Theorem dvdslcm 11543

Description: The lcm of two integers is divisible by each of them. (Contributed by Steve Rodriguez, 20-Jan-2020.)

Assertion

Ref	Expression
dvdslcm	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )

Proof of Theorem dvdslcm

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvds0 11303	. . . . 5 |- ( M e. ZZ -> M || 0 )
2	1	ad2antrr 475	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> M || 0 )
3		oveq1 5713	. . . . . . 7 |- ( M = 0 -> ( M lcm N ) = ( 0 lcm N ) )
4		0z 8917	. . . . . . . . 9 |- 0 e. ZZ
5		lcmcom 11538	. . . . . . . . 9 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N lcm 0 ) = ( 0 lcm N ) )
6	4, 5	mpan2 419	. . . . . . . 8 |- ( N e. ZZ -> ( N lcm 0 ) = ( 0 lcm N ) )
7		lcm0val 11539	. . . . . . . 8 |- ( N e. ZZ -> ( N lcm 0 ) = 0 )
8	6, 7	eqtr3d 2134	. . . . . . 7 |- ( N e. ZZ -> ( 0 lcm N ) = 0 )
9	3, 8	sylan9eqr 2154	. . . . . 6 |- ( ( N e. ZZ /\ M = 0 ) -> ( M lcm N ) = 0 )
10	9	adantll 463	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M lcm N ) = 0 )
11		oveq2 5714	. . . . . . 7 |- ( N = 0 -> ( M lcm N ) = ( M lcm 0 ) )
12		lcm0val 11539	. . . . . . 7 |- ( M e. ZZ -> ( M lcm 0 ) = 0 )
13	11, 12	sylan9eqr 2154	. . . . . 6 |- ( ( M e. ZZ /\ N = 0 ) -> ( M lcm N ) = 0 )
14	13	adantlr 464	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M lcm N ) = 0 )
15	10, 14	jaodan 752	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = 0 )
16	2, 15	breqtrrd 3901	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> M || ( M lcm N ) )
17		dvds0 11303	. . . . 5 |- ( N e. ZZ -> N || 0 )
18	17	ad2antlr 476	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> N || 0 )
19	18, 15	breqtrrd 3901	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> N || ( M lcm N ) )
20	16, 19	jca 302	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
21		lcmcllem 11541	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. { n e. NN | ( M || n /\ N || n ) } )
22		lcmn0cl 11542	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. NN )
23		breq2 3879	. . . . . 6 |- ( n = ( M lcm N ) -> ( M || n <-> M || ( M lcm N ) ) )
24		breq2 3879	. . . . . 6 |- ( n = ( M lcm N ) -> ( N || n <-> N || ( M lcm N ) ) )
25	23, 24	anbi12d 460	. . . . 5 |- ( n = ( M lcm N ) -> ( ( M || n /\ N || n ) <-> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) ) )
26	25	elrab3 2794	. . . 4 |- ( ( M lcm N ) e. NN -> ( ( M lcm N ) e. { n e. NN | ( M || n /\ N || n ) } <-> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) ) )
27	22, 26	syl 14	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( ( M lcm N ) e. { n e. NN | ( M || n /\ N || n ) } <-> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) ) )
28	21, 27	mpbid 146	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
29		lcmmndc 11536	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
30		exmiddc 788	. . 3 |- (DECID ( M = 0 \/ N = 0 ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
31	29, 30	syl 14	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 \/ N = 0 ) \/ -. ( M = 0 \/ N = 0 ) ) )
32	20, 28, 31	mpjaodan 753	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 670  DECID wdc 786   = wceq 1299   e. wcel 1448   {crab 2379   class class class wbr 3875  (class class class)co 5706   0cc0 7500   NNcn 8578   ZZcz 8906   || cdvds 11288   lcm clcm 11534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-isom 5068  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-sup 6786  df-inf 6787  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-fz 9632  df-fzo 9761  df-fl 9884  df-mod 9937  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-dvds 11289  df-lcm 11535

This theorem is referenced by:  gcddvdslcm  11547  lcmneg  11548  lcmgcdeq  11557  lcmdvdsb  11558