Theorem opprabs 32442

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 2731	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		opprabs.m	. . . . . . . . 9 ⊢ · = (.r‘𝑅)
4		opprabs.o	. . . . . . . . 9 ⊢ 𝑂 = (oppr‘𝑅)
5		eqid 2731	. . . . . . . . 9 ⊢ (.r‘𝑂) = (.r‘𝑂)
6	2, 3, 4, 5	opprmulfval 20104	. . . . . . . 8 ⊢ (.r‘𝑂) = tpos ·
7	6	tposeqi 8226	. . . . . . 7 ⊢ tpos (.r‘𝑂) = tpos tpos ·
8		fnrel 6640	. . . . . . . 8 ⊢ ( · Fn (𝐵 × 𝐵) → Rel · )
9		relxp 5687	. . . . . . . . 9 ⊢ Rel (𝐵 × 𝐵)
10		fndm 6641	. . . . . . . . . 10 ⊢ ( · Fn (𝐵 × 𝐵) → dom · = (𝐵 × 𝐵))
11	10	releqd 5770	. . . . . . . . 9 ⊢ ( · Fn (𝐵 × 𝐵) → (Rel dom · ↔ Rel (𝐵 × 𝐵)))
12	9, 11	mpbiri 257	. . . . . . . 8 ⊢ ( · Fn (𝐵 × 𝐵) → Rel dom · )
13		tpostpos2 8214	. . . . . . . 8 ⊢ ((Rel · ∧ Rel dom · ) → tpos tpos · = · )
14	8, 12, 13	syl2anc 584	. . . . . . 7 ⊢ ( · Fn (𝐵 × 𝐵) → tpos tpos · = · )
15	7, 14	eqtrid 2783	. . . . . 6 ⊢ ( · Fn (𝐵 × 𝐵) → tpos (.r‘𝑂) = · )
16	1, 15	syl 17	. . . . 5 ⊢ (𝜑 → tpos (.r‘𝑂) = · )
17	16, 3	eqtrdi 2787	. . . 4 ⊢ (𝜑 → tpos (.r‘𝑂) = (.r‘𝑅))
18	17	opeq2d 4873	. . 3 ⊢ (𝜑 → ⟨(.r‘ndx), tpos (.r‘𝑂)⟩ = ⟨(.r‘ndx), (.r‘𝑅)⟩)
19	18	oveq2d 7409	. 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 20109	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑂)
22		eqid 2731	. . . . . 6 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
23	21, 5, 22	opprval 20103	. . . . 5 ⊢ (oppr‘𝑂) = (𝑂 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
24	2, 3, 4	opprval 20103	. . . . . 6 ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
25	24	oveq1i 7403	. . . . 5 ⊢ (𝑂 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩) = ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
26	23, 25	eqtri 2759	. . . 4 ⊢ (oppr‘𝑂) = ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
27		fvex 6891	. . . . . 6 ⊢ (.r‘𝑂) ∈ V
28	27	tposex 8227	. . . . 5 ⊢ tpos (.r‘𝑂) ∈ V
29		setsabs 17094	. . . . 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 689	. . . 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 2783	. . 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 17221	. . 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 17114	. 2 ⊢ (𝜑 → 𝑅 = (𝑅 sSet ⟨(.r‘ndx), (.r‘𝑅)⟩))
37	19, 32, 36	3eqtr4rd 2782	1 ⊢ (𝜑 → 𝑅 = (oppr‘𝑂))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3473  ⟨cop 4628   × cxp 5667  dom cdm 5669  Rel wrel 5674  Fun wfun 6526   Fn wfn 6527  ‘cfv 6532  (class class class)co 7393  tpos ctpos 8192   sSet csts 17078  ndxcnx 17108  Basecbs 17126  ._rcmulr 17180  opp_rcoppr 20101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-2nd 7958  df-tpos 8193  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-3 12258  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-mulr 17193  df-oppr 20102

This theorem is referenced by: (None)