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Theorem oppgval 17984
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 6333 . . . . . . . 8 (𝑥 = 𝑅 → (+g𝑥) = (+g𝑅))
4 oppgval.2 . . . . . . . 8 + = (+g𝑅)
53, 4syl6eqr 2823 . . . . . . 7 (𝑥 = 𝑅 → (+g𝑥) = + )
65tposeqd 7511 . . . . . 6 (𝑥 = 𝑅 → tpos (+g𝑥) = tpos + )
76opeq2d 4547 . . . . 5 (𝑥 = 𝑅 → ⟨(+g‘ndx), tpos (+g𝑥)⟩ = ⟨(+g‘ndx), tpos + ⟩)
82, 7oveq12d 6814 . . . 4 (𝑥 = 𝑅 → (𝑥 sSet ⟨(+g‘ndx), tpos (+g𝑥)⟩) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
9 df-oppg 17983 . . . 4 oppg = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(+g‘ndx), tpos (+g𝑥)⟩))
10 ovex 6827 . . . 4 (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) ∈ V
118, 9, 10fvmpt 6426 . . 3 (𝑅 ∈ V → (oppg𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
12 fvprc 6327 . . . 4 𝑅 ∈ V → (oppg𝑅) = ∅)
13 reldmsets 16093 . . . . 5 Rel dom sSet
1413ovprc1 6833 . . . 4 𝑅 ∈ V → (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩) = ∅)
1512, 14eqtr4d 2808 . . 3 𝑅 ∈ V → (oppg𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩))
1611, 15pm2.61i 176 . 2 (oppg𝑅) = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)
171, 16eqtri 2793 1 𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1631  wcel 2145  Vcvv 3351  c0 4063  cop 4323  cfv 6030  (class class class)co 6796  tpos ctpos 7507  ndxcnx 16061   sSet csts 16062  +gcplusg 16149  oppgcoppg 17982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-res 5262  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-tpos 7508  df-sets 16071  df-oppg 17983
This theorem is referenced by:  oppgplusfval  17985  oppglem  17987
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