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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 6842 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
3 | dvrval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 2, 3 | eqtr4di 2794 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
5 | fveq2 6842 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
6 | dvrval.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
7 | 5, 6 | eqtr4di 2794 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
8 | fveq2 6842 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅)) | |
9 | dvrval.t | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
10 | 8, 9 | eqtr4di 2794 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · ) |
11 | eqidd 2737 | . . . . . 6 ⊢ (𝑟 = 𝑅 → 𝑥 = 𝑥) | |
12 | fveq2 6842 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = (invr‘𝑅)) | |
13 | dvrval.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝑅) | |
14 | 12, 13 | eqtr4di 2794 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = 𝐼) |
15 | 14 | fveq1d 6844 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ((invr‘𝑟)‘𝑦) = (𝐼‘𝑦)) |
16 | 10, 11, 15 | oveq123d 7378 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)) = (𝑥 · (𝐼‘𝑦))) |
17 | 4, 7, 16 | mpoeq123dv 7432 | . . . 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 20112 | . . . 4 ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)))) | |
19 | 3 | fvexi 6856 | . . . . 5 ⊢ 𝐵 ∈ V |
20 | 6 | fvexi 6856 | . . . . 5 ⊢ 𝑈 ∈ V |
21 | 19, 20 | mpoex 8012 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) ∈ V |
22 | 17, 18, 21 | fvmpt 6948 | . . 3 ⊢ (𝑅 ∈ V → (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))) |
23 | fvprc 6834 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (/r‘𝑅) = ∅) | |
24 | fvprc 6834 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅) | |
25 | 3, 24 | eqtrid 2788 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅) |
26 | 25 | orcd 871 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝐵 = ∅ ∨ 𝑈 = ∅)) |
27 | 0mpo0 7440 | . . . . 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 2779 | . . 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 2764 | 1 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 845 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ∅c0 4282 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 Basecbs 17083 .rcmulr 17134 Unitcui 20068 invrcinvr 20100 /rcdvr 20111 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-dvr 20112 |
This theorem is referenced by: dvrval 20114 cnflddiv 20827 dvrcn 23535 |
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