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Mirrors > Home > MPE Home > Th. List > dvdsrmul | Structured version Visualization version GIF version |
Description: A left-multiple of 𝑋 is divisible by 𝑋. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
dvdsr.1 | ⊢ 𝐵 = (Base‘𝑅) |
dvdsr.2 | ⊢ ∥ = (∥r‘𝑅) |
dvdsr.3 | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
dvdsrmul | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∥ (𝑌 · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
2 | simpr 488 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
3 | eqid 2737 | . . 3 ⊢ (𝑌 · 𝑋) = (𝑌 · 𝑋) | |
4 | oveq1 7220 | . . . . 5 ⊢ (𝑧 = 𝑌 → (𝑧 · 𝑋) = (𝑌 · 𝑋)) | |
5 | 4 | eqeq1d 2739 | . . . 4 ⊢ (𝑧 = 𝑌 → ((𝑧 · 𝑋) = (𝑌 · 𝑋) ↔ (𝑌 · 𝑋) = (𝑌 · 𝑋))) |
6 | 5 | rspcev 3537 | . . 3 ⊢ ((𝑌 ∈ 𝐵 ∧ (𝑌 · 𝑋) = (𝑌 · 𝑋)) → ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = (𝑌 · 𝑋)) |
7 | 2, 3, 6 | sylancl 589 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = (𝑌 · 𝑋)) |
8 | dvdsr.1 | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
9 | dvdsr.2 | . . 3 ⊢ ∥ = (∥r‘𝑅) | |
10 | dvdsr.3 | . . 3 ⊢ · = (.r‘𝑅) | |
11 | 8, 9, 10 | dvdsr 19664 | . 2 ⊢ (𝑋 ∥ (𝑌 · 𝑋) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = (𝑌 · 𝑋))) |
12 | 1, 7, 11 | sylanbrc 586 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∥ (𝑌 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃wrex 3062 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 .rcmulr 16803 ∥rcdsr 19656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-dvdsr 19659 |
This theorem is referenced by: dvdsrid 19669 dvdsrtr 19670 dvdsrmul1 19671 dvdsrneg 19672 unitmulclb 19683 unitgrp 19685 isdrng2 19777 subrguss 19815 subrgunit 19818 fidomndrnglem 20344 invrvald 21573 dvdsq1p 25058 matunitlindflem2 35511 |
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