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Theorem eqgval 19057
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 19056 . . 3 ((𝐺𝑉𝑆𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
65breqd 5160 . 2 ((𝐺𝑉𝑆𝑋) → (𝐴𝑅𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵))
7 brabv 5570 . . . 4 (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
87adantl 483 . . 3 (((𝐺𝑉𝑆𝑋) ∧ 𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9 simpr1 1195 . . . . 5 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐴𝑋)
109elexd 3495 . . . 4 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐴 ∈ V)
11 simpr2 1196 . . . . 5 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐵𝑋)
1211elexd 3495 . . . 4 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐵 ∈ V)
1310, 12jca 513 . . 3 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14 vex 3479 . . . . . . . 8 𝑥 ∈ V
15 vex 3479 . . . . . . . 8 𝑦 ∈ V
1614, 15prss 4824 . . . . . . 7 ((𝑥𝑋𝑦𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
17 eleq1 2822 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑋𝐴𝑋))
18 eleq1 2822 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑋𝐵𝑋))
1917, 18bi2anan9 638 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑋𝑦𝑋) ↔ (𝐴𝑋𝐵𝑋)))
2016, 19bitr3id 285 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → ({𝑥, 𝑦} ⊆ 𝑋 ↔ (𝐴𝑋𝐵𝑋)))
21 fveq2 6892 . . . . . . . 8 (𝑥 = 𝐴 → (𝑁𝑥) = (𝑁𝐴))
22 id 22 . . . . . . . 8 (𝑦 = 𝐵𝑦 = 𝐵)
2321, 22oveqan12d 7428 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑁𝑥) + 𝑦) = ((𝑁𝐴) + 𝐵))
2423eleq1d 2819 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (((𝑁𝑥) + 𝑦) ∈ 𝑆 ↔ ((𝑁𝐴) + 𝐵) ∈ 𝑆))
2520, 24anbi12d 632 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) ↔ ((𝐴𝑋𝐵𝑋) ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
26 df-3an 1090 . . . . 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 2733 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}
2927, 28brabga 5535 . . 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 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3475  wss 3949  {cpr 4631   class class class wbr 5149  {copab 5211  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  invgcminusg 18820   ~QG cqg 19002
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-eqg 19005
This theorem is referenced by:  eqger  19058  eqglact  19059  eqgid  19060  eqgcpbl  19062  gastacos  19174  orbstafun  19175  sylow2blem1  19488  sylow2blem3  19490  eqgabl  19702  tgpconncompeqg  23616  tgpconncomp  23617  qustgpopn  23624  qusker  32464  eqgvscpbl  32465  qusxpid  32475  nsgqusf1olem3  32526  ghmquskerlem1  32528  qsnzr  32574  qsdrngilem  32608  qsdrnglem2  32610
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