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Theorem eqgval 19056
Description: Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
eqgval.x 𝑋 = (Base‘𝐺)
eqgval.n 𝑁 = (invg𝐺)
eqgval.p + = (+g𝐺)
eqgval.r 𝑅 = (𝐺 ~QG 𝑆)
Assertion
Ref Expression
eqgval ((𝐺𝑉𝑆𝑋) → (𝐴𝑅𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))

Proof of Theorem eqgval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqgval.x . . . 4 𝑋 = (Base‘𝐺)
2 eqgval.n . . . 4 𝑁 = (invg𝐺)
3 eqgval.p . . . 4 + = (+g𝐺)
4 eqgval.r . . . 4 𝑅 = (𝐺 ~QG 𝑆)
51, 2, 3, 4eqgfval 19055 . . 3 ((𝐺𝑉𝑆𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
65breqd 5159 . 2 ((𝐺𝑉𝑆𝑋) → (𝐴𝑅𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵))
7 brabv 5569 . . . 4 (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
87adantl 482 . . 3 (((𝐺𝑉𝑆𝑋) ∧ 𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9 simpr1 1194 . . . . 5 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐴𝑋)
109elexd 3494 . . . 4 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐴 ∈ V)
11 simpr2 1195 . . . . 5 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐵𝑋)
1211elexd 3494 . . . 4 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐵 ∈ V)
1310, 12jca 512 . . 3 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14 vex 3478 . . . . . . . 8 𝑥 ∈ V
15 vex 3478 . . . . . . . 8 𝑦 ∈ V
1614, 15prss 4823 . . . . . . 7 ((𝑥𝑋𝑦𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
17 eleq1 2821 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑋𝐴𝑋))
18 eleq1 2821 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑋𝐵𝑋))
1917, 18bi2anan9 637 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑋𝑦𝑋) ↔ (𝐴𝑋𝐵𝑋)))
2016, 19bitr3id 284 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → ({𝑥, 𝑦} ⊆ 𝑋 ↔ (𝐴𝑋𝐵𝑋)))
21 fveq2 6891 . . . . . . . 8 (𝑥 = 𝐴 → (𝑁𝑥) = (𝑁𝐴))
22 id 22 . . . . . . . 8 (𝑦 = 𝐵𝑦 = 𝐵)
2321, 22oveqan12d 7427 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑁𝑥) + 𝑦) = ((𝑁𝐴) + 𝐵))
2423eleq1d 2818 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (((𝑁𝑥) + 𝑦) ∈ 𝑆 ↔ ((𝑁𝐴) + 𝐵) ∈ 𝑆))
2520, 24anbi12d 631 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) ↔ ((𝐴𝑋𝐵𝑋) ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
26 df-3an 1089 . . . . 5 ((𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆) ↔ ((𝐴𝑋𝐵𝑋) ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆))
2725, 26bitr4di 288 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
28 eqid 2732 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}
2927, 28brabga 5534 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
308, 13, 29pm5.21nd 800 . 2 ((𝐺𝑉𝑆𝑋) → (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
316, 30bitrd 278 1 ((𝐺𝑉𝑆𝑋) → (𝐴𝑅𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3474  wss 3948  {cpr 4630   class class class wbr 5148  {copab 5210  cfv 6543  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  invgcminusg 18819   ~QG cqg 19001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-eqg 19004
This theorem is referenced by:  eqger  19057  eqglact  19058  eqgid  19059  eqgcpbl  19061  gastacos  19173  orbstafun  19174  sylow2blem1  19487  sylow2blem3  19489  eqgabl  19701  tgpconncompeqg  23615  tgpconncomp  23616  qustgpopn  23623  qusker  32459  eqgvscpbl  32460  qusxpid  32470  nsgqusf1olem3  32521  ghmquskerlem1  32523  qsnzr  32569  qsdrngilem  32603  qsdrnglem2  32605
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