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Theorem lcmdvds 11606
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 19 . . . . . . 7  |-  ( 0 
||  K  ->  0  ||  K )
2 breq1 3898 . . . . . . . . 9  |-  ( M  =  0  ->  ( M  ||  K  <->  0  ||  K ) )
32adantl 273 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M  ||  K 
<->  0  ||  K ) )
4 oveq1 5735 . . . . . . . . . 10  |-  ( M  =  0  ->  ( M lcm  N )  =  ( 0 lcm  N ) )
5 0z 8969 . . . . . . . . . . . 12  |-  0  e.  ZZ
6 lcmcom 11591 . . . . . . . . . . . 12  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0 lcm  N )  =  ( N lcm  0
) )
75, 6mpan 418 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
0 lcm  N )  =  ( N lcm  0 ) )
8 lcm0val 11592 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( N lcm  0 )  =  0 )
97, 8eqtrd 2147 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
0 lcm  N )  =  0 )
104, 9sylan9eqr 2169 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M lcm  N
)  =  0 )
1110breq1d 3905 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( ( M lcm 
N )  ||  K  <->  0 
||  K ) )
123, 11imbi12d 233 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( ( M 
||  K  ->  ( M lcm  N )  ||  K
)  <->  ( 0  ||  K  ->  0  ||  K
) ) )
131, 12mpbiri 167 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M  ||  K  ->  ( M lcm  N
)  ||  K )
)
14133ad2antl3 1128 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( M  ||  K  ->  ( M lcm  N )  ||  K ) )
1514adantrd 275 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
1615ex 114 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  0  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N
)  ||  K )
) )
17 breq1 3898 . . . . . . . . 9  |-  ( N  =  0  ->  ( N  ||  K  <->  0  ||  K ) )
1817adantl 273 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( N  ||  K 
<->  0  ||  K ) )
19 oveq2 5736 . . . . . . . . . 10  |-  ( N  =  0  ->  ( M lcm  N )  =  ( M lcm  0 ) )
20 lcm0val 11592 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M lcm  0 )  =  0 )
2119, 20sylan9eqr 2169 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( M lcm  N
)  =  0 )
2221breq1d 3905 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( ( M lcm 
N )  ||  K  <->  0 
||  K ) )
2318, 22imbi12d 233 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( ( N 
||  K  ->  ( M lcm  N )  ||  K
)  <->  ( 0  ||  K  ->  0  ||  K
) ) )
241, 23mpbiri 167 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( N  ||  K  ->  ( M lcm  N
)  ||  K )
)
25243ad2antl2 1127 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( N  ||  K  ->  ( M lcm  N )  ||  K ) )
2625adantld 274 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
2726ex 114 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  =  0  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N
)  ||  K )
) )
2816, 27jaod 689 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  =  0  \/  N  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
29 neanior 2369 . . . . . 6  |-  ( ( M  =/=  0  /\  N  =/=  0 )  <->  -.  ( M  =  0  \/  N  =  0 ) )
30 lcmcl 11599 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  NN0 )
3130nn0zd 9075 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  ZZ )
32 dvds0 11356 . . . . . . . . . . . . . . . . 17  |-  ( ( M lcm  N )  e.  ZZ  ->  ( M lcm  N )  ||  0 )
3331, 32syl 14 . . . . . . . . . . . . . . . 16  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N ) 
||  0 )
3433a1d 22 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  ||  0  /\  N  ||  0
)  ->  ( M lcm  N )  ||  0 ) )
3534adantr 272 . . . . . . . . . . . . . 14  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  =  0 )  ->  ( ( M  ||  0  /\  N  ||  0 )  ->  ( M lcm  N )  ||  0
) )
36 breq2 3899 . . . . . . . . . . . . . . . . 17  |-  ( K  =  0  ->  ( M  ||  K  <->  M  ||  0
) )
37 breq2 3899 . . . . . . . . . . . . . . . . 17  |-  ( K  =  0  ->  ( N  ||  K  <->  N  ||  0
) )
3836, 37anbi12d 462 . . . . . . . . . . . . . . . 16  |-  ( K  =  0  ->  (
( M  ||  K  /\  N  ||  K )  <-> 
( M  ||  0  /\  N  ||  0 ) ) )
39 breq2 3899 . . . . . . . . . . . . . . . 16  |-  ( K  =  0  ->  (
( M lcm  N ) 
||  K  <->  ( M lcm  N )  ||  0 ) )
