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Theorem sylow2alem1 19250
Description: Lemma for sylow2a 19252. 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 3438 . . . . . 6 𝑤 ∈ V
2 simpr 484 . . . . . 6 ((𝜑𝐴𝑍) → 𝐴𝑍)
3 elecg 8561 . . . . . 6 ((𝑤 ∈ V ∧ 𝐴𝑍) → (𝑤 ∈ [𝐴] 𝐴 𝑤))
41, 2, 3sylancr 586 . . . . 5 ((𝜑𝐴𝑍) → (𝑤 ∈ [𝐴] 𝐴 𝑤))
5 sylow2a.r . . . . . . . 8 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
65gaorb 18941 . . . . . . 7 (𝐴 𝑤 ↔ (𝐴𝑌𝑤𝑌 ∧ ∃𝑘𝑋 (𝑘 𝐴) = 𝑤))
76simp3bi 1145 . . . . . 6 (𝐴 𝑤 → ∃𝑘𝑋 (𝑘 𝐴) = 𝑤)
8 oveq2 7303 . . . . . . . . . . . . . 14 (𝑢 = 𝐴 → ( 𝑢) = ( 𝐴))
9 id 22 . . . . . . . . . . . . . 14 (𝑢 = 𝐴𝑢 = 𝐴)
108, 9eqeq12d 2749 . . . . . . . . . . . . 13 (𝑢 = 𝐴 → (( 𝑢) = 𝑢 ↔ ( 𝐴) = 𝐴))
1110ralbidv 3168 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (∀𝑋 ( 𝑢) = 𝑢 ↔ ∀𝑋 ( 𝐴) = 𝐴))
12 sylow2a.z . . . . . . . . . . . 12 𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}
1311, 12elrab2 3629 . . . . . . . . . . 11 (𝐴𝑍 ↔ (𝐴𝑌 ∧ ∀𝑋 ( 𝐴) = 𝐴))
142, 13sylib 217 . . . . . . . . . 10 ((𝜑𝐴𝑍) → (𝐴𝑌 ∧ ∀𝑋 ( 𝐴) = 𝐴))
1514simprd 495 . . . . . . . . 9 ((𝜑𝐴𝑍) → ∀𝑋 ( 𝐴) = 𝐴)
16 oveq1 7302 . . . . . . . . . . 11 ( = 𝑘 → ( 𝐴) = (𝑘 𝐴))
1716eqeq1d 2735 . . . . . . . . . 10 ( = 𝑘 → (( 𝐴) = 𝐴 ↔ (𝑘 𝐴) = 𝐴))
1817rspccva 3562 . . . . . . . . 9 ((∀𝑋 ( 𝐴) = 𝐴𝑘𝑋) → (𝑘 𝐴) = 𝐴)
1915, 18sylan 579 . . . . . . . 8 (((𝜑𝐴𝑍) ∧ 𝑘𝑋) → (𝑘 𝐴) = 𝐴)
20 eqeq1 2737 . . . . . . . 8 ((𝑘 𝐴) = 𝑤 → ((𝑘 𝐴) = 𝐴𝑤 = 𝐴))
2119, 20syl5ibcom 244 . . . . . . 7 (((𝜑𝐴𝑍) ∧ 𝑘𝑋) → ((𝑘 𝐴) = 𝑤𝑤 = 𝐴))
2221rexlimdva 3146 . . . . . 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 4580 . . . 4 (𝑤 ∈ {𝐴} ↔ 𝑤 = 𝐴)
2624, 25syl6ibr 251 . . 3 ((𝜑𝐴𝑍) → (𝑤 ∈ [𝐴] 𝑤 ∈ {𝐴}))
2726ssrdv 3929 . 2 ((𝜑𝐴𝑍) → [𝐴] ⊆ {𝐴})
28 sylow2a.m . . . . . . 7 (𝜑 ∈ (𝐺 GrpAct 𝑌))
29 sylow2a.x . . . . . . . 8 𝑋 = (Base‘𝐺)
305, 29gaorber 18942 . . . . . . 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 8538 . . . 4 ((𝜑𝐴𝑍) → 𝐴 𝐴)
35 elecg 8561 . . . . 5 ((𝐴𝑍𝐴𝑍) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
362, 35sylancom 587 . . . 4 ((𝜑𝐴𝑍) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
3734, 36mpbird 256 . . 3 ((𝜑𝐴𝑍) → 𝐴 ∈ [𝐴] )
3837snssd 4745 . 2 ((𝜑𝐴𝑍) → {𝐴} ⊆ [𝐴] )
3927, 38eqssd 3940 1 ((𝜑𝐴𝑍) → [𝐴] = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1537  wcel 2101  wral 3059  wrex 3068  {crab 3221  Vcvv 3434  wss 3889  {csn 4564  {cpr 4566   class class class wbr 5077  {copab 5139  cfv 6447  (class class class)co 7295   Er wer 8515  [cec 8516  Fincfn 8753  Basecbs 16940   GrpAct cga 18923   pGrp cpgp 19162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3222  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-fv 6455  df-riota 7252  df-ov 7298  df-oprab 7299  df-mpo 7300  df-er 8518  df-ec 8520  df-map 8637  df-0g 17180  df-mgm 18354  df-sgrp 18403  df-mnd 18414  df-grp 18608  df-minusg 18609  df-ga 18924
This theorem is referenced by:  sylow2alem2  19251  sylow2a  19252
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