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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 6664 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
3 | dvrval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
4 | 2, 3 | syl6eqr 2874 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
5 | fveq2 6664 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
6 | dvrval.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
7 | 5, 6 | syl6eqr 2874 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
8 | fveq2 6664 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅)) | |
9 | dvrval.t | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
10 | 8, 9 | syl6eqr 2874 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · ) |
11 | eqidd 2822 | . . . . . 6 ⊢ (𝑟 = 𝑅 → 𝑥 = 𝑥) | |
12 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = (invr‘𝑅)) | |
13 | dvrval.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝑅) | |
14 | 12, 13 | syl6eqr 2874 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = 𝐼) |
15 | 14 | fveq1d 6666 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ((invr‘𝑟)‘𝑦) = (𝐼‘𝑦)) |
16 | 10, 11, 15 | oveq123d 7171 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)) = (𝑥 · (𝐼‘𝑦))) |
17 | 4, 7, 16 | mpoeq123dv 7223 | . . . 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 19427 | . . . 4 ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)))) | |
19 | 3 | fvexi 6678 | . . . . 5 ⊢ 𝐵 ∈ V |
20 | 6 | fvexi 6678 | . . . . 5 ⊢ 𝑈 ∈ V |
21 | 19, 20 | mpoex 7771 | . . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) ∈ V |
22 | 17, 18, 21 | fvmpt 6762 | . . 3 ⊢ (𝑅 ∈ V → (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))) |
23 | fvprc 6657 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (/r‘𝑅) = ∅) | |
24 | fvprc 6657 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅) | |
25 | 3, 24 | syl5eq 2868 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅) |
26 | 25 | orcd 869 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝐵 = ∅ ∨ 𝑈 = ∅)) |
27 | 0mpo0 7231 | . . . . 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 2859 | . . 3 ⊢ (¬ 𝑅 ∈ V → (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦)))) |
30 | 22, 29 | pm2.61i 184 | . 2 ⊢ (/r‘𝑅) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) |
31 | 1, 30 | eqtri 2844 | 1 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 843 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-dvr 19427 |
This theorem is referenced by: dvrval 19429 cnflddiv 20569 dvrcn 22786 |
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