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Mirrors > Home > ILE Home > Th. List > dvdscmul | GIF version |
Description: Multiplication by a constant maintains the divides relation. Theorem 1.1(d) in [ApostolNT] p. 14 (multiplication property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvdscmul | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∥ 𝑁 → (𝐾 · 𝑀) ∥ (𝐾 · 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpc 980 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) | |
2 | zmulcl 9100 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 · 𝑀) ∈ ℤ) | |
3 | 2 | 3adant3 1001 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 · 𝑀) ∈ ℤ) |
4 | zmulcl 9100 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 · 𝑁) ∈ ℤ) | |
5 | 4 | 3adant2 1000 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 · 𝑁) ∈ ℤ) |
6 | 3, 5 | jca 304 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) ∈ ℤ ∧ (𝐾 · 𝑁) ∈ ℤ)) |
7 | simpr 109 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℤ) | |
8 | zcn 9052 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
9 | zcn 9052 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ) | |
10 | zcn 9052 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
11 | mul12 7884 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝐾 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · (𝑥 · 𝑀))) | |
12 | 8, 9, 10, 11 | syl3an 1258 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · (𝑥 · 𝑀))) |
13 | 12 | 3coml 1188 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · (𝑥 · 𝑀))) |
14 | 13 | 3expa 1181 | . . . . . 6 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · (𝑥 · 𝑀))) |
15 | 14 | 3adantl3 1139 | . . . . 5 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · (𝑥 · 𝑀))) |
16 | oveq2 5775 | . . . . 5 ⊢ ((𝑥 · 𝑀) = 𝑁 → (𝐾 · (𝑥 · 𝑀)) = (𝐾 · 𝑁)) | |
17 | 15, 16 | sylan9eq 2190 | . . . 4 ⊢ ((((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) ∧ (𝑥 · 𝑀) = 𝑁) → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · 𝑁)) |
18 | 17 | ex 114 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝑀) = 𝑁 → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · 𝑁))) |
19 | 1, 6, 7, 18 | dvds1lem 11493 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 → (𝐾 · 𝑀) ∥ (𝐾 · 𝑁))) |
20 | 19 | 3coml 1188 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∥ 𝑁 → (𝐾 · 𝑀) ∥ (𝐾 · 𝑁))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 ℂcc 7611 · cmul 7618 ℤcz 9047 ∥ cdvds 11482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-dvds 11483 |
This theorem is referenced by: dvdscmulr 11511 mulgcd 11693 dvdsmulgcd 11702 rpmulgcd2 11765 |
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