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Theorem lcmdvds 31422
Description: The lcm of two integers divides any integer the two divide. (Contributed by Steve Rodriguez, 20-Jan-2020.)
Assertion
Ref Expression
lcmdvds  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N
)  ||  K )
)

Proof of Theorem lcmdvds
StepHypRef Expression
1 id 22 . . . . . 6  |-  ( 0 
||  K  ->  0  ||  K )
2 breq1 4387 . . . . . . . 8  |-  ( M  =  0  ->  ( M  ||  K  <->  0  ||  K ) )
32adantl 464 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M  ||  K 
<->  0  ||  K ) )
4 oveq1 6225 . . . . . . . . 9  |-  ( M  =  0  ->  ( M lcm  N )  =  ( 0 lcm  N ) )
5 0z 10814 . . . . . . . . . . 11  |-  0  e.  ZZ
6 lcmcom 31407 . . . . . . . . . . 11  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0 lcm  N )  =  ( N lcm  0
) )
75, 6mpan 668 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
0 lcm  N )  =  ( N lcm  0 ) )
8 lcm0val 31408 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  ( N lcm  0 )  =  0 )
97, 8eqtrd 2437 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
0 lcm  N )  =  0 )
104, 9sylan9eqr 2459 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M lcm  N
)  =  0 )
1110breq1d 4394 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( ( M lcm 
N )  ||  K  <->  0 
||  K ) )
123, 11imbi12d 318 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( ( M 
||  K  ->  ( M lcm  N )  ||  K
)  <->  ( 0  ||  K  ->  0  ||  K
) ) )
131, 12mpbiri 233 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M  ||  K  ->  ( M lcm  N
)  ||  K )
)
14133ad2antl3 1158 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( M  ||  K  ->  ( M lcm  N )  ||  K ) )
1514adantrd 466 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
1615ex 432 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  0  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N
)  ||  K )
) )
17 breq1 4387 . . . . . . . 8  |-  ( N  =  0  ->  ( N  ||  K  <->  0  ||  K ) )
1817adantl 464 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( N  ||  K 
<->  0  ||  K ) )
19 oveq2 6226 . . . . . . . . 9  |-  ( N  =  0  ->  ( M lcm  N )  =  ( M lcm  0 ) )
20 lcm0val 31408 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  ( M lcm  0 )  =  0 )
2119, 20sylan9eqr 2459 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( M lcm  N
)  =  0 )
2221breq1d 4394 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( ( M lcm 
N )  ||  K  <->  0 
||  K ) )
2318, 22imbi12d 318 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( ( N 
||  K  ->  ( M lcm  N )  ||  K
)  <->  ( 0  ||  K  ->  0  ||  K
) ) )
241, 23mpbiri 233 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( N  ||  K  ->  ( M lcm  N
)  ||  K )
)
25243ad2antl2 1157 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( N  ||  K  ->  ( M lcm  N )  ||  K ) )
2625adantld 465 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
2726ex 432 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  =  0  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N
)  ||  K )
) )
28 neanior 2721 . . . . . 6  |-  ( ( M  =/=  0  /\  N  =/=  0 )  <->  -.  ( M  =  0  \/  N  =  0 ) )
29 lcmcl 31415 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  NN0 )
3029nn0zd 10904 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  ZZ )
31 dvds0 14024 . . . . . . . . . . . . . . . . 17  |-  ( ( M lcm  N )  e.  ZZ  ->  ( M lcm  N )  ||  0 )
3230, 31syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N ) 
||  0 )
3332a1d 25 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  ||  0  /\  N  ||  0
)  ->  ( M lcm  N )  ||  0 ) )
3433adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  =  0 )  ->  ( ( M  ||  0  /\  N  ||  0 )  ->  ( M lcm  N )  ||  0
) )
35 breq2 4388 . . . . . . . . . . . . . . . . 17  |-  ( K  =  0  ->  ( M  ||  K  <->  M  ||  0
) )
36 breq2 4388 . . . . . . . . . . . . . . . . 17  |-  ( K  =  0  ->  ( N  ||  K  <->  N  ||  0
) )
3735, 36anbi12d 708 . . . . . . . . . . . . . . . 16  |-  ( K  =  0  ->  (
( M  ||  K  /\  N  ||  K )  <-> 
( M  ||  0  /\  N  ||  0 ) ) )
38 breq2 4388 . . . . . . . . . . . . . . . 16  |-  ( K  =  0  ->  (
( M lcm  N ) 
||  K  <->  ( M lcm  N )  ||  0 ) )
3937, 38imbi12d 318 . . . . . . . . . . . . . . 15  |-  ( K  =  0  ->  (
( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K )  <-> 
( ( M  ||  0  /\  N  ||  0
)  ->  ( M lcm  N )  ||  0 ) ) )
4039adantl 464 . . . . . . . . . . . . . 14  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  =  0 )  ->  ( (
( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N
