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Theorem lcmdvds 12093
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 4018 . . . . . . . . 9  |-  ( M  =  0  ->  ( M  ||  K  <->  0  ||  K ) )
32adantl 277 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M  ||  K 
<->  0  ||  K ) )
4 oveq1 5895 . . . . . . . . . 10  |-  ( M  =  0  ->  ( M lcm  N )  =  ( 0 lcm  N ) )
5 0z 9278 . . . . . . . . . . . 12  |-  0  e.  ZZ
6 lcmcom 12078 . . . . . . . . . . . 12  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0 lcm  N )  =  ( N lcm  0
) )
75, 6mpan 424 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
0 lcm  N )  =  ( N lcm  0 ) )
8 lcm0val 12079 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( N lcm  0 )  =  0 )
97, 8eqtrd 2220 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
0 lcm  N )  =  0 )
104, 9sylan9eqr 2242 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M lcm  N
)  =  0 )
1110breq1d 4025 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( ( M lcm 
N )  ||  K  <->  0 
||  K ) )
123, 11imbi12d 234 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( ( M 
||  K  ->  ( M lcm  N )  ||  K
)  <->  ( 0  ||  K  ->  0  ||  K
) ) )
131, 12mpbiri 168 . . . . . 6  |-  ( ( N  e.  ZZ  /\  M  =  0 )  ->  ( M  ||  K  ->  ( M lcm  N
)  ||  K )
)
14133ad2antl3 1162 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( M  ||  K  ->  ( M lcm  N )  ||  K ) )
1514adantrd 279 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  M  =  0
)  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
1615ex 115 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  =  0  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N
)  ||  K )
) )
17 breq1 4018 . . . . . . . . 9  |-  ( N  =  0  ->  ( N  ||  K  <->  0  ||  K ) )
1817adantl 277 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( N  ||  K 
<->  0  ||  K ) )
19 oveq2 5896 . . . . . . . . . 10  |-  ( N  =  0  ->  ( M lcm  N )  =  ( M lcm  0 ) )
20 lcm0val 12079 . . . . . . . . . 10  |-  ( M  e.  ZZ  ->  ( M lcm  0 )  =  0 )
2119, 20sylan9eqr 2242 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( M lcm  N
)  =  0 )
2221breq1d 4025 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( ( M lcm 
N )  ||  K  <->  0 
||  K ) )
2318, 22imbi12d 234 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( ( N 
||  K  ->  ( M lcm  N )  ||  K
)  <->  ( 0  ||  K  ->  0  ||  K
) ) )
241, 23mpbiri 168 . . . . . 6  |-  ( ( M  e.  ZZ  /\  N  =  0 )  ->  ( N  ||  K  ->  ( M lcm  N
)  ||  K )
)
25243ad2antl2 1161 . . . . 5  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( N  ||  K  ->  ( M lcm  N )  ||  K ) )
2625adantld 278 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  N  =  0
)  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
2726ex 115 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  =  0  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N
)  ||  K )
) )
2816, 27jaod 718 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  =  0  \/  N  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
29 neanior 2444 . . . . . 6  |-  ( ( M  =/=  0  /\  N  =/=  0 )  <->  -.  ( M  =  0  \/  N  =  0 ) )
30 lcmcl 12086 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  NN0 )
3130nn0zd 9387 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M lcm  N )  e.  ZZ )
32 dvds0 11827 . . . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 14  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  =  0 )  ->  ( ( M  ||  0  /\  N  ||  0 )  ->  ( M lcm  N )  ||  0
) )
36 breq2 4019 . . . . . . . . . . . . . . . . 17  |-  ( K  =  0  ->  ( M  ||  K  <->  M  ||  0
) )
37 breq2 4019 . . . . . . . . . . . . . . . . 17  |-  ( K  =  0  ->  ( N  ||  K  <->  N  ||  0
) )
3836, 37anbi12d 473 . . . . . . . . . . . . . . . 16  |-  ( K  =  0  ->  (
( M  ||  K  /\  N  ||  K )  <-> 
( M  ||  0  /\  N  ||  0 ) ) )
39 breq2 4019 . . . . . . . . . . . . . . . 16  |-  ( K  =  0  ->  (
( M lcm  N ) 
||  K  <->  ( M lcm  N )  ||  0 ) )
4038, 39imbi12d 234 . . . . . . . . . . . . . . 15  |-  ( K  =  0  ->  (
( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K )  <-> 
( ( M  ||  0  /\  N  ||  0
)  ->  ( M lcm  N )  ||  0 ) ) )
4140adantl 277 . . . . . . . . . . . . . 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 167 . . . . . . . . . . . . 13  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
4342adantrl 478 . . . . . . . . . . . 12  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  K  =  0 ) )  -> 
( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) )
4443adantllr 481 . . . . . . . . . . 11  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  K  =  0 ) )  ->  ( ( M 
||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) )
4544adantlrr 483 . . . . . . . . . 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 400 . . . . . . . . 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 11123 . . . . . . . . . . . . 13  |-  ( ( M  e.  ZZ  /\  M  =/=  0 )  -> 
( abs `  M
)  e.  NN )
48 nnabscl 11123 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  N  =/=  0 )  -> 
( abs `  N
)  e.  NN )