4038, 39imbi12d 233 . . . . . . . . . . . . . . 15  |-  ( K  =  0  ->  (
( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K )  <-> 
( ( M  ||  0  /\  N  ||  0
)  ->  ( M lcm  N )  ||  0 ) ) )
4140adantl 273 . . . . . . . . . . . . . 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 ) ) )
4235, 41mpbird 166 . . . . . . . . . . . . 13  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
4342adantrl 467 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  K  =  0 ) )  -> 
( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) )
4443adantllr 470 . . . . . . . . . . 11  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  K  =  0 ) )  ->  ( ( M 
||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
4544adantlrr 472 . . . . . . . . . 10  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  ( K  e.  ZZ  /\  K  =  0 ) )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
4645anassrs 395 . . . . . . . . 9  |-  ( ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  K  e.  ZZ )  /\  K  =  0
)  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
47 nnabscl 10764 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
48 nnabscl 10764 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
49 nnabscl 10764 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  K  =/=  0 )  -> 
( abs `  K
)  e.  NN )
50 lcmgcdlem 11604 . . . . . . . . . . . . . . 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 ) ) ) )
5150simprd 113 . . . . . . . . . . . . . 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 ) ) )
5249, 51sylani 401 . . . . . . . . . . . . 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 ) ) )
5347, 48, 52syl2an 285 . . . . . . . . . . . 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 ) ) )
5453expdimp 257 . . . . . . . . . . 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
) ) )
55 dvdsabsb 11360 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  K  <->  M 
||  ( abs `  K
) ) )
56 zabscl 10750 . . . . . . . . . . . . . . . . . . . 20  |-  ( K  e.  ZZ  ->  ( abs `  K )  e.  ZZ )
57 absdvdsb 11359 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( M  e.  ZZ  /\  ( abs `  K )  e.  ZZ )  -> 
( M  ||  ( abs `  K )  <->  ( abs `  M )  ||  ( abs `  K ) ) )
5856, 57sylan2 282 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  ( abs `  K )  <->  ( abs `  M )  ||  ( abs `  K ) ) )
5955, 58bitrd 187 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  K  <->  ( abs `  M ) 
||  ( abs `  K
) ) )
6059adantlr 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( M  ||  K 
<->  ( abs `  M
)  ||  ( abs `  K ) ) )
61 dvdsabsb 11360 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  K  <->  N 
||  ( abs `  K
) ) )
62 absdvdsb 11359 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  ZZ  /\  ( abs `  K )  e.  ZZ )  -> 
( N  ||  ( abs `  K )  <->  ( abs `  N )  ||  ( abs `  K ) ) )
6356, 62sylan2 282 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  ( abs `  K )  <->  ( abs `  N )  ||  ( abs `  K ) ) )
6461, 63bitrd 187 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  K  <->  ( abs `  N ) 
||  ( abs `  K
) ) )
6564adantll 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( N  ||  K 
<->  ( abs `  N
)  ||  ( abs `  K ) ) )
6660, 65anbi12d 462 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( M 
||  K  /\  N  ||  K )  <->  ( ( abs `  M )  ||  ( abs `  K )  /\  ( abs `  N
)  ||  ( abs `  K ) ) ) )
6766bicomd 140 . . . . . . . . . . . . . . 15  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( abs `  M ) 
||  ( abs `  K
)  /\  ( abs `  N )  ||  ( abs `  K ) )  <-> 
( M  ||  K  /\  N  ||  K ) ) )
68 lcmabs 11603 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
) lcm  ( abs `  N
) )  =  ( M lcm  N ) )
6968breq1d 3905 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N ) )  ||  ( abs `  K )  <-> 
( M lcm  N ) 
||  ( abs `  K
) ) )
7069adantr 272 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N
) )  ||  ( abs `  K )  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
71 dvdsabsb 11360 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M lcm  N )  e.  ZZ  /\  K  e.  ZZ )  ->  (
( M lcm  N ) 
||  K  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
