Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 18467 | . . 3 ⊢ Rel ∥ |
3 | brrelex12 5079 | . . 3 ⊢ ((Rel ∥ ∧ 𝑋 ∥ 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V)) | |
4 | 2, 3 | mpan 702 | . 2 ⊢ (𝑋 ∥ 𝑌 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
5 | elex 3185 | . . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ V) | |
6 | id 22 | . . . . 5 ⊢ ((𝑧 · 𝑋) = 𝑌 → (𝑧 · 𝑋) = 𝑌) | |
7 | ovex 6577 | . . . . 5 ⊢ (𝑧 · 𝑋) ∈ V | |
8 | 6, 7 | syl6eqelr 2697 | . . . 4 ⊢ ((𝑧 · 𝑋) = 𝑌 → 𝑌 ∈ V) |
9 | 8 | rexlimivw 3011 | . . 3 ⊢ (∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌 → 𝑌 ∈ V) |
10 | 5, 9 | anim12i 588 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
11 | simpl 472 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋) | |
12 | 11 | eleq1d 2672 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵)) |
13 | 11 | oveq2d 6565 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑧 · 𝑥) = (𝑧 · 𝑋)) |
14 | simpr 476 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌) | |
15 | 13, 14 | eqeq12d 2625 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑧 · 𝑥) = 𝑦 ↔ (𝑧 · 𝑋) = 𝑌)) |
16 | 15 | rexbidv 3034 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) |
17 | 12, 16 | anbi12d 743 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌))) |
18 | dvdsr.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
19 | dvdsr.3 | . . . 4 ⊢ · = (.r‘𝑅) | |
20 | 18, 1, 19 | dvdsrval 18468 | . . 3 ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} |
21 | 17, 20 | brabga 4914 | . 2 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌))) |
22 | 4, 10, 21 | pm5.21nii 367 | 1 ⊢ (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 class class class wbr 4583 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 .rcmulr 15769 ∥rcdsr 18461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-dvdsr 18464 |
This theorem is referenced by: dvdsr2 18470 dvdsrmul 18471 dvdsrcl 18472 dvdsrcl2 18473 dvdsrtr 18475 dvdsrmul1 18476 opprunit 18484 crngunit 18485 subrgdvds 18617 rhmdvdsr 29149 |
Copyright terms: Public domain | W3C validator |