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Theorem opprabs 32549
Description: The opposite ring of the opposite ring is the original ring. Note the conditions on this theorem, which makes it unpractical in case we only have e.g. 𝑅 ∈ Ring as a premise. (Contributed by Thierry Arnoux, 9-Mar-2025.)
Hypotheses
Ref Expression
opprabs.o 𝑂 = (oppr𝑅)
opprabs.m · = (.r𝑅)
opprabs.1 (𝜑𝑅𝑉)
opprabs.2 (𝜑 → Fun 𝑅)
opprabs.3 (𝜑 → (.r‘ndx) ∈ dom 𝑅)
opprabs.4 (𝜑· Fn (𝐵 × 𝐵))
Assertion
Ref Expression
opprabs (𝜑𝑅 = (oppr𝑂))

Proof of Theorem opprabs
StepHypRef Expression
1 opprabs.4 . . . . . 6 (𝜑· Fn (𝐵 × 𝐵))
2 eqid 2733 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
3 opprabs.m . . . . . . . . 9 · = (.r𝑅)
4 opprabs.o . . . . . . . . 9 𝑂 = (oppr𝑅)
5 eqid 2733 . . . . . . . . 9 (.r𝑂) = (.r𝑂)
62, 3, 4, 5opprmulfval 20141 . . . . . . . 8 (.r𝑂) = tpos ·
76tposeqi 8239 . . . . . . 7 tpos (.r𝑂) = tpos tpos ·
8 fnrel 6648 . . . . . . . 8 ( · Fn (𝐵 × 𝐵) → Rel · )
9 relxp 5693 . . . . . . . . 9 Rel (𝐵 × 𝐵)
10 fndm 6649 . . . . . . . . . 10 ( · Fn (𝐵 × 𝐵) → dom · = (𝐵 × 𝐵))
1110releqd 5776 . . . . . . . . 9 ( · Fn (𝐵 × 𝐵) → (Rel dom · ↔ Rel (𝐵 × 𝐵)))
129, 11mpbiri 258 . . . . . . . 8 ( · Fn (𝐵 × 𝐵) → Rel dom · )
13 tpostpos2 8227 . . . . . . . 8 ((Rel · ∧ Rel dom · ) → tpos tpos · = · )
148, 12, 13syl2anc 585 . . . . . . 7 ( · Fn (𝐵 × 𝐵) → tpos tpos · = · )
157, 14eqtrid 2785 . . . . . 6 ( · Fn (𝐵 × 𝐵) → tpos (.r𝑂) = · )
161, 15syl 17 . . . . 5 (𝜑 → tpos (.r𝑂) = · )
1716, 3eqtrdi 2789 . . . 4 (𝜑 → tpos (.r𝑂) = (.r𝑅))
1817opeq2d 4879 . . 3 (𝜑 → ⟨(.r‘ndx), tpos (.r𝑂)⟩ = ⟨(.r‘ndx), (.r𝑅)⟩)
1918oveq2d 7420 . 2 (𝜑 → (𝑅 sSet ⟨(.r‘ndx), tpos (.r𝑂)⟩) = (𝑅 sSet ⟨(.r‘ndx), (.r𝑅)⟩))
20 opprabs.1 . . 3 (𝜑𝑅𝑉)
214, 2opprbas 20146 . . . . . 6 (Base‘𝑅) = (Base‘𝑂)
22 eqid 2733 . . . . . 6 (oppr𝑂) = (oppr𝑂)
2321, 5, 22opprval 20140 . . . . 5 (oppr𝑂) = (𝑂 sSet ⟨(.r‘ndx), tpos (.r𝑂)⟩)
242, 3, 4opprval 20140 . . . . . 6 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
2524oveq1i 7414 . . . . 5 (𝑂 sSet ⟨(.r‘ndx), tpos (.r𝑂)⟩) = ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r𝑂)⟩)
2623, 25eqtri 2761 . . . 4 (oppr𝑂) = ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r𝑂)⟩)
27 fvex 6901 . . . . . 6 (.r𝑂) ∈ V
2827tposex 8240 . . . . 5 tpos (.r𝑂) ∈ V
29 setsabs 17108 . . . . 5 ((𝑅𝑉 ∧ tpos (.r𝑂) ∈ V) → ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r𝑂)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos (.r𝑂)⟩))
3028, 29mpan2 690 . . . 4 (𝑅𝑉 → ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r𝑂)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos (.r𝑂)⟩))
3126, 30eqtrid 2785 . . 3 (𝑅𝑉 → (oppr𝑂) = (𝑅 sSet ⟨(.r‘ndx), tpos (.r𝑂)⟩))
3220, 31syl 17 . 2 (𝜑 → (oppr𝑂) = (𝑅 sSet ⟨(.r‘ndx), tpos (.r𝑂)⟩))
33 mulridx 17235 . . 3 .r = Slot (.r‘ndx)
34 opprabs.2 . . 3 (𝜑 → Fun 𝑅)
35 opprabs.3 . . 3 (𝜑 → (.r‘ndx) ∈ dom 𝑅)
3633, 20, 34, 35setsidvald 17128 . 2 (𝜑𝑅 = (𝑅 sSet ⟨(.r‘ndx), (.r𝑅)⟩))
3719, 32, 363eqtr4rd 2784 1 (𝜑𝑅 = (oppr𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3475  cop 4633   × cxp 5673  dom cdm 5675  Rel wrel 5680  Fun wfun 6534   Fn wfn 6535  cfv 6540  (class class class)co 7404  tpos ctpos 8205   sSet csts 17092  ndxcnx 17122  Basecbs 17140  .rcmulr 17194  opprcoppr 20138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-tpos 8206  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-mulr 17207  df-oppr 20139
This theorem is referenced by: (None)
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