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Mirrors > Home > MPE Home > Th. List > muldvds1 | Structured version Visualization version GIF version |
Description: If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
muldvds1 | ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) ∥ 𝑁 → 𝐾 ∥ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zmulcl 12030 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 · 𝑀) ∈ ℤ) | |
2 | 1 | anim1i 616 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
3 | 2 | 3impa 1106 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
4 | 3simpb 1145 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ)) | |
5 | zmulcl 12030 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑥 · 𝑀) ∈ ℤ) | |
6 | 5 | ancoms 461 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝑀) ∈ ℤ) |
7 | 6 | 3ad2antl2 1182 | . 2 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → (𝑥 · 𝑀) ∈ ℤ) |
8 | zcn 11985 | . . . . . . . 8 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
9 | zcn 11985 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℂ) | |
10 | zcn 11985 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
11 | mulass 10624 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝐾 ∈ ℂ ∧ 𝑀 ∈ ℂ) → ((𝑥 · 𝐾) · 𝑀) = (𝑥 · (𝐾 · 𝑀))) | |
12 | mul32 10805 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝐾 ∈ ℂ ∧ 𝑀 ∈ ℂ) → ((𝑥 · 𝐾) · 𝑀) = ((𝑥 · 𝑀) · 𝐾)) | |
13 | 11, 12 | eqtr3d 2858 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝐾 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑥 · (𝐾 · 𝑀)) = ((𝑥 · 𝑀) · 𝐾)) |
14 | 8, 9, 10, 13 | syl3an 1156 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑥 · (𝐾 · 𝑀)) = ((𝑥 · 𝑀) · 𝐾)) |
15 | 14 | 3coml 1123 | . . . . . 6 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ) → (𝑥 · (𝐾 · 𝑀)) = ((𝑥 · 𝑀) · 𝐾)) |
16 | 15 | 3expa 1114 | . . . . 5 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → (𝑥 · (𝐾 · 𝑀)) = ((𝑥 · 𝑀) · 𝐾)) |
17 | 16 | 3adantl3 1164 | . . . 4 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → (𝑥 · (𝐾 · 𝑀)) = ((𝑥 · 𝑀) · 𝐾)) |
18 | 17 | eqeq1d 2823 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → ((𝑥 · (𝐾 · 𝑀)) = 𝑁 ↔ ((𝑥 · 𝑀) · 𝐾) = 𝑁)) |
19 | 18 | biimpd 231 | . 2 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑥 ∈ ℤ) → ((𝑥 · (𝐾 · 𝑀)) = 𝑁 → ((𝑥 · 𝑀) · 𝐾) = 𝑁)) |
20 | 3, 4, 7, 19 | dvds1lem 15620 | 1 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 · 𝑀) ∥ 𝑁 → 𝐾 ∥ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 (class class class)co 7155 ℂcc 10534 · cmul 10541 ℤcz 11980 ∥ cdvds 15606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-ltxr 10679 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-dvds 15607 |
This theorem is referenced by: 3dvds 15679 odmulg 18682 jm2.20nn 39592 jm2.27c 39602 nzss 40647 etransclem28 42546 fmtnofac2lem 43729 |
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