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Theorem oppgval 17693
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 6150 . . . . . . . 8 (𝑥 = 𝑅 → (+g𝑥) = (+g𝑅))
4 oppgval.2 . . . . . . . 8 + = (+g𝑅)
53, 4syl6eqr 2678 . . . . . . 7 (𝑥 = 𝑅 → (+g𝑥) = + )
65tposeqd 7301 . . . . . 6 (𝑥 = 𝑅 → tpos (+g𝑥) = tpos + )
76opeq2d 4382 . . . . 5 (𝑥 = 𝑅 → ⟨(+g‘ndx), tpos (+g𝑥)⟩ = ⟨(+g‘ndx), tpos + ⟩)
82, 7oveq12d 6623 . . . 4 (𝑥 = 𝑅 → (𝑥 sSet ⟨(+g‘ndx), tpos (+g𝑥)⟩) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
9 df-oppg 17692 . . . 4 oppg = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(+g‘ndx), tpos (+g𝑥)⟩))
10 ovex 6633 . . . 4 (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) ∈ V
118, 9, 10fvmpt 6240 . . 3 (𝑅 ∈ V → (oppg𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
12 fvprc 6144 . . . 4 𝑅 ∈ V → (oppg𝑅) = ∅)
13 reldmsets 15802 . . . . 5 Rel dom sSet
1413ovprc1 6638 . . . 4 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) = ∅)
1512, 14eqtr4d 2663 . . 3 𝑅 ∈ V → (oppg𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
1611, 15pm2.61i 176 . 2 (oppg𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)
171, 16eqtri 2648 1 𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1992  Vcvv 3191  c0 3896  cop 4159  cfv 5850  (class class class)co 6605  tpos ctpos 7297  ndxcnx 15773   sSet csts 15774  +gcplusg 15857  oppgcoppg 17691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-res 5091  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-tpos 7298  df-sets 15782  df-oppg 17692
This theorem is referenced by:  oppgplusfval  17694  oppglem  17696
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