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Mirrors > Home > ILE Home > Th. List > dvdsrmul1 | 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 | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
2 | 1 | a1i 9 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) → 𝐵 = (Base‘𝑅)) |
3 | dvdsr.2 | . . . . 5 ⊢ ∥ = (∥r‘𝑅) | |
4 | 3 | a1i 9 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) → ∥ = (∥r‘𝑅)) |
5 | ringsrg 13177 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing) | |
6 | 5 | adantr 276 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) → 𝑅 ∈ SRing) |
7 | dvdsrmul1.3 | . . . . 5 ⊢ · = (.r‘𝑅) | |
8 | 7 | a1i 9 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) → · = (.r‘𝑅)) |
9 | 2, 4, 6, 8 | dvdsrd 13216 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) → (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥 · 𝑋) = 𝑌))) |
10 | 1 | a1i 9 | . . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐵 = (Base‘𝑅)) |
11 | 3 | a1i 9 | . . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ∥ = (∥r‘𝑅)) |
12 | simplll 533 | . . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ Ring) | |
13 | 12, 5 | syl 14 | . . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ SRing) |
14 | 7 | a1i 9 | . . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → · = (.r‘𝑅)) |
15 | simplr 528 | . . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
16 | simpllr 534 | . . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑍 ∈ 𝐵) | |
17 | 1, 7 | ringcl 13149 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 · 𝑍) ∈ 𝐵) |
18 | 12, 15, 16, 17 | syl3anc 1238 | . . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑋 · 𝑍) ∈ 𝐵) |
19 | simpr 110 | . . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
20 | 10, 11, 13, 14, 18, 19 | dvdsrmuld 13218 | . . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑋 · 𝑍) ∥ (𝑥 · (𝑋 · 𝑍))) |
21 | 1, 7 | ringass 13152 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑥 · 𝑋) · 𝑍) = (𝑥 · (𝑋 · 𝑍))) |
22 | 12, 19, 15, 16, 21 | syl13anc 1240 | . . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 · 𝑋) · 𝑍) = (𝑥 · (𝑋 · 𝑍))) |
23 | 20, 22 | breqtrrd 4031 | . . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑋 · 𝑍) ∥ ((𝑥 · 𝑋) · 𝑍)) |
24 | oveq1 5881 | . . . . . . 7 ⊢ ((𝑥 · 𝑋) = 𝑌 → ((𝑥 · 𝑋) · 𝑍) = (𝑌 · 𝑍)) | |
25 | 24 | breq2d 4015 | . . . . . 6 ⊢ ((𝑥 · 𝑋) = 𝑌 → ((𝑋 · 𝑍) ∥ ((𝑥 · 𝑋) · 𝑍) ↔ (𝑋 · 𝑍) ∥ (𝑌 · 𝑍))) |
26 | 23, 25 | syl5ibcom 155 | . . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 · 𝑋) = 𝑌 → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍))) |
27 | 26 | rexlimdva 2594 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥 · 𝑋) = 𝑌 → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍))) |
28 | 27 | expimpd 363 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) → ((𝑋 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥 · 𝑋) = 𝑌) → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍))) |
29 | 9, 28 | sylbid 150 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵) → (𝑋 ∥ 𝑌 → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍))) |
30 | 29 | 3impia 1200 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∥ 𝑌) → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 class class class wbr 4003 ‘cfv 5216 (class class class)co 5874 Basecbs 12456 .rcmulr 12531 SRingcsrg 13099 Ringcrg 13132 ∥rcdsr 13208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-pre-ltirr 7922 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-inn 8918 df-2 8976 df-3 8977 df-ndx 12459 df-slot 12460 df-base 12462 df-sets 12463 df-plusg 12543 df-mulr 12544 df-0g 12697 df-mgm 12729 df-sgrp 12762 df-mnd 12772 df-grp 12834 df-minusg 12835 df-cmn 13043 df-abl 13044 df-mgp 13084 df-ur 13096 df-srg 13100 df-ring 13134 df-dvdsr 13211 |
This theorem is referenced by: unitmulcl 13235 |
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