)  ||  K )  <->  ( ( M  ||  0  /\  N  ||  0 )  ->  ( M lcm  N
)  ||  0 ) ) )
4134, 40mpbird 232 . . . . . . . . . . . . 13  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
4241adantrl 713 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  K  =  0 ) )  -> 
( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) )
4342adantllr 716 . . . . . . . . . . 11  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  K  =  0 ) )  ->  ( ( M 
||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
4443adantlrr 718 . . . . . . . . . 10  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  ( K  e.  ZZ  /\  K  =  0 ) )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
4544anassrs 646 . . . . . . . . 9  |-  ( ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  K  e.  ZZ )  /\  K  =  0
)  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
46 nnabscl 13183 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
47 nnabscl 13183 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
48 nnabscl 13183 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  K  =/=  0 )  -> 
( abs `  K
)  e.  NN )
49 lcmgcdlem 31420 . . . . . . . . . . . . . . 15  |-  ( ( ( abs `  M
)  e.  NN  /\  ( abs `  N )  e.  NN )  -> 
( ( ( ( abs `  M ) lcm  ( abs `  N
) )  x.  (
( abs `  M
)  gcd  ( abs `  N ) ) )  =  ( abs `  (
( abs `  M
)  x.  ( abs `  N ) ) )  /\  ( ( ( abs `  K )  e.  NN  /\  (
( abs `  M
)  ||  ( abs `  K )  /\  ( abs `  N )  ||  ( abs `  K ) ) )  ->  (
( abs `  M
) lcm  ( abs `  N
) )  ||  ( abs `  K ) ) ) )
5049simprd 461 . . . . . . . . . . . . . 14  |-  ( ( ( abs `  M
)  e.  NN  /\  ( abs `  N )  e.  NN )  -> 
( ( ( abs `  K )  e.  NN  /\  ( ( abs `  M
)  ||  ( abs `  K )  /\  ( abs `  N )  ||  ( abs `  K ) ) )  ->  (
( abs `  M
) lcm  ( abs `  N
) )  ||  ( abs `  K ) ) )
5148, 50sylani 652 . . . . . . . . . . . . 13  |-  ( ( ( abs `  M
)  e.  NN  /\  ( abs `  N )  e.  NN )  -> 
( ( ( K  e.  ZZ  /\  K  =/=  0 )  /\  (
( abs `  M
)  ||  ( abs `  K )  /\  ( abs `  N )  ||  ( abs `  K ) ) )  ->  (
( abs `  M
) lcm  ( abs `  N
) )  ||  ( abs `  K ) ) )
5246, 47, 51syl2an 475 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( ( ( K  e.  ZZ  /\  K  =/=  0 )  /\  (
( abs `  M
)  ||  ( abs `  K )  /\  ( abs `  N )  ||  ( abs `  K ) ) )  ->  (
( abs `  M
) lcm  ( abs `  N
) )  ||  ( abs `  K ) ) )
5352expdimp 435 . . . . . . . . . . 11  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  ->  ( (
( abs `  M
)  ||  ( abs `  K )  /\  ( abs `  N )  ||  ( abs `  K ) )  ->  ( ( abs `  M ) lcm  ( abs `  N ) ) 
||  ( abs `  K
) ) )
54 dvdsabsb 14028 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  K  <->  M 
||  ( abs `  K
) ) )
55 zabscl 13171 . . . . . . . . . . . . . . . . . . . 20  |-  ( K  e.  ZZ  ->  ( abs `  K )  e.  ZZ )
56 absdvdsb 14027 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( M  e.  ZZ  /\  ( abs `  K )  e.  ZZ )  -> 
( M  ||  ( abs `  K )  <->  ( abs `  M )  ||  ( abs `  K ) ) )
5755, 56sylan2 472 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  ( abs `  K )  <->  ( abs `  M )  ||  ( abs `  K ) ) )
5854, 57bitrd 253 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  K  <->  ( abs `  M ) 
||  ( abs `  K
) ) )
5958adantlr 712 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( M  ||  K 
<->  ( abs `  M
)  ||  ( abs `  K ) ) )
60 dvdsabsb 14028 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  K  <->  N 
||  ( abs `  K
) ) )
61 absdvdsb 14027 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  ZZ  /\  ( abs `  K )  e.  ZZ )  -> 
( N  ||  ( abs `  K )  <->  ( abs `  N )  ||  ( abs `  K ) ) )
6255, 61sylan2 472 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  ( abs `  K )  <->  ( abs `  N )  ||  ( abs `  K ) ) )
6360, 62bitrd 253 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  K  <->  ( abs `  N ) 
||  ( abs `  K
) ) )
6463adantll 711 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( N  ||  K 
<->  ( abs `  N
)  ||  ( abs `  K ) ) )
6559, 64anbi12d 708 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( M 
||  K  /\  N  ||  K )  <->  ( ( abs `  M )  ||  ( abs `  K )  /\  ( abs `  N
)  ||  ( abs `  K ) ) ) )
6665bicomd 201 . . . . . . . . . . . . . . 15  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( abs `  M ) 
||  ( abs `  K
)  /\  ( abs `  N )  ||  ( abs `  K ) )  <-> 
( M  ||  K  /\  N  ||  K ) ) )
67 lcmabs 31419 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
) lcm  ( abs `  N
) )  =  ( M lcm  N ) )
