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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 12034 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | |
2 | eqid 2823 | . . 3 ⊢ (𝑀 · 𝑁) = (𝑀 · 𝑁) | |
3 | dvds0lem 15622 | . . 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 1537 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 · 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: iddvdsexp 15635 dvdsmultr2 15651 dvdsfac 15678 dvdsexp 15679 fprodfvdvdsd 15685 bitsinv1lem 15792 bitsuz 15825 bitsshft 15826 bezoutlem4 15892 dvdssqim 15906 lcmcllem 15942 qredeq 16003 cncongr1 16013 hashdvds 16114 phimullem 16118 difsqpwdvds 16225 oddprmdvds 16241 4sqlem8 16283 prmdvdsprmo 16380 dec2dvds 16401 lagsubg 18344 odadd2 18971 ppiublem1 25780 perfectlem2 25808 lgsdir2lem2 25904 lgsquadlem2 25959 lgsquadlem3 25960 lgsquad2lem1 25962 lgsquad2lem2 25963 2sqlem3 25998 2sqlem8 26004 clwwlkndivn 27861 dvdspw 32984 dvdsexpim 39188 jm2.19lem2 39594 jm2.23 39600 jm2.20nn 39601 jm2.25 39603 jm2.27a 39609 lighneallem4 43782 perfectALTVlem2 43894 |
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