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Mirrors > Home > MPE Home > Th. List > dvdsmul2 | Structured version Visualization version GIF version |
Description: An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvdsmul2 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zmulcl 12358 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | |
2 | eqid 2738 | . . 3 ⊢ (𝑀 · 𝑁) = (𝑀 · 𝑁) | |
3 | dvds0lem 15965 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) ∧ (𝑀 · 𝑁) = (𝑀 · 𝑁)) → 𝑁 ∥ (𝑀 · 𝑁)) | |
4 | 2, 3 | mpan2 688 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁)) |
5 | 1, 4 | mpd3an3 1461 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 class class class wbr 5075 (class class class)co 7269 · cmul 10865 ℤcz 12308 ∥ cdvds 15952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8487 df-en 8723 df-dom 8724 df-sdom 8725 df-pnf 11000 df-mnf 11001 df-ltxr 11003 df-sub 11196 df-neg 11197 df-nn 11963 df-n0 12223 df-z 12309 df-dvds 15953 |
This theorem is referenced by: iddvdsexp 15978 dvdsmultr2 15996 dvdsfac 16024 dvdsexp2im 16025 dvdsexp 16026 fprodfvdvdsd 16032 bitsinv1lem 16137 bitsuz 16170 bitsshft 16171 bezoutlem4 16239 dvdssqim 16253 lcmcllem 16290 qredeq 16351 cncongr1 16361 hashdvds 16465 phimullem 16469 difsqpwdvds 16577 oddprmdvds 16593 4sqlem8 16635 prmdvdsprmo 16732 dec2dvds 16753 lagsubg 18807 odadd2 19439 ppiublem1 26339 perfectlem2 26367 lgsdir2lem2 26463 lgsquadlem2 26518 lgsquadlem3 26519 lgsquad2lem1 26521 lgsquad2lem2 26522 2sqlem3 26557 2sqlem8 26563 clwwlkndivn 28431 dvdsexpim 40315 jm2.19lem2 40799 jm2.23 40805 jm2.20nn 40806 jm2.25 40808 jm2.27a 40814 lighneallem4 45019 perfectALTVlem2 45131 |
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