Theorem dvdslcm 12591

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 12317	. . . . 5 |- ( M e. ZZ -> M || 0 )
2	1	ad2antrr 488	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> M || 0 )
3		oveq1 6008	. . . . . . 7 |- ( M = 0 -> ( M lcm N ) = ( 0 lcm N ) )
4		0z 9457	. . . . . . . . 9 |- 0 e. ZZ
5		lcmcom 12586	. . . . . . . . 9 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N lcm 0 ) = ( 0 lcm N ) )
6	4, 5	mpan2 425	. . . . . . . 8 |- ( N e. ZZ -> ( N lcm 0 ) = ( 0 lcm N ) )
7		lcm0val 12587	. . . . . . . 8 |- ( N e. ZZ -> ( N lcm 0 ) = 0 )
8	6, 7	eqtr3d 2264	. . . . . . 7 |- ( N e. ZZ -> ( 0 lcm N ) = 0 )
9	3, 8	sylan9eqr 2284	. . . . . 6 |- ( ( N e. ZZ /\ M = 0 ) -> ( M lcm N ) = 0 )
10	9	adantll 476	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ M = 0 ) -> ( M lcm N ) = 0 )
11		oveq2 6009	. . . . . . 7 |- ( N = 0 -> ( M lcm N ) = ( M lcm 0 ) )
12		lcm0val 12587	. . . . . . 7 |- ( M e. ZZ -> ( M lcm 0 ) = 0 )
13	11, 12	sylan9eqr 2284	. . . . . 6 |- ( ( M e. ZZ /\ N = 0 ) -> ( M lcm N ) = 0 )
14	13	adantlr 477	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ N = 0 ) -> ( M lcm N ) = 0 )
15	10, 14	jaodan 802	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) = 0 )
16	2, 15	breqtrrd 4111	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> M || ( M lcm N ) )
17		dvds0 12317	. . . . 5 |- ( N e. ZZ -> N || 0 )
18	17	ad2antlr 489	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> N || 0 )
19	18, 15	breqtrrd 4111	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> N || ( M lcm N ) )
20	16, 19	jca 306	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 \/ N = 0 ) ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
21		lcmcllem 12589	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. { n e. NN | ( M || n /\ N || n ) } )
22		lcmn0cl 12590	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M lcm N ) e. NN )
23		breq2 4087	. . . . . 6 |- ( n = ( M lcm N ) -> ( M || n <-> M || ( M lcm N ) ) )
24		breq2 4087	. . . . . 6 |- ( n = ( M lcm N ) -> ( N || n <-> N || ( M lcm N ) ) )
25	23, 24	anbi12d 473	. . . . 5 |- ( n = ( M lcm N ) -> ( ( M || n /\ N || n ) <-> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) ) )
26	25	elrab3 2960	. . . 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 147	. 2 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 \/ N = 0 ) ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )
29		lcmmndc 12584	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 \/ N = 0 ) )
30		exmiddc 841	. . 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 803	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || ( M lcm N ) /\ N || ( M lcm N ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   {crab 2512   class class class wbr 4083  (class class class)co 6001   0cc0 7999   NNcn 9110   ZZcz 9446   || cdvds 12298   lcm clcm 12582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-sup 7151  df-inf 7152  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-dvds 12299  df-lcm 12583

This theorem is referenced by:  gcddvdslcm  12595  lcmneg  12596  lcmgcdeq  12605  lcmdvdsb  12606