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Theorem lcmdvds 14572
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 4426 . . . . . . . 8  |-  ( M  =  0  ->  ( M  ||  K  <->  0  ||  K ) )
32adantl 467 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M  ||  K 
<->  0  ||  K ) )
4 oveq1 6312 . . . . . . . . 9  |-  ( M  =  0  ->  ( M lcm  N )  =  ( 0 lcm  N ) )
5 0z 10955 . . . . . . . . . . 11  |-  0  e.  ZZ
6 lcmcom 14556 . . . . . . . . . . 11  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0 lcm  N )  =  ( N lcm  0
) )
75, 6mpan 674 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
0 lcm  N )  =  ( N lcm  0 ) )
8 lcm0val 14557 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  ( N lcm  0 )  =  0 )
97, 8eqtrd 2463 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
0 lcm  N )  =  0 )
104, 9sylan9eqr 2485 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M lcm  N
)  =  0 )
1110breq1d 4433 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( ( M lcm 
N )  ||  K  <->  0 
||  K ) )
123, 11imbi12d 321 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( ( M 
||  K  ->  ( M lcm  N )  ||  K
)  <->  ( 0  ||  K  ->  0  ||  K
) ) )
131, 12mpbiri 236 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M  ||  K  ->  ( M lcm  N
)  ||  K )
)
14133ad2antl3 1169 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( M  ||  K  ->  ( M lcm  N )  ||  K ) )
1514adantrd 469 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
1615ex 435 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  0  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N
)  ||  K )
) )
17 breq1 4426 . . . . . . . 8  |-  ( N  =  0  ->  ( N  ||  K  <->  0  ||  K ) )
1817adantl 467 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( N  ||  K 
<->  0  ||  K ) )
19 oveq2 6313 . . . . . . . . 9  |-  ( N  =  0  ->  ( M lcm  N )  =  ( M lcm  0 ) )
20 lcm0val 14557 . . . . . . . . 9  |-  ( M  e.  ZZ  ->  ( M lcm  0 )  =  0 )
2119, 20sylan9eqr 2485 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( M lcm  N
)  =  0 )
2221breq1d 4433 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( ( M lcm 
N )  ||  K  <->  0 
||  K ) )
2318, 22imbi12d 321 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( ( N 
||  K  ->  ( M lcm  N )  ||  K
)  <->  ( 0  ||  K  ->  0  ||  K
) ) )
241, 23mpbiri 236 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( N  ||  K  ->  ( M lcm  N
)  ||  K )
)
25243ad2antl2 1168 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( N  ||  K  ->  ( M lcm  N )  ||  K ) )
2625adantld 468 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
2726ex 435 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  =  0  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N
)  ||  K )
) )
28 neanior 2745 . . . . . 6  |-  ( ( M  =/=  0  /\  N  =/=  0 )  <->  -.  ( M  =  0  \/  N  =  0 ) )
29 lcmcl 14565 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  NN0 )
3029nn0zd 11045 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  ZZ )
31 dvds0 14317 . . . . . . . . . . . . . . . . 17  |-  ( ( M lcm  N )  e.  ZZ  ->  ( M lcm  N )  ||  0 )
3230, 31syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N ) 
||  0 )
3332a1d 26 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  ||  0  /\  N  ||  0
)  ->  ( M lcm  N )  ||  0 ) )
3433adantr 466 . . . . . . . . . . . . . 14  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  =  0 )  ->  ( ( M  ||  0  /\  N  ||  0 )  ->  ( M lcm  N )  ||  0
) )
35 breq2 4427 . . . . . . . . . . . . . . . . 17  |-  ( K  =  0  ->  ( M  ||  K  <->  M  ||  0
) )
36 breq2 4427 . . . . . . . . . . . . . . . . 17  |-  ( K  =  0  ->  ( N  ||  K  <->  N  ||  0
) )
3735, 36anbi12d 715 . . . . . . . . . . . . . . . 16  |-  ( K  =  0  ->  (
( M  ||  K  /\  N  ||  K )  <-> 
( M  ||  0  /\  N  ||  0 ) ) )
38 breq2 4427 . . . . . . . . . . . . . . . 16  |-  ( K  =  0  ->  (
( M lcm  N ) 
||  K  <->  ( M lcm  N )  ||  0 ) )
3937, 38imbi12d 321 . . . . . . . . . . . . . . 15  |-  ( K  =  0  ->  (
( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K )  <-> 
( ( M  ||  0  /\  N  ||  0
)  ->  ( M lcm  N )  ||  0 ) ) )
4039adantl 467 . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . 13  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
4241adantrl 720 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  K  =  0 ) )  -> 
( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) )
4342adantllr 723 . . . . . . . . . . 11  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  K  =  0 ) )  ->  ( ( M 
||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
4443adantlrr 725 . . . . . . . . . 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 652 . . . . . . . . 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 13388 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
47 nnabscl 13388 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
48 nnabscl 13388 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  K  =/=  0 )  -> 
( abs `  K
)  e.  NN )
