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Theorem opprval 20002
Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
opprval.1 𝐵 = (Base‘𝑅)
opprval.2 · = (.r𝑅)
opprval.3 𝑂 = (oppr𝑅)
Assertion
Ref Expression
opprval 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)

Proof of Theorem opprval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 opprval.3 . 2 𝑂 = (oppr𝑅)
2 id 22 . . . . 5 (𝑥 = 𝑅𝑥 = 𝑅)
3 fveq2 6839 . . . . . . . 8 (𝑥 = 𝑅 → (.r𝑥) = (.r𝑅))
4 opprval.2 . . . . . . . 8 · = (.r𝑅)
53, 4eqtr4di 2795 . . . . . . 7 (𝑥 = 𝑅 → (.r𝑥) = · )
65tposeqd 8152 . . . . . 6 (𝑥 = 𝑅 → tpos (.r𝑥) = tpos · )
76opeq2d 4835 . . . . 5 (𝑥 = 𝑅 → ⟨(.r‘ndx), tpos (.r𝑥)⟩ = ⟨(.r‘ndx), tpos · ⟩)
82, 7oveq12d 7369 . . . 4 (𝑥 = 𝑅 → (𝑥 sSet ⟨(.r‘ndx), tpos (.r𝑥)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
9 df-oppr 20001 . . . 4 oppr = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(.r‘ndx), tpos (.r𝑥)⟩))
10 ovex 7384 . . . 4 (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V
118, 9, 10fvmpt 6945 . . 3 (𝑅 ∈ V → (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
12 fvprc 6831 . . . 4 𝑅 ∈ V → (oppr𝑅) = ∅)
13 reldmsets 16996 . . . . 5 Rel dom sSet
1413ovprc1 7390 . . . 4 𝑅 ∈ V → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) = ∅)
1512, 14eqtr4d 2780 . . 3 𝑅 ∈ V → (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
1611, 15pm2.61i 182 . 2 (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
171, 16eqtri 2765 1 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1541  wcel 2106  Vcvv 3443  c0 4280  cop 4590  cfv 6493  (class class class)co 7351  tpos ctpos 8148   sSet csts 16994  ndxcnx 17024  Basecbs 17042  .rcmulr 17093  opprcoppr 20000
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-tpos 8149  df-sets 16995  df-oppr 20001
This theorem is referenced by:  opprmulfval  20003  opprlem  20006  opprlemOLD  20007
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