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Theorem eqgfval 18026
 Description: Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-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
eqgfval ((𝐺𝑉𝑆𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦   𝑥, + ,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑅(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem eqgfval
Dummy variables 𝑔 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3414 . 2 (𝐺𝑉𝐺 ∈ V)
2 eqgval.x . . . 4 𝑋 = (Base‘𝐺)
32fvexi 6460 . . 3 𝑋 ∈ V
43ssex 5039 . 2 (𝑆𝑋𝑆 ∈ V)
5 eqgval.r . . 3 𝑅 = (𝐺 ~QG 𝑆)
6 simpl 476 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑔 = 𝐺)
76fveq2d 6450 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → (Base‘𝑔) = (Base‘𝐺))
87, 2syl6eqr 2832 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → (Base‘𝑔) = 𝑋)
98sseq2d 3852 . . . . . 6 ((𝑔 = 𝐺𝑠 = 𝑆) → ({𝑥, 𝑦} ⊆ (Base‘𝑔) ↔ {𝑥, 𝑦} ⊆ 𝑋))
106fveq2d 6450 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → (+g𝑔) = (+g𝐺))
11 eqgval.p . . . . . . . . 9 + = (+g𝐺)
1210, 11syl6eqr 2832 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → (+g𝑔) = + )
136fveq2d 6450 . . . . . . . . . 10 ((𝑔 = 𝐺𝑠 = 𝑆) → (invg𝑔) = (invg𝐺))
14 eqgval.n . . . . . . . . . 10 𝑁 = (invg𝐺)
1513, 14syl6eqr 2832 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → (invg𝑔) = 𝑁)
1615fveq1d 6448 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → ((invg𝑔)‘𝑥) = (𝑁𝑥))
17 eqidd 2779 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑦 = 𝑦)
1812, 16, 17oveq123d 6943 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → (((invg𝑔)‘𝑥)(+g𝑔)𝑦) = ((𝑁𝑥) + 𝑦))
19 simpr 479 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑠 = 𝑆)
2018, 19eleq12d 2853 . . . . . 6 ((𝑔 = 𝐺𝑠 = 𝑆) → ((((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠 ↔ ((𝑁𝑥) + 𝑦) ∈ 𝑆))
219, 20anbi12d 624 . . . . 5 ((𝑔 = 𝐺𝑠 = 𝑆) → (({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠) ↔ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)))
2221opabbidv 4952 . . . 4 ((𝑔 = 𝐺𝑠 = 𝑆) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
23 df-eqg 17977 . . . 4 ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠)})
243, 3xpex 7240 . . . . 5 (𝑋 × 𝑋) ∈ V
25 simpl 476 . . . . . . . 8 (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) → {𝑥, 𝑦} ⊆ 𝑋)
26 vex 3401 . . . . . . . . 9 𝑥 ∈ V
27 vex 3401 . . . . . . . . 9 𝑦 ∈ V
2826, 27prss 4582 . . . . . . . 8 ((𝑥𝑋𝑦𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
2925, 28sylibr 226 . . . . . . 7 (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) → (𝑥𝑋𝑦𝑋))
3029ssopab2i 5240 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦𝑋)}
31 df-xp 5361 . . . . . 6 (𝑋 × 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦𝑋)}
3230, 31sseqtr4i 3857 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ⊆ (𝑋 × 𝑋)
3324, 32ssexi 5040 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ∈ V
3422, 23, 33ovmpt2a 7068 . . 3 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
355, 34syl5eq 2826 . 2 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
361, 4, 35syl2an 589 1 ((𝐺𝑉𝑆𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ⊆ wss 3792  {cpr 4400  {copab 4948   × cxp 5353  ‘cfv 6135  (class class class)co 6922  Basecbs 16255  +gcplusg 16338  invgcminusg 17810   ~QG cqg 17974 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-eqg 17977 This theorem is referenced by:  eqgval  18027
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