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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 12025 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | |
2 | eqid 2821 | . . 3 ⊢ (𝑀 · 𝑁) = (𝑀 · 𝑁) | |
3 | dvds0lem 15614 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) ∧ (𝑀 · 𝑁) = (𝑀 · 𝑁)) → 𝑁 ∥ (𝑀 · 𝑁)) | |
4 | 2, 3 | mpan2 689 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁)) |
5 | 1, 4 | mpd3an3 1458 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 · 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 (class class class)co 7150 · cmul 10536 ℤcz 11975 ∥ cdvds 15601 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-dvds 15602 |
This theorem is referenced by: iddvdsexp 15627 dvdsmultr2 15643 dvdsfac 15670 dvdsexp 15671 fprodfvdvdsd 15677 bitsinv1lem 15784 bitsuz 15817 bitsshft 15818 bezoutlem4 15884 dvdssqim 15898 lcmcllem 15934 qredeq 15995 cncongr1 16005 hashdvds 16106 phimullem 16110 difsqpwdvds 16217 oddprmdvds 16233 4sqlem8 16275 prmdvdsprmo 16372 dec2dvds 16393 lagsubg 18336 odadd2 18963 ppiublem1 25772 perfectlem2 25800 lgsdir2lem2 25896 lgsquadlem2 25951 lgsquadlem3 25952 lgsquad2lem1 25954 lgsquad2lem2 25955 2sqlem3 25990 2sqlem8 25996 clwwlkndivn 27853 dvdspw 32977 dvdsexpim 39174 jm2.19lem2 39580 jm2.23 39586 jm2.20nn 39587 jm2.25 39589 jm2.27a 39595 lighneallem4 43768 perfectALTVlem2 43880 |
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