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Theorem eqgval 19125
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 19124 . . 3 ((𝐺𝑉𝑆𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
65breqd 5153 . 2 ((𝐺𝑉𝑆𝑋) → (𝐴𝑅𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵))
7 brabv 5565 . . . 4 (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
87adantl 481 . . 3 (((𝐺𝑉𝑆𝑋) ∧ 𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9 simpr1 1192 . . . . 5 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐴𝑋)
109elexd 3490 . . . 4 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐴 ∈ V)
11 simpr2 1193 . . . . 5 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐵𝑋)
1211elexd 3490 . . . 4 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐵 ∈ V)
1310, 12jca 511 . . 3 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14 vex 3473 . . . . . . . 8 𝑥 ∈ V
15 vex 3473 . . . . . . . 8 𝑦 ∈ V
1614, 15prss 4819 . . . . . . 7 ((𝑥𝑋𝑦𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
17 eleq1 2816 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑋𝐴𝑋))
18 eleq1 2816 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑋𝐵𝑋))
1917, 18bi2anan9 637 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑋𝑦𝑋) ↔ (𝐴𝑋𝐵𝑋)))
2016, 19bitr3id 285 . . . . . 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 7433 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑁𝑥) + 𝑦) = ((𝑁𝐴) + 𝐵))
2423eleq1d 2813 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (((𝑁𝑥) + 𝑦) ∈ 𝑆 ↔ ((𝑁𝐴) + 𝐵) ∈ 𝑆))
2520, 24anbi12d 630 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) ↔ ((𝐴𝑋𝐵𝑋) ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
26 df-3an 1087 . . . . 5 ((𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆) ↔ ((𝐴𝑋𝐵𝑋) ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆))
2725, 26bitr4di 289 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
28 eqid 2727 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}
2927, 28brabga 5530 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
308, 13, 29pm5.21nd 801 . 2 ((𝐺𝑉𝑆𝑋) → (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
316, 30bitrd 279 1 ((𝐺𝑉𝑆𝑋) → (𝐴𝑅𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  Vcvv 3469  wss 3944  {cpr 4626   class class class wbr 5142  {copab 5204  cfv 6542  (class class class)co 7414  Basecbs 17173  +gcplusg 17226  invgcminusg 18884   ~QG cqg 19070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-eqg 19073
This theorem is referenced by:  eqger  19126  eqglact  19127  eqgid  19128  eqgcpbl  19130  ghmquskerlem1  19227  gastacos  19254  orbstafun  19255  sylow2blem1  19568  sylow2blem3  19570  eqgabl  19782  tgpconncompeqg  24009  tgpconncomp  24010  qustgpopn  24017  qusker  33055  eqgvscpbl  33056  qusxpid  33069  nsgqusf1olem3  33119  qsnzr  33161  qsdrngilem  33195  qsdrnglem2  33197
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