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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 6906 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | |
| 3 | dvrval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
| 4 | 2, 3 | eqtr4di 2795 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵) |
| 5 | fveq2 6906 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
| 6 | dvrval.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
| 7 | 5, 6 | eqtr4di 2795 | . . . . 5 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
| 8 | fveq2 6906 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅)) | |
| 9 | dvrval.t | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
| 10 | 8, 9 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · ) |
| 11 | eqidd 2738 | . . . . . 6 ⊢ (𝑟 = 𝑅 → 𝑥 = 𝑥) | |
| 12 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = (invr‘𝑅)) | |
| 13 | dvrval.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝑅) | |
| 14 | 12, 13 | eqtr4di 2795 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (invr‘𝑟) = 𝐼) |
| 15 | 14 | fveq1d 6908 | . . . . . 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 20401 | . . . 4 ⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)))) | |
| 19 | 3 | fvexi 6920 | . . . . 5 ⊢ 𝐵 ∈ V |
| 20 | 6 | fvexi 6920 | . . . . 5 ⊢ 𝑈 ∈ V |
| 21 | 19, 20 | mpoex 8104 | . . . 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 6898 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (/r‘𝑅) = ∅) | |
| 24 | fvprc 6898 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅) | |
| 25 | 3, 24 | eqtrid 2789 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅) |
| 26 | 25 | orcd 874 | . . . . 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 2780 | . . 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 2765 | 1 ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17247 .rcmulr 17298 Unitcui 20355 invrcinvr 20387 /rcdvr 20400 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-dvr 20401 |
| This theorem is referenced by: dvrval 20403 cnflddiv 21413 cnflddivOLD 21414 dvrcn 24192 |
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