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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 6907 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
3 | dvrval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 2, 3 | eqtr4di 2793 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
5 | fveq2 6907 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
6 | dvrval.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
7 | 5, 6 | eqtr4di 2793 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
8 | fveq2 6907 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅)) | |
9 | dvrval.t | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
10 | 8, 9 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · ) |
11 | eqidd 2736 | . . . . . 6 ⊢ (𝑟 = 𝑅 → 𝑥 = 𝑥) | |
12 | fveq2 6907 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = (invr‘𝑅)) | |
13 | dvrval.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝑅) | |
14 | 12, 13 | eqtr4di 2793 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = 𝐼) |
15 | 14 | fveq1d 6909 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ((invr‘𝑟)‘𝑦) = (𝐼‘𝑦)) |
16 | 10, 11, 15 | oveq123d 7452 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)) = (𝑥 · (𝐼‘𝑦))) |
17 | 4, 7, 16 | mpoeq123dv 7508 | . . . 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 20418 | . . . 4 ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)))) | |
19 | 3 | fvexi 6921 | . . . . 5 ⊢ 𝐵 ∈ V |
20 | 6 | fvexi 6921 | . . . . 5 ⊢ 𝑈 ∈ V |
21 | 19, 20 | mpoex 8103 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) ∈ V |
22 | 17, 18, 21 | fvmpt 7016 | . . 3 ⊢ (𝑅 ∈ V → (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))) |
23 | fvprc 6899 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (/r‘𝑅) = ∅) | |
24 | fvprc 6899 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅) | |
25 | 3, 24 | eqtrid 2787 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅) |
26 | 25 | orcd 873 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝐵 = ∅ ∨ 𝑈 = ∅)) |
27 | 0mpo0 7516 | . . . . 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 2778 | . . 3 ⊢ (¬ 𝑅 ∈ V → (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))) |
30 | 22, 29 | pm2.61i 182 | . 2 ⊢ (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) |
31 | 1, 30 | eqtri 2763 | 1 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 847 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17245 .rcmulr 17299 Unitcui 20372 invrcinvr 20404 /rcdvr 20417 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-dvr 20418 |
This theorem is referenced by: dvrval 20420 cnflddiv 21431 cnflddivOLD 21432 dvrcn 24208 |
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