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

Theorem eqgfval 18979
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 3464 . 2 (𝐺𝑉𝐺 ∈ V)
2 eqgval.x . . . 4 𝑋 = (Base‘𝐺)
32fvexi 6857 . . 3 𝑋 ∈ V
43ssex 5279 . 2 (𝑆𝑋𝑆 ∈ V)
5 eqgval.r . . 3 𝑅 = (𝐺 ~QG 𝑆)
6 simpl 484 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑔 = 𝐺)
76fveq2d 6847 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → (Base‘𝑔) = (Base‘𝐺))
87, 2eqtr4di 2795 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → (Base‘𝑔) = 𝑋)
98sseq2d 3977 . . . . . 6 ((𝑔 = 𝐺𝑠 = 𝑆) → ({𝑥, 𝑦} ⊆ (Base‘𝑔) ↔ {𝑥, 𝑦} ⊆ 𝑋))
106fveq2d 6847 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → (+g𝑔) = (+g𝐺))
11 eqgval.p . . . . . . . . 9 + = (+g𝐺)
1210, 11eqtr4di 2795 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → (+g𝑔) = + )
136fveq2d 6847 . . . . . . . . . 10 ((𝑔 = 𝐺𝑠 = 𝑆) → (invg𝑔) = (invg𝐺))
14 eqgval.n . . . . . . . . . 10 𝑁 = (invg𝐺)
1513, 14eqtr4di 2795 . . . . . . . . 9 ((𝑔 = 𝐺𝑠 = 𝑆) → (invg𝑔) = 𝑁)
1615fveq1d 6845 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → ((invg𝑔)‘𝑥) = (𝑁𝑥))
17 eqidd 2738 . . . . . . . 8 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑦 = 𝑦)
1812, 16, 17oveq123d 7379 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → (((invg𝑔)‘𝑥)(+g𝑔)𝑦) = ((𝑁𝑥) + 𝑦))
19 simpr 486 . . . . . . 7 ((𝑔 = 𝐺𝑠 = 𝑆) → 𝑠 = 𝑆)
2018, 19eleq12d 2832 . . . . . 6 ((𝑔 = 𝐺𝑠 = 𝑆) → ((((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠 ↔ ((𝑁𝑥) + 𝑦) ∈ 𝑆))
219, 20anbi12d 632 . . . . 5 ((𝑔 = 𝐺𝑠 = 𝑆) → (({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠) ↔ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)))
2221opabbidv 5172 . . . 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 18928 . . . 4 ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg𝑔)‘𝑥)(+g𝑔)𝑦) ∈ 𝑠)})
243, 3xpex 7688 . . . . 5 (𝑋 × 𝑋) ∈ V
25 simpl 484 . . . . . . . 8 (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) → {𝑥, 𝑦} ⊆ 𝑋)
26 vex 3450 . . . . . . . . 9 𝑥 ∈ V
27 vex 3450 . . . . . . . . 9 𝑦 ∈ V
2826, 27prss 4781 . . . . . . . 8 ((𝑥𝑋𝑦𝑋) ↔ {𝑥, 𝑦} ⊆ 𝑋)
2925, 28sylibr 233 . . . . . . 7 (({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆) → (𝑥𝑋𝑦𝑋))
3029ssopab2i 5508 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦𝑋)}
31 df-xp 5640 . . . . . 6 (𝑋 × 𝑋) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑋𝑦𝑋)}
3230, 31sseqtrri 3982 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ⊆ (𝑋 × 𝑋)
3324, 32ssexi 5280 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)} ∈ V
3422, 23, 33ovmpoa 7511 . . 3 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝐺 ~QG 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
355, 34eqtrid 2789 . 2 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
361, 4, 35syl2an 597 1 ((𝐺𝑉𝑆𝑋) → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑋 ∧ ((𝑁𝑥) + 𝑦) ∈ 𝑆)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3446  wss 3911  {cpr 4589  {copab 5168   × cxp 5632  cfv 6497  (class class class)co 7358  Basecbs 17084  +gcplusg 17134  invgcminusg 18750   ~QG cqg 18925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-eqg 18928
This theorem is referenced by:  eqgval  18980  quslsm  32189
  Copyright terms: Public domain W3C validator