6867breq1d 4394 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N ) )  ||  ( abs `  K )  <-> 
( M lcm  N ) 
||  ( abs `  K
) ) )
6968adantr 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N
) )  ||  ( abs `  K )  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
70 dvdsabsb 14028 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M lcm  N )  e.  ZZ  /\  K  e.  ZZ )  ->  (
( M lcm  N ) 
||  K  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
7130, 70sylan 469 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( M lcm 
N )  ||  K  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
7269, 71bitr4d 256 . . . . . . . . . . . . . . 15  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N
) )  ||  ( abs `  K )  <->  ( M lcm  N )  ||  K ) )
7366, 72imbi12d 318 . . . . . . . . . . . . . 14  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( ( abs `  M
)  ||  ( abs `  K )  /\  ( abs `  N )  ||  ( abs `  K ) )  ->  ( ( abs `  M ) lcm  ( abs `  N ) ) 
||  ( abs `  K
) )  <->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
7473adantrr 714 . . . . . . . . . . . . 13  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( ( ( abs `  M ) 
||  ( abs `  K
)  /\  ( abs `  N )  ||  ( abs `  K ) )  ->  ( ( abs `  M ) lcm  ( abs `  N ) )  ||  ( abs `  K ) )  <->  ( ( M 
||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
7574adantllr 716 . . . . . . . . . . . 12  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  -> 
( ( ( ( abs `  M ) 
||  ( abs `  K
)  /\  ( abs `  N )  ||  ( abs `  K ) )  ->  ( ( abs `  M ) lcm  ( abs `  N ) )  ||  ( abs `  K ) )  <->  ( ( M 
||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
7675adantlrr 718 . . . . . . . . . . 11  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  ->  ( (
( ( abs `  M
)  ||  ( abs `  K )  /\  ( abs `  N )  ||  ( abs `  K ) )  ->  ( ( abs `  M ) lcm  ( abs `  N ) ) 
||  ( abs `  K
) )  <->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
7753, 76mpbid 210 . . . . . . . . . 10  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
7877anassrs 646 . . . . . . . . 9  |-  ( ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  K  e.  ZZ )  /\  K  =/=  0
)  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
7945, 78pm2.61dane 2714 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  K  e.  ZZ )  ->  ( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) )
8079ex 432 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( K  e.  ZZ  ->  ( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) ) )
8180an4s 824 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  -> 
( K  e.  ZZ  ->  ( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) ) )
8228, 81sylan2br 474 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( K  e.  ZZ  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
8382impancom 438 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( ( M 
||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
84833impa 1189 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
85843comr 1202 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
8616, 27, 85ecase3d 941 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N
)  ||  K )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836    =/= wne 2591   class class class wbr 4384   ` cfv 5513  (class class class)co 6218   0cc0 9425    x. cmul 9430   NNcn 10474   ZZcz 10803   abscabs 13092    || cdvds 14011    gcd cgcd 14169   lcm clcm 31404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513  ax-cnex 9481  ax-resscn 9482  ax-1cn 9483  ax-icn 9484  ax-addcl 9485  ax-addrcl 9486  ax-mulcl 9487  ax-mulrcl 9488  ax-mulcom 9489  ax-addass 9490  ax-mulass 9491  ax-distr 9492  ax-i2m1 9493  ax-1ne0 9494  ax-1rid 9495  ax-rnegex 9496  ax-rrecex 9497  ax-cnre 9498  ax-pre-lttri 9499  ax-pre-lttrn 9500  ax-pre-ltadd 9501  ax-pre-mulgt0 9502  ax-pre-sup 9503
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-nel 2594  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-2nd 6722  df-recs 6982  df-rdg 7016  df-er 7251  df-en 7458  df-dom 7459  df-sdom 7460  df-sup 7838  df-pnf 9563  df-mnf 9564  df-xr 9565  df-ltxr 9566  df-le 9567  df-sub 9742  df-neg 9743  df-div 10146  df-nn 10475  df-2 10533  df-3 10534  df-n0 10735  df-z 10804  df-uz 11024  df-rp 11162  df-fl 11851  df-mod 11920  df-seq 12034  df-exp 12093  df-cj 12957  df-re 12958  df-im 12959  df-sqrt 13093  df-abs 13094  df-dvds 14012  df-gcd 14170  df-lcm 31405
This theorem is referenced by:  lcmdvdsb  31425  nzin  31431
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