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Mirrors > Home > MPE Home > Th. List > dvdsr2 | 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 |
---|---|
dvdsr2 | ⊢ (𝑋 ∈ 𝐵 → (𝑋 ∥ 𝑌 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsr.1 | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | dvdsr.2 | . . 3 ⊢ ∥ = (∥r‘𝑅) | |
3 | dvdsr.3 | . . 3 ⊢ · = (.r‘𝑅) | |
4 | 1, 2, 3 | dvdsr 18469 | . 2 ⊢ (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) |
5 | 4 | baib 942 | 1 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ∥ 𝑌 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 ‘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: dvdsr01 18478 dvdsr02 18479 unitgrp 18490 rspsn 19075 znunit 19731 dvdsq1p 23724 rhmdvdsr 29149 |
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