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 = (oppr‘𝑅)
opprabs.m	⊢ · = (.r‘𝑅)
opprabs.1	⊢ (𝜑 → 𝑅 ∈ 𝑉)
opprabs.2	⊢ (𝜑 → Fun 𝑅)
opprabs.3	⊢ (𝜑 → (.r‘ndx) ∈ dom 𝑅)
opprabs.4	⊢ (𝜑 → · Fn (𝐵 × 𝐵))

Assertion

Ref	Expression
opprabs	⊢ (𝜑 → 𝑅 = (oppr‘𝑂))

Proof of Theorem opprabs

Step	Hyp	Ref	Expression
1		opprabs.4	. . . . . 6 ⊢ (𝜑 → · Fn (𝐵 × 𝐵))
2		eqid 2733	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		opprabs.m	. . . . . . . . 9 ⊢ · = (.r‘𝑅)
4		opprabs.o	. . . . . . . . 9 ⊢ 𝑂 = (oppr‘𝑅)
5		eqid 2733	. . . . . . . . 9 ⊢ (.r‘𝑂) = (.r‘𝑂)
6	2, 3, 4, 5	opprmulfval 20141	. . . . . . . 8 ⊢ (.r‘𝑂) = tpos ·
7	6	tposeqi 8239	. . . . . . 7 ⊢ tpos (.r‘𝑂) = 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))
11	10	releqd 5776	. . . . . . . . 9 ⊢ ( · Fn (𝐵 × 𝐵) → (Rel dom · ↔ Rel (𝐵 × 𝐵)))
12	9, 11	mpbiri 258	. . . . . . . 8 ⊢ ( · Fn (𝐵 × 𝐵) → Rel dom · )
13		tpostpos2 8227	. . . . . . . 8 ⊢ ((Rel · ∧ Rel dom · ) → tpos tpos · = · )
14	8, 12, 13	syl2anc 585	. . . . . . 7 ⊢ ( · Fn (𝐵 × 𝐵) → tpos tpos · = · )
15	7, 14	eqtrid 2785	. . . . . 6 ⊢ ( · Fn (𝐵 × 𝐵) → tpos (.r‘𝑂) = · )
16	1, 15	syl 17	. . . . 5 ⊢ (𝜑 → tpos (.r‘𝑂) = · )
17	16, 3	eqtrdi 2789	. . . 4 ⊢ (𝜑 → tpos (.r‘𝑂) = (.r‘𝑅))
18	17	opeq2d 4879	. . 3 ⊢ (𝜑 → ⟨(.r‘ndx), tpos (.r‘𝑂)⟩ = ⟨(.r‘ndx), (.r‘𝑅)⟩)
19	18	oveq2d 7420	. 2 ⊢ (𝜑 → (𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩) = (𝑅 sSet ⟨(.r‘ndx), (.r‘𝑅)⟩))
20		opprabs.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
21	4, 2	opprbas 20146	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑂)
22		eqid 2733	. . . . . 6 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
23	21, 5, 22	opprval 20140	. . . . 5 ⊢ (oppr‘𝑂) = (𝑂 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
24	2, 3, 4	opprval 20140	. . . . . 6 ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
25	24	oveq1i 7414	. . . . 5 ⊢ (𝑂 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩) = ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
26	23, 25	eqtri 2761	. . . 4 ⊢ (oppr‘𝑂) = ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
27		fvex 6901	. . . . . 6 ⊢ (.r‘𝑂) ∈ V
28	27	tposex 8240	. . . . 5 ⊢ tpos (.r‘𝑂) ∈ V
29		setsabs 17108	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ tpos (.r‘𝑂) ∈ V) → ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩))
30	28, 29	mpan2 690	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩) = (𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩))
31	26, 30	eqtrid 2785	. . 3 ⊢ (𝑅 ∈ 𝑉 → (oppr‘𝑂) = (𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩))
32	20, 31	syl 17	. 2 ⊢ (𝜑 → (oppr‘𝑂) = (𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩))
33		mulridx 17235	. . 3 ⊢ .r = Slot (.r‘ndx)
34		opprabs.2	. . 3 ⊢ (𝜑 → Fun 𝑅)
35		opprabs.3	. . 3 ⊢ (𝜑 → (.r‘ndx) ∈ dom 𝑅)
36	33, 20, 34, 35	setsidvald 17128	. 2 ⊢ (𝜑 → 𝑅 = (𝑅 sSet ⟨(.r‘ndx), (.r‘𝑅)⟩))
37	19, 32, 36	3eqtr4rd 2784	1 ⊢ (𝜑 → 𝑅 = (oppr‘𝑂))

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  opp_rcoppr 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)