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Theorem opprval 19352
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 6643 . . . . . . . 8 (𝑥 = 𝑅 → (.r𝑥) = (.r𝑅))
4 opprval.2 . . . . . . . 8 · = (.r𝑅)
53, 4syl6eqr 2874 . . . . . . 7 (𝑥 = 𝑅 → (.r𝑥) = · )
65tposeqd 7870 . . . . . 6 (𝑥 = 𝑅 → tpos (.r𝑥) = tpos · )
76opeq2d 4783 . . . . 5 (𝑥 = 𝑅 → ⟨(.r‘ndx), tpos (.r𝑥)⟩ = ⟨(.r‘ndx), tpos · ⟩)
82, 7oveq12d 7148 . . . 4 (𝑥 = 𝑅 → (𝑥 sSet ⟨(.r‘ndx), tpos (.r𝑥)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
9 df-oppr 19351 . . . 4 oppr = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(.r‘ndx), tpos (.r𝑥)⟩))
10 ovex 7163 . . . 4 (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V
118, 9, 10fvmpt 6741 . . 3 (𝑅 ∈ V → (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
12 fvprc 6636 . . . 4 𝑅 ∈ V → (oppr𝑅) = ∅)
13 reldmsets 16489 . . . . 5 Rel dom sSet
1413ovprc1 7169 . . . 4 𝑅 ∈ V → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) = ∅)
1512, 14eqtr4d 2859 . . 3 𝑅 ∈ V → (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
1611, 15pm2.61i 185 . 2 (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
171, 16eqtri 2844 1 𝑂 = (𝑅 sSet ⟨(.r‘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  Basecbs 16461  .rcmulr 16544  opprcoppr 19350
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-oppr 19351
This theorem is referenced by:  opprmulfval  19353  opprlem  19356
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