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Mirrors > Home > MPE Home > Th. List > dvdscmul | Structured version Visualization version 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 1146 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) | |
2 | zmulcl 12034 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 · 𝑀) ∈ ℤ) | |
3 | 2 | 3adant3 1128 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 · 𝑀) ∈ ℤ) |
4 | zmulcl 12034 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 · 𝑁) ∈ ℤ) | |
5 | 4 | 3adant2 1127 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 · 𝑁) ∈ ℤ) |
6 | 3, 5 | jca 514 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) ∈ ℤ ∧ (𝐾 · 𝑁) ∈ ℤ)) |
7 | simpr 487 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → 𝑥 ∈ ℤ) | |
8 | zcn 11989 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
9 | zcn 11989 | . . . . . . . . 9 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ) | |
10 | zcn 11989 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
11 | mul12 10807 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝐾 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · (𝑥 · 𝑀))) | |
12 | 8, 9, 10, 11 | syl3an 1156 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · (𝑥 · 𝑀))) |
13 | 12 | 3coml 1123 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · (𝑥 · 𝑀))) |
14 | 13 | 3expa 1114 | . . . . . 6 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · (𝑥 · 𝑀))) |
15 | 14 | 3adantl3 1164 | . . . . 5 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · (𝑥 · 𝑀))) |
16 | oveq2 7166 | . . . . 5 ⊢ ((𝑥 · 𝑀) = 𝑁 → (𝐾 · (𝑥 · 𝑀)) = (𝐾 · 𝑁)) | |
17 | 15, 16 | sylan9eq 2878 | . . . 4 ⊢ ((((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) ∧ (𝑥 · 𝑀) = 𝑁) → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · 𝑁)) |
18 | 17 | ex 415 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → ((𝑥 · 𝑀) = 𝑁 → (𝑥 · (𝐾 · 𝑀)) = (𝐾 · 𝑁))) |
19 | 1, 6, 7, 18 | dvds1lem 15623 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 → (𝐾 · 𝑀) ∥ (𝐾 · 𝑁))) |
20 | 19 | 3coml 1123 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∥ 𝑁 → (𝐾 · 𝑀) ∥ (𝐾 · 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℂcc 10537 · cmul 10544 ℤcz 11984 ∥ cdvds 15609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-dvds 15610 |
This theorem is referenced by: dvdscmulr 15640 mulgcd 15898 dvdsmulgcd 15907 rpmulgcd2 16002 pcprendvds2 16180 pcpremul 16182 prmreclem1 16254 sylow3lem4 18757 ablfacrp2 19191 dvdsmulf1o 25773 jm2.27a 39609 jm2.27c 39611 |
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