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Theorem lcmdvds 11789
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 3938 . . . . . . . . 9  |-  ( M  =  0  ->  ( M  ||  K  <->  0  ||  K ) )
32adantl 275 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M  ||  K 
<->  0  ||  K ) )
4 oveq1 5787 . . . . . . . . . 10  |-  ( M  =  0  ->  ( M lcm  N )  =  ( 0 lcm  N ) )
5 0z 9087 . . . . . . . . . . . 12  |-  0  e.  ZZ
6 lcmcom 11774 . . . . . . . . . . . 12  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0 lcm  N )  =  ( N lcm  0
) )
75, 6mpan 421 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
0 lcm  N )  =  ( N lcm  0 ) )
8 lcm0val 11775 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( N lcm  0 )  =  0 )
97, 8eqtrd 2173 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
0 lcm  N )  =  0 )
104, 9sylan9eqr 2195 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M lcm  N
)  =  0 )
1110breq1d 3945 . . . . . . . 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 1146 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( M  ||  K  ->  ( M lcm  N )  ||  K ) )
1514adantrd 277 . . . 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 3938 . . . . . . . . 9  |-  ( N  =  0  ->  ( N  ||  K  <->  0  ||  K ) )
1817adantl 275 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( N  ||  K 
<->  0  ||  K ) )
19 oveq2 5788 . . . . . . . . . 10  |-  ( N  =  0  ->  ( M lcm  N )  =  ( M lcm  0 ) )
20 lcm0val 11775 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M lcm  0 )  =  0 )
2119, 20sylan9eqr 2195 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( M lcm  N
)  =  0 )
2221breq1d 3945 . . . . . . . 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 1145 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( N  ||  K  ->  ( M lcm  N )  ||  K ) )
2625adantld 276 . . . 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 707 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  =  0  \/  N  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
29 neanior 2396 . . . . . 6  |-  ( ( M  =/=  0  /\  N  =/=  0 )  <->  -.  ( M  =  0  \/  N  =  0 ) )
30 lcmcl 11782 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  NN0 )
3130nn0zd 9193 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  ZZ )
32 dvds0 11537 . . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . 14  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  =  0 )  ->  ( ( M  ||  0  /\  N  ||  0 )  ->  ( M lcm  N )  ||  0
) )
36 breq2 3939 . . . . . . . . . . . . . . . . 17  |-  ( K  =  0  ->  ( M  ||  K  <->  M  ||  0
) )
37 breq2 3939 . . . . . . . . . . . . . . . . 17  |-  ( K  =  0  ->  ( N  ||  K  <->  N  ||  0
) )
3836, 37anbi12d 465 . . . . . . . . . . . . . . . 16  |-  ( K  =  0  ->  (
( M  ||  K  /\  N  ||  K )  <-> 
( M  ||  0  /\  N  ||  0 ) ) )
39 breq2 3939 . . . . . . . . . . . . . . . 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 275 . . . . . . . . . . . . . 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 470 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  K  =  0 ) )  -> 
( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) )
4443adantllr 473 . . . . . . . . . . 11  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  K  =  0 ) )  ->  ( ( M 
||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
4544adantlrr 475 . . . . . . . . . 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 398 . . . . . . . . 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 10902 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
48 nnabscl 10902 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
49 nnabscl 10902 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  K  =/=  0 )  -> 
( abs `  K
)  e.  NN )
50 lcmgcdlem 11787 . . . . . . . . . . . . . . 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 404 . . . . . . . . . . . . 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 287 . . . . . . . . . . . 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 11541 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  K  <->  M 
||  ( abs `  K
) ) )
56 zabscl 10888 . . . . . . . . . . . . . . . . . . . 20  |-  ( K  e.  ZZ  ->  ( abs `  K )  e.  ZZ )
57 absdvdsb 11540 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( M  e.  ZZ  /\  ( abs `  K )  e.  ZZ )  -> 
( M  ||  ( abs `  K )  <->  ( abs `  M )  ||  ( abs `  K ) ) )
5856, 57sylan2 284 . . . . . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( M  ||  K 
<->  ( abs `  M
)  ||  ( abs `  K ) ) )
61 dvdsabsb 11541 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  K  <->  N 
||  ( abs `  K
) ) )
