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Theorem eqgval 18082
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 18081 . . 3 ((𝐺𝑉𝑆𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
65breqd 4973 . 2 ((𝐺𝑉𝑆𝑋) → (𝐴𝑅𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵))
7 brabv 5341 . . . 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 1187 . . . . 5 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐴𝑋)
109elexd 3457 . . . 4 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐴 ∈ V)
11 simpr2 1188 . . . . 5 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐵𝑋)
1211elexd 3457 . . . 4 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → 𝐵 ∈ V)
1310, 12jca 512 . . 3 (((𝐺𝑉𝑆𝑋) ∧ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14 vex 3440 . . . . . . . 8 𝑥 ∈ V
15 vex 3440 . . . . . . . 8 𝑦 ∈ V
1614, 15prss 4660 . . . . . . 7 ((𝑥𝑋𝑦𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
17 eleq1 2870 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑋𝐴𝑋))
18 eleq1 2870 . . . . . . . 8 (𝑦 = 𝐵 → (𝑦𝑋𝐵𝑋))
1917, 18bi2anan9 635 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑋𝑦𝑋) ↔ (𝐴𝑋𝐵𝑋)))
2016, 19syl5bbr 286 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → ({𝑥, 𝑦} ⊆ 𝑋 ↔ (𝐴𝑋𝐵𝑋)))
21 fveq2 6538 . . . . . . . 8 (𝑥 = 𝐴 → (𝑁𝑥) = (𝑁𝐴))
22 id 22 . . . . . . . 8 (𝑦 = 𝐵𝑦 = 𝐵)
2321, 22oveqan12d 7035 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑁𝑥) + 𝑦) = ((𝑁𝐴) + 𝐵))
2423eleq1d 2867 . . . . . 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 1082 . . . . 5 ((𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆) ↔ ((𝐴𝑋𝐵𝑋) ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆))
2725, 26syl6bbr 290 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
28 eqid 2795 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}
2927, 28brabga 5311 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
308, 13, 29pm5.21nd 798 . 2 ((𝐺𝑉𝑆𝑋) → (𝐴{⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)}𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
316, 30bitrd 280 1 ((𝐺𝑉𝑆𝑋) → (𝐴𝑅𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ ((𝑁𝐴) + 𝐵) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  Vcvv 3437  wss 3859  {cpr 4474   class class class wbr 4962  {copab 5024  cfv 6225  (class class class)co 7016  Basecbs 16312  +gcplusg 16394  invgcminusg 17862   ~QG cqg 18029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-iota 6189  df-fun 6227  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-eqg 18032
This theorem is referenced by:  eqger  18083  eqglact  18084  eqgid  18085  eqgcpbl  18087  gastacos  18181  orbstafun  18182  sylow2blem1  18475  sylow2blem3  18477  eqgabl  18680  tgpconncompeqg  22403  tgpconncomp  22404  qustgpopn  22411  qusker  30572  eqgvscpbl  30573
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