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Mirrors > Home > ILE Home > Th. List > dvdsmulc | GIF version |
Description: Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvdsmulc | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∥ 𝑁 → (𝑀 · 𝐾) ∥ (𝑁 · 𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpc 998 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) | |
2 | zmulcl 9341 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 · 𝐾) ∈ ℤ) | |
3 | 2 | 3adant2 1018 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 · 𝐾) ∈ ℤ) |
4 | zmulcl 9341 | . . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 · 𝐾) ∈ ℤ) | |
5 | 4 | 3adant1 1017 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 · 𝐾) ∈ ℤ) |
6 | 3, 5 | jca 306 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑀 · 𝐾) ∈ ℤ ∧ (𝑁 · 𝐾) ∈ ℤ)) |
7 | 6 | 3comr 1213 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 · 𝐾) ∈ ℤ ∧ (𝑁 · 𝐾) ∈ ℤ)) |
8 | simpr 110 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℤ) | |
9 | zcn 9293 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
10 | zcn 9293 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
11 | zcn 9293 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ) | |
12 | mulass 7977 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝐾 ∈ ℂ) → ((𝑥 · 𝑀) · 𝐾) = (𝑥 · (𝑀 · 𝐾))) | |
13 | 9, 10, 11, 12 | syl3an 1291 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑥 · 𝑀) · 𝐾) = (𝑥 · (𝑀 · 𝐾))) |
14 | 13 | 3com13 1210 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝑀) · 𝐾) = (𝑥 · (𝑀 · 𝐾))) |
15 | 14 | 3expa 1205 | . . . . . 6 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝑀) · 𝐾) = (𝑥 · (𝑀 · 𝐾))) |
16 | 15 | 3adantl3 1157 | . . . . 5 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝑀) · 𝐾) = (𝑥 · (𝑀 · 𝐾))) |
17 | oveq1 5907 | . . . . 5 ⊢ ((𝑥 · 𝑀) = 𝑁 → ((𝑥 · 𝑀) · 𝐾) = (𝑁 · 𝐾)) | |
18 | 16, 17 | sylan9req 2243 | . . . 4 ⊢ ((((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) ∧ (𝑥 · 𝑀) = 𝑁) → (𝑥 · (𝑀 · 𝐾)) = (𝑁 · 𝐾)) |
19 | 18 | ex 115 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝑀) = 𝑁 → (𝑥 · (𝑀 · 𝐾)) = (𝑁 · 𝐾))) |
20 | 1, 7, 8, 19 | dvds1lem 11850 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 → (𝑀 · 𝐾) ∥ (𝑁 · 𝐾))) |
21 | 20 | 3coml 1212 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∥ 𝑁 → (𝑀 · 𝐾) ∥ (𝑁 · 𝐾))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 class class class wbr 4021 (class class class)co 5900 ℂcc 7844 · cmul 7851 ℤcz 9288 ∥ cdvds 11835 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-cnre 7957 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-iota 5199 df-fun 5240 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-sub 8165 df-neg 8166 df-inn 8955 df-n0 9212 df-z 9289 df-dvds 11836 |
This theorem is referenced by: dvdsmulcr 11869 coprmdvds2 12136 mulgcddvds 12137 rpmulgcd2 12138 pcpremul 12336 |
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