7231, 71sylan 279 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( M lcm 
N )  ||  K  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
7370, 72bitr4d 190 . . . . . . . . . . . . . . 15  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N
) )  ||  ( abs `  K )  <->  ( M lcm  N )  ||  K ) )
7467, 73imbi12d 233 . . . . . . . . . . . . . 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
) ) )
7574adantrr 468 . . . . . . . . . . . . 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
) ) )
7675adantllr 470 . . . . . . . . . . . 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
) ) )
7776adantlrr 472 . . . . . . . . . . 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
) ) )
7854, 77mpbid 146 . . . . . . . . . 10  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  ( K  e.  ZZ  /\  K  =/=  0 ) )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
7978anassrs 395 . . . . . . . . 9  |-  ( ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  K  e.  ZZ )  /\  K  =/=  0
)  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
80 zdceq 9030 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  0  e.  ZZ )  -> DECID  K  =  0 )
815, 80mpan2 419 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  -> DECID  K  =  0
)
82 exmiddc 804 . . . . . . . . . . . 12  |-  (DECID  K  =  0  ->  ( K  =  0  \/  -.  K  =  0 ) )
8381, 82syl 14 . . . . . . . . . . 11  |-  ( K  e.  ZZ  ->  ( K  =  0  \/  -.  K  =  0
) )
84 df-ne 2283 . . . . . . . . . . . 12  |-  ( K  =/=  0  <->  -.  K  =  0 )
8584orbi2i 734 . . . . . . . . . . 11  |-  ( ( K  =  0  \/  K  =/=  0 )  <-> 
( K  =  0  \/  -.  K  =  0 ) )
8683, 85sylibr 133 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  ( K  =  0  \/  K  =/=  0 ) )
8786adantl 273 . . . . . . . . 9  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  K  e.  ZZ )  ->  ( K  =  0  \/  K  =/=  0
) )
8846, 79, 87mpjaodan 770 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  K  e.  ZZ )  ->  ( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) )
8988ex 114 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( K  e.  ZZ  ->  ( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) ) )
9089an4s 560 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  -> 
( K  e.  ZZ  ->  ( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) ) )
9129, 90sylan2br 284 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( K  e.  ZZ  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
9291impancom 258 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( ( M 
||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
93923impa 1159 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
94933comr 1172 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
95 lcmmndc 11589 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  \/  N  =  0 ) )
96 exmiddc 804 . . . 4  |-  (DECID  ( M  =  0  \/  N  =  0 )  -> 
( ( M  =  0  \/  N  =  0 )  \/  -.  ( M  =  0  \/  N  =  0
) ) )
9795, 96syl 14 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M  =  0  \/  N  =  0 )  \/  -.  ( M  =  0  \/  N  =  0
) ) )
98973adant1 982 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  =  0  \/  N  =  0 )  \/  -.  ( M  =  0  \/  N  =  0 ) ) )
9928, 94, 98mpjaod 690 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N
)  ||  K )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680  DECID wdc 802    /\ w3a 945    = wceq 1314    e. wcel 1463    =/= wne 2282   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   0cc0 7547    x. cmul 7552   NNcn 8630   ZZcz 8958   abscabs 10661    || cdvds 11341    gcd cgcd 11483   lcm clcm 11587
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664  ax-caucvg 7665
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-isom 5090  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-sup 6823  df-inf 6824  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-3 8690  df-4 8691  df-n0 8882  df-z 8959  df-uz 9229  df-q 9314  df-rp 9344  df-fz 9684  df-fzo 9813  df-fl 9936  df-mod 9989  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663  df-dvds 11342  df-gcd 11484  df-lcm 11588
This theorem is referenced by:  lcmdvdsb  11611
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