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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 7157 | . 2 ⊢ (𝑥 = 𝑋 → (𝑥 · (𝐼‘𝑦)) = (𝑋 · (𝐼‘𝑦))) | |
2 | fveq2 6665 | . . 3 ⊢ (𝑦 = 𝑌 → (𝐼‘𝑦) = (𝐼‘𝑌)) | |
3 | 2 | oveq2d 7166 | . 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 19428 | . 2 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) |
10 | ovex 7183 | . 2 ⊢ (𝑋 · (𝐼‘𝑌)) ∈ V | |
11 | 1, 3, 9, 10 | ovmpo 7304 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 .rcmulr 16560 Unitcui 19383 invrcinvr 19415 /rcdvr 19426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-dvr 19427 |
This theorem is referenced by: dvrcl 19430 unitdvcl 19431 dvrid 19432 dvr1 19433 dvrass 19434 dvrcan1 19435 ringinvdv 19438 subrgdv 19546 abvdiv 19602 cnflddiv 20569 nmdvr 23273 sum2dchr 25844 dvrdir 30856 rdivmuldivd 30857 dvrcan5 30859 |
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