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Mirrors > Home > MPE Home > Th. List > dvrval | Structured version Visualization version GIF version |
Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
dvrval.b | ⊢ 𝐵 = (Base‘𝑅) |
dvrval.t | ⊢ · = (.r‘𝑅) |
dvrval.u | ⊢ 𝑈 = (Unit‘𝑅) |
dvrval.i | ⊢ 𝐼 = (invr‘𝑅) |
dvrval.d | ⊢ / = (/r‘𝑅) |
Ref | Expression |
---|---|
dvrval | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7142 | . 2 ⊢ (𝑥 = 𝑋 → (𝑥 · (𝐼‘𝑦)) = (𝑋 · (𝐼‘𝑦))) | |
2 | fveq2 6645 | . . 3 ⊢ (𝑦 = 𝑌 → (𝐼‘𝑦) = (𝐼‘𝑌)) | |
3 | 2 | oveq2d 7151 | . 2 ⊢ (𝑦 = 𝑌 → (𝑋 · (𝐼‘𝑦)) = (𝑋 · (𝐼‘𝑌))) |
4 | dvrval.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
5 | dvrval.t | . . 3 ⊢ · = (.r‘𝑅) | |
6 | dvrval.u | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
7 | dvrval.i | . . 3 ⊢ 𝐼 = (invr‘𝑅) | |
8 | dvrval.d | . . 3 ⊢ / = (/r‘𝑅) | |
9 | 4, 5, 6, 7, 8 | dvrfval 19430 | . 2 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) |
10 | ovex 7168 | . 2 ⊢ (𝑋 · (𝐼‘𝑌)) ∈ V | |
11 | 1, 3, 9, 10 | ovmpo 7289 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 .rcmulr 16558 Unitcui 19385 invrcinvr 19417 /rcdvr 19428 |
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-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 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-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 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-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-1st 7671 df-2nd 7672 df-dvr 19429 |
This theorem is referenced by: dvrcl 19432 unitdvcl 19433 dvrid 19434 dvr1 19435 dvrass 19436 dvrcan1 19437 ringinvdv 19440 subrgdv 19545 abvdiv 19601 cnflddiv 20121 nmdvr 23276 sum2dchr 25858 dvrdir 30912 rdivmuldivd 30913 dvrcan5 30915 |
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