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Theorem dvdslcm 11761
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.
StepHypRef Expression
1 dvds0 11519 . . . . 5  |-  ( M  e.  ZZ  ->  M  ||  0 )
21ad2antrr 479 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  \/  N  =  0 ) )  ->  M  ||  0 )
3 oveq1 5781 . . . . . . 7  |-  ( M  =  0  ->  ( M lcm  N )  =  ( 0 lcm  N ) )
4 0z 9077 . . . . . . . . 9  |-  0  e.  ZZ
5 lcmcom 11756 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  0  e.  ZZ )  ->  ( N lcm  0 )  =  ( 0 lcm  N
) )
64, 5mpan2 421 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N lcm  0 )  =  ( 0 lcm  N ) )
7 lcm0val 11757 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N lcm  0 )  =  0 )
86, 7eqtr3d 2174 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
0 lcm  N )  =  0 )
93, 8sylan9eqr 2194 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M lcm  N
)  =  0 )
109adantll 467 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0 )  ->  ( M lcm  N )  =  0 )
11 oveq2 5782 . . . . . . 7  |-  ( N  =  0  ->  ( M lcm  N )  =  ( M lcm  0 ) )
12 lcm0val 11757 . . . . . . 7  |-  ( M  e.  ZZ  ->  ( M lcm  0 )  =  0 )
1311, 12sylan9eqr 2194 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( M lcm  N
)  =  0 )
1413adantlr 468 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0 )  ->  ( M lcm  N )  =  0 )
1510, 14jaodan 786 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  \/  N  =  0 ) )  -> 
( M lcm  N )  =  0 )
162, 15breqtrrd 3956 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  \/  N  =  0 ) )  ->  M  ||  ( M lcm  N
) )
17 dvds0 11519 . . . . 5  |-  ( N  e.  ZZ  ->  N  ||  0 )
1817ad2antlr 480 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  \/  N  =  0 ) )  ->  N  ||  0 )
1918, 15breqtrrd 3956 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  \/  N  =  0 ) )  ->  N  ||  ( M lcm  N
) )
2016, 19jca 304 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =  0  \/  N  =  0 ) )  -> 
( M  ||  ( M lcm  N )  /\  N  ||  ( M lcm  N ) ) )
21 lcmcllem 11759 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N
)  e.  { n  e.  NN  |  ( M 
||  n  /\  N  ||  n ) } )
22 lcmn0cl 11760 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M lcm  N
)  e.  NN )
23 breq2 3933 . . . . . 6  |-  ( n  =  ( M lcm  N
)  ->  ( M  ||  n  <->  M  ||  ( M lcm 
N ) ) )
24 breq2 3933 . . . . . 6  |-  ( n  =  ( M lcm  N
)  ->  ( N  ||  n  <->  N  ||  ( M lcm 
N ) ) )
2523, 24anbi12d 464 . . . . 5  |-  ( n  =  ( M lcm  N
)  ->  ( ( M  ||  n  /\  N  ||  n )  <->  ( M  ||  ( M lcm  N )  /\  N  ||  ( M lcm  N ) ) ) )
2625elrab3 2841 . . . 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 ) ) ) )
2722, 26syl 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 ) ) ) )
2821, 27mpbid 146 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( M  ||  ( M lcm  N )  /\  N  ||  ( M lcm 
N ) ) )
29 lcmmndc 11754 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  \/  N  =  0 ) )
30 exmiddc 821 . . 3  |-  (DECID  ( M  =  0  \/  N  =  0 )  -> 
( ( M  =  0  \/  N  =  0 )  \/  -.  ( M  =  0  \/  N  =  0
) ) )
3129, 30syl 14 . 2  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  =  0  \/  N  =  0 )  \/  -.  ( M  =  0  \/  N  =  0
) ) )
3220, 28, 31mpjaodan 787 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  ||  ( M lcm  N )  /\  N  ||  ( M lcm  N ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697  DECID wdc 819    = wceq 1331    e. wcel 1480   {crab 2420   class class class wbr 3929  (class class class)co 5774   0cc0 7632   NNcn 8732   ZZcz 9066    || cdvds 11504   lcm clcm 11752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-sup 6871  df-inf 6872  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-fz 9803  df-fzo 9932  df-fl 10055  df-mod 10108  df-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-dvds 11505  df-lcm 11753
This theorem is referenced by:  gcddvdslcm  11765  lcmneg  11766  lcmgcdeq  11775  lcmdvdsb  11776
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