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Mirrors > Home > MPE Home > Th. List > dvrfval | 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.) (Proof shortened by AV, 2-Mar-2024.) |
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 |
---|---|
dvrfval | ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvrval.d | . 2 ⊢ / = (/r‘𝑅) | |
2 | fveq2 6663 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
3 | dvrval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 2, 3 | syl6eqr 2871 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
5 | fveq2 6663 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
6 | dvrval.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
7 | 5, 6 | syl6eqr 2871 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
8 | fveq2 6663 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅)) | |
9 | dvrval.t | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
10 | 8, 9 | syl6eqr 2871 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · ) |
11 | eqidd 2819 | . . . . . 6 ⊢ (𝑟 = 𝑅 → 𝑥 = 𝑥) | |
12 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = (invr‘𝑅)) | |
13 | dvrval.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝑅) | |
14 | 12, 13 | syl6eqr 2871 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = 𝐼) |
15 | 14 | fveq1d 6665 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ((invr‘𝑟)‘𝑦) = (𝐼‘𝑦)) |
16 | 10, 11, 15 | oveq123d 7166 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)) = (𝑥 · (𝐼‘𝑦))) |
17 | 4, 7, 16 | mpoeq123dv 7218 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))) |
18 | df-dvr 19362 | . . . 4 ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)))) | |
19 | 3 | fvexi 6677 | . . . . 5 ⊢ 𝐵 ∈ V |
20 | 6 | fvexi 6677 | . . . . 5 ⊢ 𝑈 ∈ V |
21 | 19, 20 | mpoex 7766 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) ∈ V |
22 | 17, 18, 21 | fvmpt 6761 | . . 3 ⊢ (𝑅 ∈ V → (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))) |
23 | fvprc 6656 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (/r‘𝑅) = ∅) | |
24 | fvprc 6656 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅) | |
25 | 3, 24 | syl5eq 2865 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅) |
26 | 25 | orcd 869 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝐵 = ∅ ∨ 𝑈 = ∅)) |
27 | 0mpo0 7226 | . . . . 5 ⊢ ((𝐵 = ∅ ∨ 𝑈 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) = ∅) | |
28 | 26, 27 | syl 17 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) = ∅) |
29 | 23, 28 | eqtr4d 2856 | . . 3 ⊢ (¬ 𝑅 ∈ V → (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))) |
30 | 22, 29 | pm2.61i 183 | . 2 ⊢ (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) |
31 | 1, 30 | eqtri 2841 | 1 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 841 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 Basecbs 16471 .rcmulr 16554 Unitcui 19318 invrcinvr 19350 /rcdvr 19361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-dvr 19362 |
This theorem is referenced by: dvrval 19364 cnflddiv 20503 dvrcn 22719 |
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