Theorem opprabs 33453

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 2729	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		opprabs.m	. . . . . . . . 9 ⊢ · = (.r‘𝑅)
4		opprabs.o	. . . . . . . . 9 ⊢ 𝑂 = (oppr‘𝑅)
5		eqid 2729	. . . . . . . . 9 ⊢ (.r‘𝑂) = (.r‘𝑂)
6	2, 3, 4, 5	opprmulfval 20248	. . . . . . . 8 ⊢ (.r‘𝑂) = tpos ·
7	6	tposeqi 8238	. . . . . . 7 ⊢ tpos (.r‘𝑂) = tpos tpos ·
8		fnrel 6620	. . . . . . . 8 ⊢ ( · Fn (𝐵 × 𝐵) → Rel · )
9		relxp 5656	. . . . . . . . 9 ⊢ Rel (𝐵 × 𝐵)
10		fndm 6621	. . . . . . . . . 10 ⊢ ( · Fn (𝐵 × 𝐵) → dom · = (𝐵 × 𝐵))
11	10	releqd 5741	. . . . . . . . 9 ⊢ ( · Fn (𝐵 × 𝐵) → (Rel dom · ↔ Rel (𝐵 × 𝐵)))
12	9, 11	mpbiri 258	. . . . . . . 8 ⊢ ( · Fn (𝐵 × 𝐵) → Rel dom · )
13		tpostpos2 8226	. . . . . . . 8 ⊢ ((Rel · ∧ Rel dom · ) → tpos tpos · = · )
14	8, 12, 13	syl2anc 584	. . . . . . 7 ⊢ ( · Fn (𝐵 × 𝐵) → tpos tpos · = · )
15	7, 14	eqtrid 2776	. . . . . 6 ⊢ ( · Fn (𝐵 × 𝐵) → tpos (.r‘𝑂) = · )
16	1, 15	syl 17	. . . . 5 ⊢ (𝜑 → tpos (.r‘𝑂) = · )
17	16, 3	eqtrdi 2780	. . . 4 ⊢ (𝜑 → tpos (.r‘𝑂) = (.r‘𝑅))
18	17	opeq2d 4844	. . 3 ⊢ (𝜑 → ⟨(.r‘ndx), tpos (.r‘𝑂)⟩ = ⟨(.r‘ndx), (.r‘𝑅)⟩)
19	18	oveq2d 7403	. 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 20252	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑂)
22		eqid 2729	. . . . . 6 ⊢ (oppr‘𝑂) = (oppr‘𝑂)
23	21, 5, 22	opprval 20247	. . . . 5 ⊢ (oppr‘𝑂) = (𝑂 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
24	2, 3, 4	opprval 20247	. . . . . 6 ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)
25	24	oveq1i 7397	. . . . 5 ⊢ (𝑂 sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩) = ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
26	23, 25	eqtri 2752	. . . 4 ⊢ (oppr‘𝑂) = ((𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) sSet ⟨(.r‘ndx), tpos (.r‘𝑂)⟩)
27		fvex 6871	. . . . . 6 ⊢ (.r‘𝑂) ∈ V
28	27	tposex 8239	. . . . 5 ⊢ tpos (.r‘𝑂) ∈ V
29		setsabs 17149	. . . . 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 691	. . . 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 2776	. . 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 17258	. . 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 17169	. 2 ⊢ (𝜑 → 𝑅 = (𝑅 sSet ⟨(.r‘ndx), (.r‘𝑅)⟩))
37	19, 32, 36	3eqtr4rd 2775	1 ⊢ (𝜑 → 𝑅 = (oppr‘𝑂))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⟨cop 4595   × cxp 5636  dom cdm 5638  Rel wrel 5643  Fun wfun 6505   Fn wfn 6506  ‘cfv 6511  (class class class)co 7387  tpos ctpos 8204   sSet csts 17133  ndxcnx 17163  Basecbs 17179  ._rcmulr 17221  opp_rcoppr 20245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-mulr 17234  df-oppr 20246

This theorem is referenced by: (None)