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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 | fvex 6239 | . . . . . . 7 ⊢ (.r‘𝑅) ∈ V | |
4 | 2, 3 | eqeltri 2726 | . . . . . 6 ⊢ · ∈ V |
5 | 4 | tposex 7431 | . . . . 5 ⊢ tpos · ∈ V |
6 | mulrid 16044 | . . . . . 6 ⊢ .r = Slot (.r‘ndx) | |
7 | 6 | setsid 15961 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ tpos · ∈ V) → tpos · = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉))) |
8 | 5, 7 | mpan2 707 | . . . 4 ⊢ (𝑅 ∈ V → tpos · = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉))) |
9 | opprval.1 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
10 | opprval.3 | . . . . . 6 ⊢ 𝑂 = (oppr‘𝑅) | |
11 | 9, 2, 10 | opprval 18670 | . . . . 5 ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
12 | 11 | fveq2i 6232 | . . . 4 ⊢ (.r‘𝑂) = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
13 | 8, 12 | syl6reqr 2704 | . . 3 ⊢ (𝑅 ∈ V → (.r‘𝑂) = tpos · ) |
14 | tpos0 7427 | . . . . 5 ⊢ tpos ∅ = ∅ | |
15 | 6 | str0 15958 | . . . . 5 ⊢ ∅ = (.r‘∅) |
16 | 14, 15 | eqtr2i 2674 | . . . 4 ⊢ (.r‘∅) = tpos ∅ |
17 | fvprc 6223 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = ∅) | |
18 | 10, 17 | syl5eq 2697 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
19 | 18 | fveq2d 6233 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑂) = (.r‘∅)) |
20 | fvprc 6223 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑅) = ∅) | |
21 | 2, 20 | syl5eq 2697 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → · = ∅) |
22 | 21 | tposeqd 7400 | . . . 4 ⊢ (¬ 𝑅 ∈ V → tpos · = tpos ∅) |
23 | 16, 19, 22 | 3eqtr4a 2711 | . . 3 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑂) = tpos · ) |
24 | 13, 23 | pm2.61i 176 | . 2 ⊢ (.r‘𝑂) = tpos · |
25 | 1, 24 | eqtri 2673 | 1 ⊢ ∙ = tpos · |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∅c0 3948 〈cop 4216 ‘cfv 5926 (class class class)co 6690 tpos ctpos 7396 ndxcnx 15901 sSet csts 15902 Basecbs 15904 .rcmulr 15989 opprcoppr 18668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-i2m1 10042 ax-1ne0 10043 ax-rrecex 10046 ax-cnre 10047 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-nn 11059 df-2 11117 df-3 11118 df-ndx 15907 df-slot 15908 df-sets 15911 df-mulr 16002 df-oppr 18669 |
This theorem is referenced by: opprmul 18672 |
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