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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 7165 | . 2 ⊢ (𝑥 = 𝑋 → (𝑥 · (𝐼‘𝑦)) = (𝑋 · (𝐼‘𝑦))) | |
2 | fveq2 6672 | . . 3 ⊢ (𝑦 = 𝑌 → (𝐼‘𝑦) = (𝐼‘𝑌)) | |
3 | 2 | oveq2d 7174 | . 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 19436 | . 2 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) |
10 | ovex 7191 | . 2 ⊢ (𝑋 · (𝐼‘𝑌)) ∈ V | |
11 | 1, 3, 9, 10 | ovmpo 7312 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 .rcmulr 16568 Unitcui 19391 invrcinvr 19423 /rcdvr 19434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-dvr 19435 |
This theorem is referenced by: dvrcl 19438 unitdvcl 19439 dvrid 19440 dvr1 19441 dvrass 19442 dvrcan1 19443 ringinvdv 19446 subrgdv 19554 abvdiv 19610 cnflddiv 20577 nmdvr 23281 sum2dchr 25852 dvrdir 30863 rdivmuldivd 30864 dvrcan5 30866 |
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