Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqgfval Structured version   Visualization version   GIF version

Theorem eqgfval 17465
 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 3185 . 2 (𝐺𝑉𝐺 ∈ V)
2 eqgval.x . . . 4 𝑋 = (Base‘𝐺)
3 fvex 6113 . . . 4 (Base‘𝐺) ∈ V
42, 3eqeltri 2684 . . 3 𝑋 ∈ V
54ssex 4730 . 2 (𝑆𝑋𝑆 ∈ V)
6 eqgval.r . . 3 𝑅 = (𝐺 ~QG 𝑆)
7 simpl 472 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑔 = 𝐺)
87fveq2d 6107 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → (Base‘𝑔) = (Base‘𝐺))
98, 2syl6eqr 2662 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → (Base‘𝑔) = 𝑋)
109sseq2d 3596 . . . . . 6 ((𝑔 = 𝐺𝑠 = 𝑆) → ({𝑥, 𝑦} ⊆ (Base‘𝑔) ↔ {𝑥, 𝑦} ⊆ 𝑋))
117fveq2d 6107 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → (+g𝑔) = (+g𝐺))
12 eqgval.p . . . . . . . . 9 + = (+g𝐺)
1311, 12syl6eqr 2662 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → (+g𝑔) = + )
147fveq2d 6107 . . . . . . . . . 10 ((𝑔 = 𝐺𝑠 = 𝑆) → (invg𝑔) = (invg𝐺))
15 eqgval.n . . . . . . . . . 10 𝑁 = (invg𝐺)
1614, 15syl6eqr 2662 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → (invg𝑔) = 𝑁)
1716fveq1d 6105 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → ((invg𝑔)‘𝑥) = (𝑁𝑥))
18 eqidd 2611 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑦 = 𝑦)
1913, 17, 18oveq123d 6570 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → (((invg𝑔)‘𝑥)(+g𝑔)𝑦) = ((𝑁𝑥) + 𝑦))
20 simpr 476 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑠 = 𝑆)
2119, 20eleq12d 2682 . . . . . 6 ((𝑔 = 𝐺𝑠 = 𝑆) → ((((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠 ↔ ((𝑁𝑥) + 𝑦) ∈ 𝑆))
2210, 21anbi12d 743 . . . . 5 ((𝑔 = 𝐺𝑠 = 𝑆) → (({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠) ↔ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)))
2322opabbidv 4648 . . . 4 ((𝑔 = 𝐺𝑠 = 𝑆) → {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
24 df-eqg 17416 . . . 4 ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠)})
254, 4xpex 6860 . . . . 5 (𝑋 × 𝑋) ∈ V
26 simpl 472 . . . . . . . 8 (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) → {𝑥, 𝑦} ⊆ 𝑋)
27 vex 3176 . . . . . . . . 9 𝑥 ∈ V
28 vex 3176 . . . . . . . . 9 𝑦 ∈ V
2927, 28prss 4291 . . . . . . . 8 ((𝑥𝑋𝑦𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
3026, 29sylibr 223 . . . . . . 7 (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) → (𝑥𝑋𝑦𝑋))
3130ssopab2i 4928 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦𝑋)}
32 df-xp 5044 . . . . . 6 (𝑋 × 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦𝑋)}
3331, 32sseqtr4i 3601 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ⊆ (𝑋 × 𝑋)
3425, 33ssexi 4731 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ∈ V
3523, 24, 34ovmpt2a 6689 . . 3 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
366, 35syl5eq 2656 . 2 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
371, 5, 36syl2an 493 1 ((𝐺𝑉𝑆𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  {cpr 4127  {copab 4642   × cxp 5036  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  invgcminusg 17246   ~QG cqg 17413 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-eqg 17416 This theorem is referenced by:  eqgval  17466
 Copyright terms: Public domain W3C validator