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Theorem oppgval 18453
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 6643 . . . . . . . 8 (𝑥 = 𝑅 → (+g𝑥) = (+g𝑅))
4 oppgval.2 . . . . . . . 8 + = (+g𝑅)
53, 4syl6eqr 2874 . . . . . . 7 (𝑥 = 𝑅 → (+g𝑥) = + )
65tposeqd 7870 . . . . . 6 (𝑥 = 𝑅 → tpos (+g𝑥) = tpos + )
76opeq2d 4783 . . . . 5 (𝑥 = 𝑅 → ⟨(+g‘ndx), tpos (+g𝑥)⟩ = ⟨(+g‘ndx), tpos + ⟩)
82, 7oveq12d 7148 . . . 4 (𝑥 = 𝑅 → (𝑥 sSet ⟨(+g‘ndx), tpos (+g𝑥)⟩) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
9 df-oppg 18452 . . . 4 oppg = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(+g‘ndx), tpos (+g𝑥)⟩))
10 ovex 7163 . . . 4 (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) ∈ V
118, 9, 10fvmpt 6741 . . 3 (𝑅 ∈ V → (oppg𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
12 fvprc 6636 . . . 4 𝑅 ∈ V → (oppg𝑅) = ∅)
13 reldmsets 16489 . . . . 5 Rel dom sSet
1413ovprc1 7169 . . . 4 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) = ∅)
1512, 14eqtr4d 2859 . . 3 𝑅 ∈ V → (oppg𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
1611, 15pm2.61i 185 . 2 (oppg𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)
171, 16eqtri 2844 1 𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2115  Vcvv 3471  c0 4266  cop 4546  cfv 6328  (class class class)co 7130  tpos ctpos 7866  ndxcnx 16458   sSet csts 16459  +gcplusg 16543  oppgcoppg 18451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-res 5540  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-tpos 7867  df-sets 16468  df-oppg 18452
This theorem is referenced by:  oppgplusfval  18454  oppglem  18456
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