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Mirrors > Home > MPE Home > Th. List > opprmulfval | Structured version Visualization version GIF version |
Description: Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
opprval.1 | ⊢ 𝐵 = (Base‘𝑅) |
opprval.2 | ⊢ · = (.r‘𝑅) |
opprval.3 | ⊢ 𝑂 = (oppr‘𝑅) |
opprmulfval.4 | ⊢ ∙ = (.r‘𝑂) |
Ref | Expression |
---|---|
opprmulfval | ⊢ ∙ = tpos · |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprmulfval.4 | . 2 ⊢ ∙ = (.r‘𝑂) | |
2 | opprval.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
3 | opprval.2 | . . . . . 6 ⊢ · = (.r‘𝑅) | |
4 | opprval.3 | . . . . . 6 ⊢ 𝑂 = (oppr‘𝑅) | |
5 | 2, 3, 4 | opprval 19370 | . . . . 5 ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
6 | 5 | fveq2i 6648 | . . . 4 ⊢ (.r‘𝑂) = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
7 | 3 | fvexi 6659 | . . . . . 6 ⊢ · ∈ V |
8 | 7 | tposex 7909 | . . . . 5 ⊢ tpos · ∈ V |
9 | mulrid 16608 | . . . . . 6 ⊢ .r = Slot (.r‘ndx) | |
10 | 9 | setsid 16530 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ tpos · ∈ V) → tpos · = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉))) |
11 | 8, 10 | mpan2 690 | . . . 4 ⊢ (𝑅 ∈ V → tpos · = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉))) |
12 | 6, 11 | eqtr4id 2852 | . . 3 ⊢ (𝑅 ∈ V → (.r‘𝑂) = tpos · ) |
13 | tpos0 7905 | . . . . 5 ⊢ tpos ∅ = ∅ | |
14 | 9 | str0 16527 | . . . . 5 ⊢ ∅ = (.r‘∅) |
15 | 13, 14 | eqtr2i 2822 | . . . 4 ⊢ (.r‘∅) = tpos ∅ |
16 | fvprc 6638 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = ∅) | |
17 | 4, 16 | syl5eq 2845 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
18 | 17 | fveq2d 6649 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑂) = (.r‘∅)) |
19 | fvprc 6638 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑅) = ∅) | |
20 | 3, 19 | syl5eq 2845 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → · = ∅) |
21 | 20 | tposeqd 7878 | . . . 4 ⊢ (¬ 𝑅 ∈ V → tpos · = tpos ∅) |
22 | 15, 18, 21 | 3eqtr4a 2859 | . . 3 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑂) = tpos · ) |
23 | 12, 22 | pm2.61i 185 | . 2 ⊢ (.r‘𝑂) = tpos · |
24 | 1, 23 | eqtri 2821 | 1 ⊢ ∙ = tpos · |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 〈cop 4531 ‘cfv 6324 (class class class)co 7135 tpos ctpos 7874 ndxcnx 16472 sSet csts 16473 Basecbs 16475 .rcmulr 16558 opprcoppr 19368 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-nn 11626 df-2 11688 df-3 11689 df-ndx 16478 df-slot 16479 df-sets 16482 df-mulr 16571 df-oppr 19369 |
This theorem is referenced by: opprmul 19372 |
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