Theorem opprabs 33557

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 2737	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		opprabs.m	. . . . . . . . 9 ⊢ · = (.r‘𝑅)
4		opprabs.o	. . . . . . . . 9 ⊢ 𝑂 = (oppr‘𝑅)
5		eqid 2737	. . . . . . . . 9 ⊢ (.r‘𝑂) = (.r‘𝑂)
6	2, 3, 4, 5	opprmulfval 20310	. . . . . . . 8 ⊢ (.r‘𝑂) = tpos ·
7	6	tposeqi 8202	. . . . . . 7 ⊢ tpos (.r‘𝑂) = tpos tpos ·
8		fnrel 6594	. . . . . . . 8 ⊢ ( · Fn (𝐵 × 𝐵) → Rel · )
9		relxp 5642	. . . . . . . . 9 ⊢ Rel (𝐵 × 𝐵)
10		fndm 6595	. . . . . . . . . 10 ⊢ ( · Fn (𝐵 × 𝐵) → dom · = (𝐵 × 𝐵))
11	10	releqd 5728	. . . . . . . . 9 ⊢ ( · Fn (𝐵 × 𝐵) → (Rel dom · ↔ Rel (𝐵 × 𝐵)))
12	9, 11	mpbiri 258	. . . . . . . 8 ⊢ ( · Fn (𝐵 × 𝐵) → Rel dom · )
13		tpostpos2 8190	. . . . . . . 8 ⊢ ((Rel · ∧ Rel dom · ) → tpos tpos · = · )
14	8, 12, 13	syl2anc 585	. . . . . . 7 ⊢ ( · Fn (𝐵 × 𝐵) → tpos tpos · = · )
15	7, 14	eqtrid 2784	. . . . . 6 ⊢ ( · Fn (𝐵 × 𝐵) → tpos (.r‘𝑂) = · )
16	1, 15	syl 17	. . . . 5 ⊢ (𝜑 → tpos (.r‘𝑂) = · )
17	16, 3	eqtrdi 2788	. . . 4 ⊢ (𝜑 → tpos (.r‘𝑂) = (.r‘𝑅))
18	17	opeq2d 4824	. . 3 ⊢ (𝜑 → ⟨(.r‘ndx), tpos (.r‘𝑂)⟩ = ⟨(.r‘ndx), (.r‘𝑅)⟩)
19	18	oveq2d 7376	. 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 20314	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑂)
22		eqid 2737	. . . . . 6 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
23	21, 5, 22	opprval 20309	. . . . 5 ⊢ (oppr‘𝑂) = (𝑂 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
24	2, 3, 4	opprval 20309	. . . . . 6 ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
25	24	oveq1i 7370	. . . . 5 ⊢ (𝑂 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩) = ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
26	23, 25	eqtri 2760	. . . 4 ⊢ (oppr‘𝑂) = ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
27		fvex 6847	. . . . . 6 ⊢ (.r‘𝑂) ∈ V
28	27	tposex 8203	. . . . 5 ⊢ tpos (.r‘𝑂) ∈ V
29		setsabs 17140	. . . . 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 692	. . . 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 2784	. . 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 17249	. . 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 17160	. 2 ⊢ (𝜑 → 𝑅 = (𝑅 sSet ⟨(.r‘ndx), (.r‘𝑅)⟩))
37	19, 32, 36	3eqtr4rd 2783	1 ⊢ (𝜑 → 𝑅 = (oppr‘𝑂))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   × cxp 5622  dom cdm 5624  Rel wrel 5629  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360  tpos ctpos 8168   sSet csts 17124  ndxcnx 17154  Basecbs 17170  ._rcmulr 17212  opp_rcoppr 20307

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-tpos 8169  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-mulr 17225  df-oppr 20308

This theorem is referenced by: (None)