49 lcmgcdlem 14570 . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . 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 658 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 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 438 . . . . . . . . . . 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 14321 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  K  <->  M 
||  ( abs `  K
) ) )
55 zabscl 13376 . . . . . . . . . . . . . . . . . . . 20  |-  ( K  e.  ZZ  ->  ( abs `  K )  e.  ZZ )
56 absdvdsb 14320 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( M  e.  ZZ  /\  ( abs `  K )  e.  ZZ )  -> 
( M  ||  ( abs `  K )  <->  ( abs `  M )  ||  ( abs `  K ) ) )
5755, 56sylan2 476 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  ( abs `  K )  <->  ( abs `  M )  ||  ( abs `  K ) ) )
5854, 57bitrd 256 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  K  <->  ( abs `  M ) 
||  ( abs `  K
) ) )
5958adantlr 719 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( M  ||  K 
<->  ( abs `  M
)  ||  ( abs `  K ) ) )
60 dvdsabsb 14321 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  K  <->  N 
||  ( abs `  K
) ) )
61 absdvdsb 14320 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  ZZ  /\  ( abs `  K )  e.  ZZ )  -> 
( N  ||  ( abs `  K )  <->  ( abs `  N )  ||  ( abs `  K ) ) )
6255, 61sylan2 476 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  ( abs `  K )  <->  ( abs `  N )  ||  ( abs `  K ) ) )
6360, 62bitrd 256 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  K  <->  ( abs `  N ) 
||  ( abs `  K
) ) )
6463adantll 718 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( N  ||  K 
<->  ( abs `  N
)  ||  ( abs `  K ) ) )
6559, 64anbi12d 715 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( M 
||  K  /\  N  ||  K )  <->  ( ( abs `  M )  ||  ( abs `  K )  /\  ( abs `  N
)  ||  ( abs `  K ) ) ) )
6665bicomd 204 . . . . . . . . . . . . . . 15  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( abs `  M ) 
||  ( abs `  K
)  /\  ( abs `  N )  ||  ( abs `  K ) )  <-> 
( M  ||  K  /\  N  ||  K ) ) )
67 lcmabs 14569 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
) lcm  ( abs `  N
) )  =  ( M lcm  N ) )
6867breq1d 4433 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N ) )  ||  ( abs `  K )  <-> 
( M lcm  N ) 
||  ( abs `  K
) ) )
6968adantr 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N
) )  ||  ( abs `  K )  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
70 dvdsabsb 14321 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M lcm  N )  e.  ZZ  /\  K  e.  ZZ )  ->  (
( M lcm  N ) 
||  K  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
7130, 70sylan 473 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( M lcm 
N )  ||  K  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
7269, 71bitr4d 259 . . . . . . . . . . . . . . 15  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N
) )  ||  ( abs `  K )  <->  ( M lcm  N )  ||  K ) )
7366, 72imbi12d 321 . . . . . . . . . . . . . 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 721 . . . . . . . . . . . . 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 723 . . . . . . . . . . . 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 725 . . . . . . . . . . 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 213 . . . . . . . . . 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 652 . . . . . . . . 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 2738 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  K  e.  ZZ )  ->  ( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) )
8079ex 435 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( K  e.  ZZ  ->  ( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) ) )
8180an4s 833 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  -> 
( K  e.  ZZ  ->  ( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) ) )
8228, 81sylan2br 478 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( K  e.  ZZ  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
8382impancom 441 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( ( M 
||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
84833impa 1200 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
85843comr 1213 . 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 951 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   class class class wbr 4423   ` cfv 5601  (class class class)co 6305   0cc0 9546    x. cmul 9551   NNcn 10616   ZZcz 10944   abscabs 13297    || cdvds 14304    gcd cgcd 14467   lcm clcm 14546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-inf 7966  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fl 12034  df-mod 12103  df-seq 12220  df-exp 12279  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-dvds 14305  df-gcd 14468  df-lcm 14550
This theorem is referenced by:  lcmdvdsb  14577  lcmftp  14608  lcmfunsnlem1  14609  lcmfunsnlem2lem1  14610  nzin  36637
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