Theorem opprabs 34110

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 2763	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		opprabs.m	. . . . . . . . 9 ⊢ · = (.r‘𝑅)
4		opprabs.o	. . . . . . . . 9 ⊢ 𝑂 = (oppr‘𝑅)
5		eqid 2763	. . . . . . . . 9 ⊢ (.r‘𝑂) = (.r‘𝑂)
6	2, 3, 4, 5	opprmulfval 20620	. . . . . . . 8 ⊢ (.r‘𝑂) = tpos ·
7	6	tposeqi 8314	. . . . . . 7 ⊢ tpos (.r‘𝑂) = tpos tpos ·
8		fnrel 6671	. . . . . . . 8 ⊢ ( · Fn (𝐵 × 𝐵) → Rel · )
9		relxp 5676	. . . . . . . . 9 ⊢ Rel (𝐵 × 𝐵)
10		fndm 6672	. . . . . . . . . 10 ⊢ ( · Fn (𝐵 × 𝐵) → dom · = (𝐵 × 𝐵))
11	10	releqd 5763	. . . . . . . . 9 ⊢ ( · Fn (𝐵 × 𝐵) → (Rel dom · ↔ Rel (𝐵 × 𝐵)))
12	9, 11	mpbiri 261	. . . . . . . 8 ⊢ ( · Fn (𝐵 × 𝐵) → Rel dom · )
13		tpostpos2 8302	. . . . . . . 8 ⊢ ((Rel · ∧ Rel dom · ) → tpos tpos · = · )
14	8, 12, 13	syl2anc 596	. . . . . . 7 ⊢ ( · Fn (𝐵 × 𝐵) → tpos tpos · = · )
15	7, 14	eqtrid 2810	. . . . . 6 ⊢ ( · Fn (𝐵 × 𝐵) → tpos (.r‘𝑂) = · )
16	1, 15	syl 18	. . . . 5 ⊢ (𝜑 → tpos (.r‘𝑂) = · )
17	16, 3	eqtrdi 2814	. . . 4 ⊢ (𝜑 → tpos (.r‘𝑂) = (.r‘𝑅))
18	17	opeq2d 4852	. . 3 ⊢ (𝜑 → ⟨(.r‘ndx), tpos (.r‘𝑂)⟩ = ⟨(.r‘ndx), (.r‘𝑅)⟩)
19	18	oveq2d 7467	. 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 20624	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑂)
22		eqid 2763	. . . . . 6 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
23	21, 5, 22	opprval 20619	. . . . 5 ⊢ (oppr‘𝑂) = (𝑂 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
24	2, 3, 4	opprval 20619	. . . . . 6 ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
25	24	oveq1i 7461	. . . . 5 ⊢ (𝑂 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩) = ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
26	23, 25	eqtri 2786	. . . 4 ⊢ (oppr‘𝑂) = ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
27		fvex 6929	. . . . . 6 ⊢ (.r‘𝑂) ∈ V
28	27	tposex 8315	. . . . 5 ⊢ tpos (.r‘𝑂) ∈ V
29		setsabs 17401	. . . . 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 704	. . . 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 2810	. . 3 ⊢ (𝑅 ∈ 𝑉 → (oppr‘𝑂) = (𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩))
32	20, 31	syl 18	. 2 ⊢ (𝜑 → (oppr‘𝑂) = (𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩))
33		mulridx 17512	. . 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 17421	. 2 ⊢ (𝜑 → 𝑅 = (𝑅 sSet ⟨(.r‘ndx), (.r‘𝑅)⟩))
37	19, 32, 36	3eqtr4rd 2809	1 ⊢ (𝜑 → 𝑅 = (oppr‘𝑂))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4840   × cxp 5656  dom cdm 5658  Rel wrel 5663  Fun wfun 6560   Fn wfn 6561  ‘cfv 6566  (class class class)co 7451  tpos ctpos 8280   sSet csts 17385  ndxcnx 17415  Basecbs 17431  ._rcmulr 17473  opp_rcoppr 20617

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-tpos 8281  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-mulr 17487  df-oppr 20618

This theorem is used by: (None)