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Theorem opprval 18816
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 6344 . . . . . . . 8 (𝑥 = 𝑅 → (.r𝑥) = (.r𝑅))
4 opprval.2 . . . . . . . 8 · = (.r𝑅)
53, 4syl6eqr 2804 . . . . . . 7 (𝑥 = 𝑅 → (.r𝑥) = · )
65tposeqd 7516 . . . . . 6 (𝑥 = 𝑅 → tpos (.r𝑥) = tpos · )
76opeq2d 4552 . . . . 5 (𝑥 = 𝑅 → ⟨(.r‘ndx), tpos (.r𝑥)⟩ = ⟨(.r‘ndx), tpos · ⟩)
82, 7oveq12d 6823 . . . 4 (𝑥 = 𝑅 → (𝑥 sSet ⟨(.r‘ndx), tpos (.r𝑥)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
9 df-oppr 18815 . . . 4 oppr = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(.r‘ndx), tpos (.r𝑥)⟩))
10 ovex 6833 . . . 4 (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V
118, 9, 10fvmpt 6436 . . 3 (𝑅 ∈ V → (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
12 fvprc 6338 . . . 4 𝑅 ∈ V → (oppr𝑅) = ∅)
13 reldmsets 16080 . . . . 5 Rel dom sSet
1413ovprc1 6839 . . . 4 𝑅 ∈ V → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) = ∅)
1512, 14eqtr4d 2789 . . 3 𝑅 ∈ V → (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
1611, 15pm2.61i 176 . 2 (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
171, 16eqtri 2774 1 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1624  wcel 2131  Vcvv 3332  c0 4050  cop 4319  cfv 6041  (class class class)co 6805  tpos ctpos 7512  ndxcnx 16048   sSet csts 16049  Basecbs 16051  .rcmulr 16136  opprcoppr 18814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-res 5270  df-iota 6004  df-fun 6043  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-tpos 7513  df-sets 16058  df-oppr 18815
This theorem is referenced by:  opprmulfval  18817  opprlem  18820
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