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Mirrors > Home > MPE Home > Th. List > dvdsrmul1 | Structured version Visualization version GIF version |
Description: The divisibility relation is preserved under right-multiplication. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
dvdsr.1 | ⊢ 𝐵 = (Base‘𝑅) |
dvdsr.2 | ⊢ ∥ = (∥r‘𝑅) |
dvdsrmul1.3 | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
dvdsrmul1 | ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∥ 𝑌) → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsr.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | dvdsr.2 | . . . 4 ⊢ ∥ = (∥r‘𝑅) | |
3 | dvdsrmul1.3 | . . . 4 ⊢ · = (.r‘𝑅) | |
4 | 1, 2, 3 | dvdsr 19392 | . . 3 ⊢ (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥 · 𝑋) = 𝑌)) |
5 | simplll 774 | . . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ Ring) | |
6 | simplr 768 | . . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
7 | simpllr 775 | . . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑍 ∈ 𝐵) | |
8 | 1, 3 | ringcl 19307 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 · 𝑍) ∈ 𝐵) |
9 | 5, 6, 7, 8 | syl3anc 1368 | . . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑋 · 𝑍) ∈ 𝐵) |
10 | 1, 2, 3 | dvdsrmul 19394 | . . . . . . . 8 ⊢ (((𝑋 · 𝑍) ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑋 · 𝑍) ∥ (𝑥 · (𝑋 · 𝑍))) |
11 | 9, 10 | sylancom 591 | . . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑋 · 𝑍) ∥ (𝑥 · (𝑋 · 𝑍))) |
12 | simpr 488 | . . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
13 | 1, 3 | ringass 19310 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑥 · 𝑋) · 𝑍) = (𝑥 · (𝑋 · 𝑍))) |
14 | 5, 12, 6, 7, 13 | syl13anc 1369 | . . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 · 𝑋) · 𝑍) = (𝑥 · (𝑋 · 𝑍))) |
15 | 11, 14 | breqtrrd 5058 | . . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑋 · 𝑍) ∥ ((𝑥 · 𝑋) · 𝑍)) |
16 | oveq1 7142 | . . . . . . 7 ⊢ ((𝑥 · 𝑋) = 𝑌 → ((𝑥 · 𝑋) · 𝑍) = (𝑌 · 𝑍)) | |
17 | 16 | breq2d 5042 | . . . . . 6 ⊢ ((𝑥 · 𝑋) = 𝑌 → ((𝑋 · 𝑍) ∥ ((𝑥 · 𝑋) · 𝑍) ↔ (𝑋 · 𝑍) ∥ (𝑌 · 𝑍))) |
18 | 15, 17 | syl5ibcom 248 | . . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 · 𝑋) = 𝑌 → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍))) |
19 | 18 | rexlimdva 3243 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥 · 𝑋) = 𝑌 → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍))) |
20 | 19 | expimpd 457 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) → ((𝑋 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥 · 𝑋) = 𝑌) → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍))) |
21 | 4, 20 | syl5bi 245 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) → (𝑋 ∥ 𝑌 → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍))) |
22 | 21 | 3impia 1114 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∥ 𝑌) → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 .rcmulr 16558 Ringcrg 19290 ∥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 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 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-nel 3092 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-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 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-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 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-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mgp 19233 df-ring 19292 df-dvdsr 19387 |
This theorem is referenced by: unitmulcl 19410 |
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