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Mirrors > Home > MPE Home > Th. List > dvdsr | Structured version Visualization version GIF version |
Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
dvdsr.1 | ⊢ 𝐵 = (Base‘𝑅) |
dvdsr.2 | ⊢ ∥ = (∥r‘𝑅) |
dvdsr.3 | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
dvdsr | ⊢ (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsr.2 | . . . 4 ⊢ ∥ = (∥r‘𝑅) | |
2 | 1 | reldvdsr 19390 | . . 3 ⊢ Rel ∥ |
3 | 2 | brrelex12i 5571 | . 2 ⊢ (𝑋 ∥ 𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
4 | elex 3459 | . . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ V) | |
5 | id 22 | . . . . 5 ⊢ ((𝑧 · 𝑋) = 𝑌 → (𝑧 · 𝑋) = 𝑌) | |
6 | ovex 7168 | . . . . 5 ⊢ (𝑧 · 𝑋) ∈ V | |
7 | 5, 6 | eqeltrrdi 2899 | . . . 4 ⊢ ((𝑧 · 𝑋) = 𝑌 → 𝑌 ∈ V) |
8 | 7 | rexlimivw 3241 | . . 3 ⊢ (∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌 → 𝑌 ∈ V) |
9 | 4, 8 | anim12i 615 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
10 | simpl 486 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) | |
11 | 10 | eleq1d 2874 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵)) |
12 | 10 | oveq2d 7151 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑧 · 𝑥) = (𝑧 · 𝑋)) |
13 | simpr 488 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
14 | 12, 13 | eqeq12d 2814 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧 · 𝑋) = 𝑌)) |
15 | 14 | rexbidv 3256 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) |
16 | 11, 15 | anbi12d 633 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌))) |
17 | dvdsr.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
18 | dvdsr.3 | . . . 4 ⊢ · = (.r‘𝑅) | |
19 | 17, 1, 18 | dvdsrval 19391 | . . 3 ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} |
20 | 16, 19 | brabga 5386 | . 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌))) |
21 | 3, 9, 20 | pm5.21nii 383 | 1 ⊢ (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 .rcmulr 16558 ∥rcdsr 19384 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-dvdsr 19387 |
This theorem is referenced by: dvdsr2 19393 dvdsrmul 19394 dvdsrcl 19395 dvdsrcl2 19396 dvdsrtr 19398 dvdsrmul1 19399 opprunit 19407 crngunit 19408 subrgdvds 19542 rhmdvdsr 30942 |
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