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Theorem opprval 18447
 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 6103 . . . . . . . 8 (𝑥 = 𝑅 → (.r𝑥) = (.r𝑅))
4 opprval.2 . . . . . . . 8 · = (.r𝑅)
53, 4syl6eqr 2662 . . . . . . 7 (𝑥 = 𝑅 → (.r𝑥) = · )
65tposeqd 7242 . . . . . 6 (𝑥 = 𝑅 → tpos (.r𝑥) = tpos · )
76opeq2d 4347 . . . . 5 (𝑥 = 𝑅 → ⟨(.r‘ndx), tpos (.r𝑥)⟩ = ⟨(.r‘ndx), tpos · ⟩)
82, 7oveq12d 6567 . . . 4 (𝑥 = 𝑅 → (𝑥 sSet ⟨(.r‘ndx), tpos (.r𝑥)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
9 df-oppr 18446 . . . 4 oppr = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(.r‘ndx), tpos (.r𝑥)⟩))
10 ovex 6577 . . . 4 (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V
118, 9, 10fvmpt 6191 . . 3 (𝑅 ∈ V → (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
12 fvprc 6097 . . . 4 𝑅 ∈ V → (oppr𝑅) = ∅)
13 reldmsets 15718 . . . . 5 Rel dom sSet
1413ovprc1 6582 . . . 4 𝑅 ∈ V → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) = ∅)
1512, 14eqtr4d 2647 . . 3 𝑅 ∈ V → (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
1611, 15pm2.61i 175 . 2 (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
171, 16eqtri 2632 1 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549  tpos ctpos 7238  ndxcnx 15692   sSet csts 15693  Basecbs 15695  .rcmulr 15769  opprcoppr 18445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-tpos 7239  df-sets 15701  df-oppr 18446 This theorem is referenced by:  opprmulfval  18448  opprlem  18451
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