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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 12596 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | zcn 12596 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
3 | mulcom 11226 | . . 3 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑁 · 𝑀) = (𝑀 · 𝑁)) | |
4 | 1, 2, 3 | syl2anr 595 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 · 𝑀) = (𝑀 · 𝑁)) |
5 | zmulcl 12644 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | |
6 | dvds0lem 16247 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) ∧ (𝑁 · 𝑀) = (𝑀 · 𝑁)) → 𝑀 ∥ (𝑀 · 𝑁)) | |
7 | 6 | ex 411 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → ((𝑁 · 𝑀) = (𝑀 · 𝑁) → 𝑀 ∥ (𝑀 · 𝑁))) |
8 | 7 | 3com12 1120 | . . 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 (class class class)co 7419 ℂcc 11138 · cmul 11145 ℤcz 12591 ∥ cdvds 16234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-dvds 16235 |
This theorem is referenced by: dvdsmultr1 16276 3dvdsdec 16312 3dvds2dec 16313 2teven 16335 opoe 16343 omoe 16344 z4even 16352 ndvdsi 16392 bits0e 16407 bits0o 16408 mulgcd 16527 dvdsmulgcd 16534 lcmcllem 16570 lcmgcdlem 16580 qredeq 16631 cncongr2 16642 nprm 16662 exprmfct 16678 prmdiv 16757 iserodd 16807 difsqpwdvds 16859 expnprm 16874 pockthlem 16877 prmreclem3 16890 4sqlem14 16930 odmulg2 19522 odbezout 19525 gexdvds 19551 sylow2alem2 19585 odadd1 19815 odadd2 19816 gexexlem 19819 prmirredlem 21415 znunit 21514 wilthlem2 27046 dvdsflf1o 27164 mpodvdsmulf1o 27171 dvdsmulf1o 27173 ppiublem1 27180 perfectlem1 27207 bposlem3 27264 lgsdir 27310 lgsquadlem1 27358 lgsquad2lem1 27362 lgsquad2lem2 27363 2lgsoddprmlem2 27387 2lgsoddprmlem3 27392 2sqlem4 27399 2sqblem 27409 2sqmod 27414 dchrisumlem1 27467 ex-ind-dvds 30343 jm2.23 42559 jm2.27c 42570 inductionexd 43727 fouriersw 45757 etransclem24 45784 etransclem28 45788 2pwp1prm 47066 m2even 47131 perfectALTVlem1 47198 |
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