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Theorem eqgfval 18324
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 3498 . 2 (𝐺𝑉𝐺 ∈ V)
2 eqgval.x . . . 4 𝑋 = (Base‘𝐺)
32fvexi 6672 . . 3 𝑋 ∈ V
43ssex 5211 . 2 (𝑆𝑋𝑆 ∈ V)
5 eqgval.r . . 3 𝑅 = (𝐺 ~QG 𝑆)
6 simpl 486 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑔 = 𝐺)
76fveq2d 6662 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → (Base‘𝑔) = (Base‘𝐺))
87, 2syl6eqr 2877 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → (Base‘𝑔) = 𝑋)
98sseq2d 3984 . . . . . 6 ((𝑔 = 𝐺𝑠 = 𝑆) → ({𝑥, 𝑦} ⊆ (Base‘𝑔) ↔ {𝑥, 𝑦} ⊆ 𝑋))
106fveq2d 6662 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → (+g𝑔) = (+g𝐺))
11 eqgval.p . . . . . . . . 9 + = (+g𝐺)
1210, 11syl6eqr 2877 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → (+g𝑔) = + )
136fveq2d 6662 . . . . . . . . . 10 ((𝑔 = 𝐺𝑠 = 𝑆) → (invg𝑔) = (invg𝐺))
14 eqgval.n . . . . . . . . . 10 𝑁 = (invg𝐺)
1513, 14syl6eqr 2877 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → (invg𝑔) = 𝑁)
1615fveq1d 6660 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → ((invg𝑔)‘𝑥) = (𝑁𝑥))
17 eqidd 2825 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑦 = 𝑦)
1812, 16, 17oveq123d 7166 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → (((invg𝑔)‘𝑥)(+g𝑔)𝑦) = ((𝑁𝑥) + 𝑦))
19 simpr 488 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑠 = 𝑆)
2018, 19eleq12d 2910 . . . . . 6 ((𝑔 = 𝐺𝑠 = 𝑆) → ((((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠 ↔ ((𝑁𝑥) + 𝑦) ∈ 𝑆))
219, 20anbi12d 633 . . . . 5 ((𝑔 = 𝐺𝑠 = 𝑆) → (({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠) ↔ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)))
2221opabbidv 5118 . . . 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 18274 . . . 4 ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠)})
243, 3xpex 7466 . . . . 5 (𝑋 × 𝑋) ∈ V
25 simpl 486 . . . . . . . 8 (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) → {𝑥, 𝑦} ⊆ 𝑋)
26 vex 3483 . . . . . . . . 9 𝑥 ∈ V
27 vex 3483 . . . . . . . . 9 𝑦 ∈ V
2826, 27prss 4737 . . . . . . . 8 ((𝑥𝑋𝑦𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
2925, 28sylibr 237 . . . . . . 7 (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) → (𝑥𝑋𝑦𝑋))
3029ssopab2i 5424 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦𝑋)}
31 df-xp 5548 . . . . . 6 (𝑋 × 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦𝑋)}
3230, 31sseqtrri 3989 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ⊆ (𝑋 × 𝑋)
3324, 32ssexi 5212 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ∈ V
3422, 23, 33ovmpoa 7294 . . 3 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
355, 34syl5eq 2871 . 2 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
361, 4, 35syl2an 598 1 ((𝐺𝑉𝑆𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  wss 3919  {cpr 4551  {copab 5114   × cxp 5540  cfv 6343  (class class class)co 7145  Basecbs 16479  +gcplusg 16561  invgcminusg 18100   ~QG cqg 18271
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150  df-eqg 18274
This theorem is referenced by:  eqgval  18325
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