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Theorem oppgval 19365
Description: Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
Hypotheses
Ref Expression
oppgval.2 + = (+g𝑅)
oppgval.3 𝑂 = (oppg𝑅)
Assertion
Ref Expression
oppgval 𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)

Proof of Theorem oppgval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oppgval.3 . 2 𝑂 = (oppg𝑅)
2 id 22 . . . . 5 (𝑥 = 𝑅𝑥 = 𝑅)
3 fveq2 6906 . . . . . . . 8 (𝑥 = 𝑅 → (+g𝑥) = (+g𝑅))
4 oppgval.2 . . . . . . . 8 + = (+g𝑅)
53, 4eqtr4di 2795 . . . . . . 7 (𝑥 = 𝑅 → (+g𝑥) = + )
65tposeqd 8254 . . . . . 6 (𝑥 = 𝑅 → tpos (+g𝑥) = tpos + )
76opeq2d 4880 . . . . 5 (𝑥 = 𝑅 → ⟨(+g‘ndx), tpos (+g𝑥)⟩ = ⟨(+g‘ndx), tpos + ⟩)
82, 7oveq12d 7449 . . . 4 (𝑥 = 𝑅 → (𝑥 sSet ⟨(+g‘ndx), tpos (+g𝑥)⟩) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
9 df-oppg 19364 . . . 4 oppg = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(+g‘ndx), tpos (+g𝑥)⟩))
10 ovex 7464 . . . 4 (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) ∈ V
118, 9, 10fvmpt 7016 . . 3 (𝑅 ∈ V → (oppg𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
12 fvprc 6898 . . . 4 𝑅 ∈ V → (oppg𝑅) = ∅)
13 reldmsets 17202 . . . . 5 Rel dom sSet
1413ovprc1 7470 . . . 4 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) = ∅)
1512, 14eqtr4d 2780 . . 3 𝑅 ∈ V → (oppg𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
1611, 15pm2.61i 182 . 2 (oppg𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)
171, 16eqtri 2765 1 𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cop 4632  cfv 6561  (class class class)co 7431  tpos ctpos 8250   sSet csts 17200  ndxcnx 17230  +gcplusg 17297  oppgcoppg 19363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-tpos 8251  df-sets 17201  df-oppg 19364
This theorem is referenced by:  oppgplusfval  19366  oppglemOLD  19369  oppgbas  19370  oppgtset  19371  oppgle  32951
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