62 absdvdsb 11540 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  ZZ  /\  ( abs `  K )  e.  ZZ )  -> 
( N  ||  ( abs `  K )  <->  ( abs `  N )  ||  ( abs `  K ) ) )
6356, 62sylan2 284 . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( N  ||  K 
<->  ( abs `  N
)  ||  ( abs `  K ) ) )
6660, 65anbi12d 465 . . . . . . . . . . . . . . . 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 11786 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
) lcm  ( abs `  N
) )  =  ( M lcm  N ) )
6968breq1d 3945 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N ) )  ||  ( abs `  K )  <-> 
( M lcm  N ) 
||  ( abs `  K
) ) )
7069adantr 274 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N
) )  ||  ( abs `  K )  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
71 dvdsabsb 11541 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M lcm  N )  e.  ZZ  /\  K  e.  ZZ )  ->  (
( M lcm  N ) 
||  K  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
7231, 71sylan 281 . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 475 . . . . . . . . . . 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 398 . . . . . . . . 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 9148 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  0  e.  ZZ )  -> DECID  K  =  0 )
815, 80mpan2 422 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  -> DECID  K  =  0
)
82 exmiddc 822 . . . . . . . . . . . 12  |-  (DECID  K  =  0  ->  ( K  =  0  \/  -.  K  =  0 ) )
8381, 82syl 14 . . . . . . . . . . 11  |-  ( K  e.  ZZ  ->  ( K  =  0  \/  -.  K  =  0
) )
84 df-ne 2310 . . . . . . . . . . . 12  |-  ( K  =/=  0  <->  -.  K  =  0 )
8584orbi2i 752 . . . . . . . . . . 11  |-  ( ( K  =  0  \/  K  =/=  0 )  <-> 
( K  =  0  \/  -.  K  =  0 ) )
8683, 85sylibr 133 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  ( K  =  0  \/  K  =/=  0 ) )
8786adantl 275 . . . . . . . . 9  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  K  e.  ZZ )  ->  ( K  =  0  \/  K  =/=  0
) )
8846, 79, 87mpjaodan 788 . . . . . . . 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 578 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  -> 
( K  e.  ZZ  ->  ( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) ) )
9129, 90sylan2br 286 . . . . 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 1177 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
94933comr 1190 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
95 lcmmndc 11772 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  \/  N  =  0 ) )
96 exmiddc 822 . . . 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 1000 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  =  0  \/  N  =  0 )  \/  -.  ( M  =  0  \/  N  =  0 ) ) )
9928, 94, 98mpjaod 708 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 698  DECID wdc 820    /\ w3a 963    = wceq 1332    e. wcel 1481    =/= wne 2309   class class class wbr 3935   ` cfv 5129  (class class class)co 5780   0cc0 7642    x. cmul 7647   NNcn 8742   ZZcz 9076   abscabs 10799    || cdvds 11522    gcd cgcd 11664   lcm clcm 11770
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4049  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-iinf 4508  ax-cnex 7733  ax-resscn 7734  ax-1cn 7735  ax-1re 7736  ax-icn 7737  ax-addcl 7738  ax-addrcl 7739  ax-mulcl 7740  ax-mulrcl 7741  ax-addcom 7742  ax-mulcom 7743  ax-addass 7744  ax-mulass 7745  ax-distr 7746  ax-i2m1 7747  ax-0lt1 7748  ax-1rid 7749  ax-0id 7750  ax-rnegex 7751  ax-precex 7752  ax-cnre 7753  ax-pre-ltirr 7754  ax-pre-ltwlin 7755  ax-pre-lttrn 7756  ax-pre-apti 7757  ax-pre-ltadd 7758  ax-pre-mulgt0 7759  ax-pre-mulext 7760  ax-arch 7761  ax-caucvg 7762
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-if 3478  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-iun 3821  df-br 3936  df-opab 3996  df-mpt 3997  df-tr 4033  df-id 4221  df-po 4224  df-iso 4225  df-iord 4294  df-on 4296  df-ilim 4297  df-suc 4299  df-iom 4511  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-isom 5138  df-riota 5736  df-ov 5783  df-oprab 5784  df-mpo 5785  df-1st 6044  df-2nd 6045  df-recs 6208  df-frec 6294  df-sup 6877  df-inf 6878  df-pnf 7824  df-mnf 7825  df-xr 7826  df-ltxr 7827  df-le 7828  df-sub 7957  df-neg 7958  df-reap 8359  df-ap 8366  df-div 8455  df-inn 8743  df-2 8801  df-3 8802  df-4 8803  df-n0 9000  df-z 9077  df-uz 9349  df-q 9437  df-rp 9469  df-fz 9820  df-fzo 9949  df-fl 10072  df-mod 10125  df-seqfrec 10248  df-exp 10322  df-cj 10644  df-re 10645  df-im 10646  df-rsqrt 10800  df-abs 10801  df-dvds 11523  df-gcd 11665  df-lcm 11771
This theorem is referenced by:  lcmdvdsb  11794
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