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Theorem opprabs 33027
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 = (opprβ€˜π‘…)
opprabs.m Β· = (.rβ€˜π‘…)
opprabs.1 (πœ‘ β†’ 𝑅 ∈ 𝑉)
opprabs.2 (πœ‘ β†’ Fun 𝑅)
opprabs.3 (πœ‘ β†’ (.rβ€˜ndx) ∈ dom 𝑅)
opprabs.4 (πœ‘ β†’ Β· Fn (𝐡 Γ— 𝐡))
Assertion
Ref Expression
opprabs (πœ‘ β†’ 𝑅 = (opprβ€˜π‘‚))

Proof of Theorem opprabs
StepHypRef Expression
1 opprabs.4 . . . . . 6 (πœ‘ β†’ Β· Fn (𝐡 Γ— 𝐡))
2 eqid 2724 . . . . . . . . 9 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
3 opprabs.m . . . . . . . . 9 Β· = (.rβ€˜π‘…)
4 opprabs.o . . . . . . . . 9 𝑂 = (opprβ€˜π‘…)
5 eqid 2724 . . . . . . . . 9 (.rβ€˜π‘‚) = (.rβ€˜π‘‚)
62, 3, 4, 5opprmulfval 20223 . . . . . . . 8 (.rβ€˜π‘‚) = tpos Β·
76tposeqi 8239 . . . . . . 7 tpos (.rβ€˜π‘‚) = tpos tpos Β·
8 fnrel 6641 . . . . . . . 8 ( Β· Fn (𝐡 Γ— 𝐡) β†’ Rel Β· )
9 relxp 5684 . . . . . . . . 9 Rel (𝐡 Γ— 𝐡)
10 fndm 6642 . . . . . . . . . 10 ( Β· Fn (𝐡 Γ— 𝐡) β†’ dom Β· = (𝐡 Γ— 𝐡))
1110releqd 5768 . . . . . . . . 9 ( Β· Fn (𝐡 Γ— 𝐡) β†’ (Rel dom Β· ↔ Rel (𝐡 Γ— 𝐡)))
129, 11mpbiri 258 . . . . . . . 8 ( Β· Fn (𝐡 Γ— 𝐡) β†’ Rel dom Β· )
13 tpostpos2 8227 . . . . . . . 8 ((Rel Β· ∧ Rel dom Β· ) β†’ tpos tpos Β· = Β· )
148, 12, 13syl2anc 583 . . . . . . 7 ( Β· Fn (𝐡 Γ— 𝐡) β†’ tpos tpos Β· = Β· )
157, 14eqtrid 2776 . . . . . 6 ( Β· Fn (𝐡 Γ— 𝐡) β†’ tpos (.rβ€˜π‘‚) = Β· )
161, 15syl 17 . . . . 5 (πœ‘ β†’ tpos (.rβ€˜π‘‚) = Β· )
1716, 3eqtrdi 2780 . . . 4 (πœ‘ β†’ tpos (.rβ€˜π‘‚) = (.rβ€˜π‘…))
1817opeq2d 4872 . . 3 (πœ‘ β†’ ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘‚)⟩ = ⟨(.rβ€˜ndx), (.rβ€˜π‘…)⟩)
1918oveq2d 7417 . 2 (πœ‘ β†’ (𝑅 sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘‚)⟩) = (𝑅 sSet ⟨(.rβ€˜ndx), (.rβ€˜π‘…)⟩))
20 opprabs.1 . . 3 (πœ‘ β†’ 𝑅 ∈ 𝑉)
214, 2opprbas 20228 . . . . . 6 (Baseβ€˜π‘…) = (Baseβ€˜π‘‚)
22 eqid 2724 . . . . . 6 (opprβ€˜π‘‚) = (opprβ€˜π‘‚)
2321, 5, 22opprval 20222 . . . . 5 (opprβ€˜π‘‚) = (𝑂 sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘‚)⟩)
242, 3, 4opprval 20222 . . . . . 6 𝑂 = (𝑅 sSet ⟨(.rβ€˜ndx), tpos Β· ⟩)
2524oveq1i 7411 . . . . 5 (𝑂 sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘‚)⟩) = ((𝑅 sSet ⟨(.rβ€˜ndx), tpos Β· ⟩) sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘‚)⟩)
2623, 25eqtri 2752 . . . 4 (opprβ€˜π‘‚) = ((𝑅 sSet ⟨(.rβ€˜ndx), tpos Β· ⟩) sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘‚)⟩)
27 fvex 6894 . . . . . 6 (.rβ€˜π‘‚) ∈ V
2827tposex 8240 . . . . 5 tpos (.rβ€˜π‘‚) ∈ V
29 setsabs 17108 . . . . 5 ((𝑅 ∈ 𝑉 ∧ tpos (.rβ€˜π‘‚) ∈ V) β†’ ((𝑅 sSet ⟨(.rβ€˜ndx), tpos Β· ⟩) sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘‚)⟩) = (𝑅 sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘‚)⟩))
3028, 29mpan2 688 . . . 4 (𝑅 ∈ 𝑉 β†’ ((𝑅 sSet ⟨(.rβ€˜ndx), tpos Β· ⟩) sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘‚)⟩) = (𝑅 sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘‚)⟩))
3126, 30eqtrid 2776 . . 3 (𝑅 ∈ 𝑉 β†’ (opprβ€˜π‘‚) = (𝑅 sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘‚)⟩))
3220, 31syl 17 . 2 (πœ‘ β†’ (opprβ€˜π‘‚) = (𝑅 sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘‚)⟩))
33 mulridx 17235 . . 3 .r = Slot (.rβ€˜ndx)
34 opprabs.2 . . 3 (πœ‘ β†’ Fun 𝑅)
35 opprabs.3 . . 3 (πœ‘ β†’ (.rβ€˜ndx) ∈ dom 𝑅)
3633, 20, 34, 35setsidvald 17128 . 2 (πœ‘ β†’ 𝑅 = (𝑅 sSet ⟨(.rβ€˜ndx), (.rβ€˜π‘…)⟩))
3719, 32, 363eqtr4rd 2775 1 (πœ‘ β†’ 𝑅 = (opprβ€˜π‘‚))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3466  βŸ¨cop 4626   Γ— cxp 5664  dom cdm 5666  Rel wrel 5671  Fun wfun 6527   Fn wfn 6528  β€˜cfv 6533  (class class class)co 7401  tpos ctpos 8205   sSet csts 17092  ndxcnx 17122  Basecbs 17140  .rcmulr 17194  opprcoppr 20220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-tpos 8206  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  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 20221
This theorem is referenced by: (None)
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