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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 11716 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | |
2 | eqid 2799 | . . 3 ⊢ (𝑀 · 𝑁) = (𝑀 · 𝑁) | |
3 | dvds0lem 15331 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) ∧ (𝑀 · 𝑁) = (𝑀 · 𝑁)) → 𝑁 ∥ (𝑀 · 𝑁)) | |
4 | 2, 3 | mpan2 683 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁)) |
5 | 1, 4 | mpd3an3 1587 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 class class class wbr 4843 (class class class)co 6878 · cmul 10229 ℤcz 11666 ∥ cdvds 15319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-ltxr 10368 df-sub 10558 df-neg 10559 df-nn 11313 df-n0 11581 df-z 11667 df-dvds 15320 |
This theorem is referenced by: iddvdsexp 15344 dvdsmultr2 15360 dvdsfac 15387 dvdsexp 15388 fprodfvdvdsd 15394 bitsinv1lem 15498 bitsuz 15531 bitsshft 15532 bezoutlem4 15594 dvdssqim 15608 lcmcllem 15644 qredeq 15705 cncongr1 15715 hashdvds 15813 phimullem 15817 difsqpwdvds 15924 oddprmdvds 15940 4sqlem8 15982 prmdvdsprmo 16079 dec2dvds 16100 lagsubg 17969 odadd2 18567 ppiublem1 25279 perfectlem2 25307 lgsdir2lem2 25403 lgsquadlem2 25458 lgsquadlem3 25459 lgsquad2lem1 25461 lgsquad2lem2 25462 2sqlem3 25497 2sqlem8 25503 clwwlkndivn 27398 dvdspw 32150 jm2.19lem2 38342 jm2.23 38348 jm2.20nn 38349 jm2.25 38351 jm2.27a 38357 lighneallem4 42309 perfectALTVlem2 42413 |
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