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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.2 | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
3 | 2 | fvexi 6677 | . . . . . 6 ⊢ · ∈ V |
4 | 3 | tposex 7915 | . . . . 5 ⊢ tpos · ∈ V |
5 | mulrid 16604 | . . . . . 6 ⊢ .r = Slot (.r‘ndx) | |
6 | 5 | setsid 16526 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ tpos · ∈ V) → tpos · = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉))) |
7 | 4, 6 | mpan2 687 | . . . 4 ⊢ (𝑅 ∈ V → tpos · = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉))) |
8 | opprval.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
9 | opprval.3 | . . . . . 6 ⊢ 𝑂 = (oppr‘𝑅) | |
10 | 8, 2, 9 | opprval 19303 | . . . . 5 ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
11 | 10 | fveq2i 6666 | . . . 4 ⊢ (.r‘𝑂) = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
12 | 7, 11 | syl6reqr 2872 | . . 3 ⊢ (𝑅 ∈ V → (.r‘𝑂) = tpos · ) |
13 | tpos0 7911 | . . . . 5 ⊢ tpos ∅ = ∅ | |
14 | 5 | str0 16523 | . . . . 5 ⊢ ∅ = (.r‘∅) |
15 | 13, 14 | eqtr2i 2842 | . . . 4 ⊢ (.r‘∅) = tpos ∅ |
16 | fvprc 6656 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = ∅) | |
17 | 9, 16 | syl5eq 2865 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
18 | 17 | fveq2d 6667 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑂) = (.r‘∅)) |
19 | fvprc 6656 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑅) = ∅) | |
20 | 2, 19 | syl5eq 2865 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → · = ∅) |
21 | 20 | tposeqd 7884 | . . . 4 ⊢ (¬ 𝑅 ∈ V → tpos · = tpos ∅) |
22 | 15, 18, 21 | 3eqtr4a 2879 | . . 3 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑂) = tpos · ) |
23 | 12, 22 | pm2.61i 183 | . 2 ⊢ (.r‘𝑂) = tpos · |
24 | 1, 23 | eqtri 2841 | 1 ⊢ ∙ = tpos · |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 〈cop 4563 ‘cfv 6348 (class class class)co 7145 tpos ctpos 7880 ndxcnx 16468 sSet csts 16469 Basecbs 16471 .rcmulr 16554 opprcoppr 19301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-1cn 10583 ax-addcl 10585 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-sets 16478 df-mulr 16567 df-oppr 19302 |
This theorem is referenced by: opprmul 19305 |
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