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

Theorem eqgfval 17689
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 3243 . 2 (𝐺𝑉𝐺 ∈ V)
2 eqgval.x . . . 4 𝑋 = (Base‘𝐺)
3 fvex 6239 . . . 4 (Base‘𝐺) ∈ V
42, 3eqeltri 2726 . . 3 𝑋 ∈ V
54ssex 4835 . 2 (𝑆𝑋𝑆 ∈ V)
6 eqgval.r . . 3 𝑅 = (𝐺 ~QG 𝑆)
7 simpl 472 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑔 = 𝐺)
87fveq2d 6233 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → (Base‘𝑔) = (Base‘𝐺))
98, 2syl6eqr 2703 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → (Base‘𝑔) = 𝑋)
109sseq2d 3666 . . . . . 6 ((𝑔 = 𝐺𝑠 = 𝑆) → ({𝑥, 𝑦} ⊆ (Base‘𝑔) ↔ {𝑥, 𝑦} ⊆ 𝑋))
117fveq2d 6233 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → (+g𝑔) = (+g𝐺))
12 eqgval.p . . . . . . . . 9 + = (+g𝐺)
1311, 12syl6eqr 2703 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → (+g𝑔) = + )
147fveq2d 6233 . . . . . . . . . 10 ((𝑔 = 𝐺𝑠 = 𝑆) → (invg𝑔) = (invg𝐺))
15 eqgval.n . . . . . . . . . 10 𝑁 = (invg𝐺)
1614, 15syl6eqr 2703 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → (invg𝑔) = 𝑁)
1716fveq1d 6231 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → ((invg𝑔)‘𝑥) = (𝑁𝑥))
18 eqidd 2652 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑦 = 𝑦)
1913, 17, 18oveq123d 6711 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → (((invg𝑔)‘𝑥)(+g𝑔)𝑦) = ((𝑁𝑥) + 𝑦))
20 simpr 476 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑠 = 𝑆)
2119, 20eleq12d 2724 . . . . . 6 ((𝑔 = 𝐺𝑠 = 𝑆) → ((((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠 ↔ ((𝑁𝑥) + 𝑦) ∈ 𝑆))
2210, 21anbi12d 747 . . . . 5 ((𝑔 = 𝐺𝑠 = 𝑆) → (({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠) ↔ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)))
2322opabbidv 4749 . . . 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 17640 . . . 4 ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠)})
254, 4xpex 7004 . . . . 5 (𝑋 × 𝑋) ∈ V
26 simpl 472 . . . . . . . 8 (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) → {𝑥, 𝑦} ⊆ 𝑋)
27 vex 3234 . . . . . . . . 9 𝑥 ∈ V
28 vex 3234 . . . . . . . . 9 𝑦 ∈ V
2927, 28prss 4383 . . . . . . . 8 ((𝑥𝑋𝑦𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
3026, 29sylibr 224 . . . . . . 7 (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) → (𝑥𝑋𝑦𝑋))
3130ssopab2i 5032 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦𝑋)}
32 df-xp 5149 . . . . . 6 (𝑋 × 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦𝑋)}
3331, 32sseqtr4i 3671 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ⊆ (𝑋 × 𝑋)
3425, 33ssexi 4836 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ∈ V
3523, 24, 34ovmpt2a 6833 . . 3 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
366, 35syl5eq 2697 . 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 1523  wcel 2030  Vcvv 3231  wss 3607  {cpr 4212  {copab 4745   × cxp 5141  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  invgcminusg 17470   ~QG cqg 17637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-eqg 17640
This theorem is referenced by:  eqgval  17690
  Copyright terms: Public domain W3C validator