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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opprabs | Structured version Visualization version GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
opprabs | ⊢ (𝜑 → 𝑅 = (oppr‘𝑂)) |
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)) |
Colors of variables: wff setvar class |
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 opprcoppr 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) |
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