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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 20335 | . . . . 5 ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
| 6 | 5 | fveq2i 6909 | . . . 4 ⊢ (.r‘𝑂) = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
| 7 | 3 | fvexi 6920 | . . . . . 6 ⊢ · ∈ V |
| 8 | 7 | tposex 8285 | . . . . 5 ⊢ tpos · ∈ V |
| 9 | mulridx 17338 | . . . . . 6 ⊢ .r = Slot (.r‘ndx) | |
| 10 | 9 | setsid 17244 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ tpos · ∈ V) → tpos · = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉))) |
| 11 | 8, 10 | mpan2 691 | . . . 4 ⊢ (𝑅 ∈ V → tpos · = (.r‘(𝑅 sSet 〈(.r‘ndx), tpos · 〉))) |
| 12 | 6, 11 | eqtr4id 2796 | . . 3 ⊢ (𝑅 ∈ V → (.r‘𝑂) = tpos · ) |
| 13 | tpos0 8281 | . . . . 5 ⊢ tpos ∅ = ∅ | |
| 14 | 9 | str0 17226 | . . . . 5 ⊢ ∅ = (.r‘∅) |
| 15 | 13, 14 | eqtr2i 2766 | . . . 4 ⊢ (.r‘∅) = tpos ∅ |
| 16 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = ∅) | |
| 17 | 4, 16 | eqtrid 2789 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅) |
| 18 | 17 | fveq2d 6910 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑂) = (.r‘∅)) |
| 19 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑅) = ∅) | |
| 20 | 3, 19 | eqtrid 2789 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → · = ∅) |
| 21 | 20 | tposeqd 8254 | . . . 4 ⊢ (¬ 𝑅 ∈ V → tpos · = tpos ∅) |
| 22 | 15, 18, 21 | 3eqtr4a 2803 | . . 3 ⊢ (¬ 𝑅 ∈ V → (.r‘𝑂) = tpos · ) |
| 23 | 12, 22 | pm2.61i 182 | . 2 ⊢ (.r‘𝑂) = tpos · |
| 24 | 1, 23 | eqtri 2765 | 1 ⊢ ∙ = tpos · |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 〈cop 4632 ‘cfv 6561 (class class class)co 7431 tpos ctpos 8250 sSet csts 17200 ndxcnx 17230 Basecbs 17247 .rcmulr 17298 opprcoppr 20333 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-mulr 17311 df-oppr 20334 |
| This theorem is referenced by: opprmul 20337 opprabs 33510 |
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