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Mirrors > Home > MPE Home > Th. List > dvdsmul1 | Structured version Visualization version GIF version |
Description: An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvdsmul1 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 · 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 11987 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | zcn 11987 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
3 | mulcom 10623 | . . 3 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑁 · 𝑀) = (𝑀 · 𝑁)) | |
4 | 1, 2, 3 | syl2anr 598 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 · 𝑀) = (𝑀 · 𝑁)) |
5 | zmulcl 12032 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | |
6 | dvds0lem 15620 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) ∧ (𝑁 · 𝑀) = (𝑀 · 𝑁)) → 𝑀 ∥ (𝑀 · 𝑁)) | |
7 | 6 | ex 415 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → ((𝑁 · 𝑀) = (𝑀 · 𝑁) → 𝑀 ∥ (𝑀 · 𝑁))) |
8 | 7 | 3com12 1119 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → ((𝑁 · 𝑀) = (𝑀 · 𝑁) → 𝑀 ∥ (𝑀 · 𝑁))) |
9 | 5, 8 | mpd3an3 1458 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 · 𝑀) = (𝑀 · 𝑁) → 𝑀 ∥ (𝑀 · 𝑁))) |
10 | 4, 9 | mpd 15 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 · 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℂcc 10535 · cmul 10542 ℤcz 11982 ∥ cdvds 15607 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
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 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 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-dvds 15608 |
This theorem is referenced by: dvdsmultr1 15647 3dvdsdec 15681 3dvds2dec 15682 2teven 15704 opoe 15712 omoe 15713 z4even 15723 ndvdsi 15763 bits0e 15778 bits0o 15779 mulgcd 15896 dvdsmulgcd 15905 lcmcllem 15940 lcmgcdlem 15950 qredeq 16001 cncongr2 16012 nprm 16032 exprmfct 16048 prmdiv 16122 iserodd 16172 difsqpwdvds 16223 expnprm 16238 pockthlem 16241 prmreclem3 16254 4sqlem14 16294 odmulg2 18682 odbezout 18685 gexdvds 18709 sylow2alem2 18743 odadd1 18968 odadd2 18969 gexexlem 18972 prmirredlem 20640 znunit 20710 wilthlem2 25646 dvdsflf1o 25764 dvdsmulf1o 25771 ppiublem1 25778 perfectlem1 25805 bposlem3 25862 lgsdir 25908 lgsquadlem1 25956 lgsquad2lem1 25960 lgsquad2lem2 25961 2lgsoddprmlem2 25985 2lgsoddprmlem3 25990 2sqlem4 25997 2sqblem 26007 2sqmod 26012 dchrisumlem1 26065 ex-ind-dvds 28240 jm2.23 39613 jm2.27c 39624 inductionexd 40525 fouriersw 42536 etransclem24 42563 etransclem28 42567 2pwp1prm 43771 m2even 43839 perfectALTVlem1 43906 |
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