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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 12618 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
| 2 | zcn 12618 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
| 3 | mulcom 11241 | . . 3 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑁 · 𝑀) = (𝑀 · 𝑁)) | |
| 4 | 1, 2, 3 | syl2anr 597 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 · 𝑀) = (𝑀 · 𝑁)) |
| 5 | zmulcl 12666 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | |
| 6 | dvds0lem 16304 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) ∧ (𝑁 · 𝑀) = (𝑀 · 𝑁)) → 𝑀 ∥ (𝑀 · 𝑁)) | |
| 7 | 6 | ex 412 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → ((𝑁 · 𝑀) = (𝑀 · 𝑁) → 𝑀 ∥ (𝑀 · 𝑁))) |
| 8 | 7 | 3com12 1124 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → ((𝑁 · 𝑀) = (𝑀 · 𝑁) → 𝑀 ∥ (𝑀 · 𝑁))) |
| 9 | 5, 8 | mpd3an3 1464 | . 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℂcc 11153 · cmul 11160 ℤcz 12613 ∥ cdvds 16290 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-dvds 16291 |
| This theorem is referenced by: dvdsmultr1 16333 3dvdsdec 16369 3dvds2dec 16370 2teven 16392 opoe 16400 omoe 16401 z4even 16409 ndvdsi 16449 bits0e 16466 bits0o 16467 mulgcd 16585 dvdsmulgcd 16593 lcmcllem 16633 lcmgcdlem 16643 qredeq 16694 cncongr2 16705 nprm 16725 exprmfct 16741 prmdiv 16822 iserodd 16873 difsqpwdvds 16925 expnprm 16940 pockthlem 16943 prmreclem3 16956 4sqlem14 16996 odmulg2 19573 odbezout 19576 gexdvds 19602 sylow2alem2 19636 odadd1 19866 odadd2 19867 gexexlem 19870 prmirredlem 21483 znunit 21582 wilthlem2 27112 dvdsflf1o 27230 mpodvdsmulf1o 27237 dvdsmulf1o 27239 ppiublem1 27246 perfectlem1 27273 bposlem3 27330 lgsdir 27376 lgsquadlem1 27424 lgsquad2lem1 27428 lgsquad2lem2 27429 2lgsoddprmlem2 27453 2lgsoddprmlem3 27458 2sqlem4 27465 2sqblem 27475 2sqmod 27480 dchrisumlem1 27533 ex-ind-dvds 30480 jm2.23 43008 jm2.27c 43019 inductionexd 44168 fouriersw 46246 etransclem24 46273 etransclem28 46277 2pwp1prm 47576 m2even 47641 perfectALTVlem1 47708 |
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