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Theorem sylow2alem1 18721
Description: Lemma for sylow2a 18723. 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 3476 . . . . . 6 𝑤 ∈ V
2 simpr 487 . . . . . 6 ((𝜑𝐴𝑍) → 𝐴𝑍)
3 elecg 8310 . . . . . 6 ((𝑤 ∈ V ∧ 𝐴𝑍) → (𝑤 ∈ [𝐴] 𝐴 𝑤))
41, 2, 3sylancr 589 . . . . 5 ((𝜑𝐴𝑍) → (𝑤 ∈ [𝐴] 𝐴 𝑤))
5 sylow2a.r . . . . . . . 8 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
65gaorb 18416 . . . . . . 7 (𝐴 𝑤 ↔ (𝐴𝑌𝑤𝑌 ∧ ∃𝑘𝑋 (𝑘 𝐴) = 𝑤))
76simp3bi 1143 . . . . . 6 (𝐴 𝑤 → ∃𝑘𝑋 (𝑘 𝐴) = 𝑤)
8 oveq2 7141 . . . . . . . . . . . . . 14 (𝑢 = 𝐴 → ( 𝑢) = ( 𝐴))
9 id 22 . . . . . . . . . . . . . 14 (𝑢 = 𝐴𝑢 = 𝐴)
108, 9eqeq12d 2836 . . . . . . . . . . . . 13 (𝑢 = 𝐴 → (( 𝑢) = 𝑢 ↔ ( 𝐴) = 𝐴))
1110ralbidv 3184 . . . . . . . . . . . 12 (𝑢 = 𝐴 → (∀𝑋 ( 𝑢) = 𝑢 ↔ ∀𝑋 ( 𝐴) = 𝐴))
12 sylow2a.z . . . . . . . . . . . 12 𝑍 = {𝑢𝑌 ∣ ∀𝑋 ( 𝑢) = 𝑢}
1311, 12elrab2 3663 . . . . . . . . . . 11 (𝐴𝑍 ↔ (𝐴𝑌 ∧ ∀𝑋 ( 𝐴) = 𝐴))
142, 13sylib 220 . . . . . . . . . 10 ((𝜑𝐴𝑍) → (𝐴𝑌 ∧ ∀𝑋 ( 𝐴) = 𝐴))
1514simprd 498 . . . . . . . . 9 ((𝜑𝐴𝑍) → ∀𝑋 ( 𝐴) = 𝐴)
16 oveq1 7140 . . . . . . . . . . 11 ( = 𝑘 → ( 𝐴) = (𝑘 𝐴))
1716eqeq1d 2822 . . . . . . . . . 10 ( = 𝑘 → (( 𝐴) = 𝐴 ↔ (𝑘 𝐴) = 𝐴))
1817rspccva 3601 . . . . . . . . 9 ((∀𝑋 ( 𝐴) = 𝐴𝑘𝑋) → (𝑘 𝐴) = 𝐴)
1915, 18sylan 582 . . . . . . . 8 (((𝜑𝐴𝑍) ∧ 𝑘𝑋) → (𝑘 𝐴) = 𝐴)
20 eqeq1 2824 . . . . . . . 8 ((𝑘 𝐴) = 𝑤 → ((𝑘 𝐴) = 𝐴𝑤 = 𝐴))
2119, 20syl5ibcom 247 . . . . . . 7 (((𝜑𝐴𝑍) ∧ 𝑘𝑋) → ((𝑘 𝐴) = 𝑤𝑤 = 𝐴))
2221rexlimdva 3271 . . . . . 6 ((𝜑𝐴𝑍) → (∃𝑘𝑋 (𝑘 𝐴) = 𝑤𝑤 = 𝐴))
237, 22syl5 34 . . . . 5 ((𝜑𝐴𝑍) → (𝐴 𝑤𝑤 = 𝐴))
244, 23sylbid 242 . . . 4 ((𝜑𝐴𝑍) → (𝑤 ∈ [𝐴] 𝑤 = 𝐴))
25 velsn 4559 . . . 4 (𝑤 ∈ {𝐴} ↔ 𝑤 = 𝐴)
2624, 25syl6ibr 254 . . 3 ((𝜑𝐴𝑍) → (𝑤 ∈ [𝐴] 𝑤 ∈ {𝐴}))
2726ssrdv 3952 . 2 ((𝜑𝐴𝑍) → [𝐴] ⊆ {𝐴})
28 sylow2a.m . . . . . . 7 (𝜑 ∈ (𝐺 GrpAct 𝑌))
29 sylow2a.x . . . . . . . 8 𝑋 = (Base‘𝐺)
305, 29gaorber 18417 . . . . . . 7 ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)
3128, 30syl 17 . . . . . 6 (𝜑 Er 𝑌)
3231adantr 483 . . . . 5 ((𝜑𝐴𝑍) → Er 𝑌)
3314simpld 497 . . . . 5 ((𝜑𝐴𝑍) → 𝐴𝑌)
3432, 33erref 8287 . . . 4 ((𝜑𝐴𝑍) → 𝐴 𝐴)
35 elecg 8310 . . . . 5 ((𝐴𝑍𝐴𝑍) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
362, 35sylancom 590 . . . 4 ((𝜑𝐴𝑍) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
3734, 36mpbird 259 . . 3 ((𝜑𝐴𝑍) → 𝐴 ∈ [𝐴] )
3837snssd 4718 . 2 ((𝜑𝐴𝑍) → {𝐴} ⊆ [𝐴] )
3927, 38eqssd 3963 1 ((𝜑𝐴𝑍) → [𝐴] = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  wral 3125  wrex 3126  {crab 3129  Vcvv 3473  wss 3913  {csn 4543  {cpr 4545   class class class wbr 5042  {copab 5104  cfv 6331  (class class class)co 7133   Er wer 8264  [cec 8265  Fincfn 8487  Basecbs 16462   GrpAct cga 18398   pGrp cpgp 18633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-er 8267  df-ec 8269  df-map 8386  df-0g 16694  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-grp 18085  df-minusg 18086  df-ga 18399
This theorem is referenced by:  sylow2alem2  18722  sylow2a  18723
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