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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 20311 | . . 3 ⊢ Rel ∥ |
3 | 2 | brrelex12i 5733 | . 2 ⊢ (𝑋 ∥ 𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
4 | elex 3480 | . . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ V) | |
5 | id 22 | . . . . 5 ⊢ ((𝑧 · 𝑋) = 𝑌 → (𝑧 · 𝑋) = 𝑌) | |
6 | ovex 7452 | . . . . 5 ⊢ (𝑧 · 𝑋) ∈ V | |
7 | 5, 6 | eqeltrrdi 2834 | . . . 4 ⊢ ((𝑧 · 𝑋) = 𝑌 → 𝑌 ∈ V) |
8 | 7 | rexlimivw 3140 | . . 3 ⊢ (∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌 → 𝑌 ∈ V) |
9 | 4, 8 | anim12i 611 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
10 | simpl 481 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) | |
11 | 10 | eleq1d 2810 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵)) |
12 | 10 | oveq2d 7435 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑧 · 𝑥) = (𝑧 · 𝑋)) |
13 | simpr 483 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
14 | 12, 13 | eqeq12d 2741 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧 · 𝑋) = 𝑌)) |
15 | 14 | rexbidv 3168 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) |
16 | 11, 15 | anbi12d 630 | . . 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 20312 | . . 3 ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} |
20 | 16, 19 | brabga 5536 | . 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌))) |
21 | 3, 9, 20 | pm5.21nii 377 | 1 ⊢ (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 Vcvv 3461 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 .rcmulr 17237 ∥rcdsr 20305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-dvdsr 20308 |
This theorem is referenced by: dvdsr2 20314 dvdsrmul 20315 dvdsrcl 20316 dvdsrcl2 20317 dvdsrtr 20319 dvdsrmul1 20320 opprunit 20328 crngunit 20329 rhmdvdsr 20459 subrgdvds 20537 dvdsruassoi 33196 dvdsruasso 33197 dvdsrspss 33199 rprmasso2 33338 unitmulrprm 33340 rprmirredlem 33342 1arithufdlem3 33361 rhmqusspan 41788 |
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