐴𝑍) → (𝑤 ∈ [𝐴] 𝑤 = 𝐴))
25 velsn 4577 . . . 4 (𝑤 ∈ {𝐴} ↔ 𝑤 = 𝐴)
2624, 25syl6ibr 251 . . 3 ((𝜑𝐴𝑍) → (𝑤 ∈ [𝐴] 𝑤 ∈ {𝐴}))
2726ssrdv 3926 . 2 ((𝜑𝐴𝑍) → [𝐴] ⊆ {𝐴})
28 sylow2a.m . . . . . . 7 (𝜑 ∈ (𝐺 GrpAct 𝑌))
29 sylow2a.x . . . . . . . 8 𝑋 = (Base‘𝐺)
305, 29gaorber 18924 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)
3128, 30syl 17 . . . . . 6 (𝜑 Er 𝑌)
3231adantr 481 . . . . 5 ((𝜑𝐴𝑍) → Er 𝑌)
3314simpld 495 . . . . 5 ((𝜑𝐴𝑍) → 𝐴𝑌)
3432, 33erref 8505 . . . 4 ((𝜑𝐴𝑍) → 𝐴 𝐴)
35 elecg 8528 . . . . 5 ((𝐴𝑍𝐴𝑍) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
362, 35sylancom 588 . . . 4 ((𝜑𝐴𝑍) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
3734, 36mpbird 256 . . 3 ((𝜑𝐴𝑍) → 𝐴 ∈ [𝐴] )
3837snssd 4742 . 2 ((𝜑𝐴𝑍) → {𝐴} ⊆ [𝐴] )
3927, 38eqssd 3937 1 ((𝜑𝐴𝑍) → [𝐴] = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064  wrex 3065  {crab 3068  Vcvv 3429  wss 3886  {csn 4561  {cpr 4563   class class class wbr 5073  {copab 5135  cfv 6426  (class class class)co 7267   Er wer 8482  [cec 8483  Fincfn 8720  Basecbs 16922   GrpAct cga 18905   pGrp cpgp 19144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-fv 6434  df-riota 7224  df-ov 7270  df-oprab 7271  df-mpo 7272  df-er 8485  df-ec 8487  df-map 8604  df-0g 17162  df-mgm 18336  df-sgrp 18385  df-mnd 18396  df-grp 18590  df-minusg 18591  df-ga 18906
This theorem is referenced by:  sylow2alem2  19233  sylow2a  19234
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