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Theorem opprvalg 13194
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
opprvalg (𝑅𝑉𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))

Proof of Theorem opprvalg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 opprval.3 . 2 𝑂 = (oppr𝑅)
2 df-oppr 13193 . . 3 oppr = (𝑥 ∈ V ↦ (𝑥 sSet ⟨(.r‘ndx), tpos (.r𝑥)⟩))
3 id 19 . . . 4 (𝑥 = 𝑅𝑥 = 𝑅)
4 fveq2 5515 . . . . . . 7 (𝑥 = 𝑅 → (.r𝑥) = (.r𝑅))
5 opprval.2 . . . . . . 7 · = (.r𝑅)
64, 5eqtr4di 2228 . . . . . 6 (𝑥 = 𝑅 → (.r𝑥) = · )
76tposeqd 6248 . . . . 5 (𝑥 = 𝑅 → tpos (.r𝑥) = tpos · )
87opeq2d 3785 . . . 4 (𝑥 = 𝑅 → ⟨(.r‘ndx), tpos (.r𝑥)⟩ = ⟨(.r‘ndx), tpos · ⟩)
93, 8oveq12d 5892 . . 3 (𝑥 = 𝑅 → (𝑥 sSet ⟨(.r‘ndx), tpos (.r𝑥)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
10 elex 2748 . . 3 (𝑅𝑉𝑅 ∈ V)
11 mulrslid 12584 . . . . . 6 (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ)
1211simpri 113 . . . . 5 (.r‘ndx) ∈ ℕ
1312a1i 9 . . . 4 (𝑅𝑉 → (.r‘ndx) ∈ ℕ)
1411slotex 12483 . . . . . 6 (𝑅𝑉 → (.r𝑅) ∈ V)
155, 14eqeltrid 2264 . . . . 5 (𝑅𝑉· ∈ V)
16 tposexg 6258 . . . . 5 ( · ∈ V → tpos · ∈ V)
1715, 16syl 14 . . . 4 (𝑅𝑉 → tpos · ∈ V)
18 setsex 12488 . . . 4 ((𝑅 ∈ V ∧ (.r‘ndx) ∈ ℕ ∧ tpos · ∈ V) → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V)
1910, 13, 17, 18syl3anc 1238 . . 3 (𝑅𝑉 → (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) ∈ V)
202, 9, 10, 19fvmptd3 5609 . 2 (𝑅𝑉 → (oppr𝑅) = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
211, 20eqtrid 2222 1 (𝑅𝑉𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  Vcvv 2737  cop 3595  cfv 5216  (class class class)co 5874  tpos ctpos 6244  cn 8917  ndxcnx 12453   sSet csts 12454  Slot cslot 12455  Basecbs 12456  .rcmulr 12531  opprcoppr 13192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-tpos 6245  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-sets 12463  df-mulr 12544  df-oppr 13193
This theorem is referenced by:  opprmulfvalg  13195  opprex  13198  opprsllem  13199
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