49 nnabscl 11123 . . . . . . . . . . . . . 14  |-  ( ( K  e.  ZZ  /\  K  =/=  0 )  -> 
( abs `  K
)  e.  NN )
50 lcmgcdlem 12091 . . . . . . . . . . . . . . 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 114 . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . 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 289 . . . . . . . . . . . 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 259 . . . . . . . . . . 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 11831 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  K  <->  M 
||  ( abs `  K
) ) )
56 zabscl 11109 . . . . . . . . . . . . . . . . . . . 20  |-  ( K  e.  ZZ  ->  ( abs `  K )  e.  ZZ )
57 absdvdsb 11830 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( M  e.  ZZ  /\  ( abs `  K )  e.  ZZ )  -> 
( M  ||  ( abs `  K )  <->  ( abs `  M )  ||  ( abs `  K ) ) )
5856, 57sylan2 286 . . . . . . . . . . . . . . . . . . 19  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  ( abs `  K )  <->  ( abs `  M )  ||  ( abs `  K ) ) )
5955, 58bitrd 188 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  K  e.  ZZ )  ->  ( M  ||  K  <->  ( abs `  M ) 
||  ( abs `  K
) ) )
6059adantlr 477 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( M  ||  K 
<->  ( abs `  M
)  ||  ( abs `  K ) ) )
61 dvdsabsb 11831 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  K  <->  N 
||  ( abs `  K
) ) )
62 absdvdsb 11830 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( N  e.  ZZ  /\  ( abs `  K )  e.  ZZ )  -> 
( N  ||  ( abs `  K )  <->  ( abs `  N )  ||  ( abs `  K ) ) )
6356, 62sylan2 286 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  ( abs `  K )  <->  ( abs `  N )  ||  ( abs `  K ) ) )
6461, 63bitrd 188 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  ->  ( N  ||  K  <->  ( abs `  N ) 
||  ( abs `  K
) ) )
6564adantll 476 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( N  ||  K 
<->  ( abs `  N
)  ||  ( abs `  K ) ) )
6660, 65anbi12d 473 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( M 
||  K  /\  N  ||  K )  <->  ( ( abs `  M )  ||  ( abs `  K )  /\  ( abs `  N
)  ||  ( abs `  K ) ) ) )
6766bicomd 141 . . . . . . . . . . . . . . 15  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( abs `  M ) 
||  ( abs `  K
)  /\  ( abs `  N )  ||  ( abs `  K ) )  <-> 
( M  ||  K  /\  N  ||  K ) ) )
68 lcmabs 12090 . . . . . . . . . . . . . . . . . 18  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  M
) lcm  ( abs `  N
) )  =  ( M lcm  N ) )
6968breq1d 4025 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N ) )  ||  ( abs `  K )  <-> 
( M lcm  N ) 
||  ( abs `  K
) ) )
7069adantr 276 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N
) )  ||  ( abs `  K )  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
71 dvdsabsb 11831 . . . . . . . . . . . . . . . . 17  |-  ( ( ( M lcm  N )  e.  ZZ  /\  K  e.  ZZ )  ->  (
( M lcm  N ) 
||  K  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
7231, 71sylan 283 . . . . . . . . . . . . . . . 16  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( M lcm 
N )  ||  K  <->  ( M lcm  N )  ||  ( abs `  K ) ) )
7370, 72bitr4d 191 . . . . . . . . . . . . . . 15  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( ( abs `  M ) lcm  ( abs `  N
) )  ||  ( abs `  K )  <->  ( M lcm  N )  ||  K ) )
7467, 73imbi12d 234 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 483 . . . . . . . . . . 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 147 . . . . . . . . . 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 400 . . . . . . . . 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 9342 . . . . . . . . . . . . 13  |-  ( ( K  e.  ZZ  /\  0  e.  ZZ )  -> DECID  K  =  0 )
815, 80mpan2 425 . . . . . . . . . . . 12  |-  ( K  e.  ZZ  -> DECID  K  =  0
)
82 exmiddc 837 . . . . . . . . . . . 12  |-  (DECID  K  =  0  ->  ( K  =  0  \/  -.  K  =  0 ) )
8381, 82syl 14 . . . . . . . . . . 11  |-  ( K  e.  ZZ  ->  ( K  =  0  \/  -.  K  =  0
) )
84 df-ne 2358 . . . . . . . . . . . 12  |-  ( K  =/=  0  <->  -.  K  =  0 )
8584orbi2i 763 . . . . . . . . . . 11  |-  ( ( K  =  0  \/  K  =/=  0 )  <-> 
( K  =  0  \/  -.  K  =  0 ) )
8683, 85sylibr 134 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  ( K  =  0  \/  K  =/=  0 ) )
8786adantl 277 . . . . . . . . 9  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  K  e.  ZZ )  ->  ( K  =  0  \/  K  =/=  0
) )
8846, 79, 87mpjaodan 799 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  /\  K  e.  ZZ )  ->  ( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) )
8988ex 115 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  M  =/=  0 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( K  e.  ZZ  ->  ( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) ) )
9089an4s 588 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( M  =/=  0  /\  N  =/=  0 ) )  -> 
( K  e.  ZZ  ->  ( ( M  ||  K  /\  N  ||  K
)  ->  ( M lcm  N )  ||  K ) ) )
9129, 90sylan2br 288 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( M  =  0  \/  N  =  0 ) )  ->  ( K  e.  ZZ  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
9291impancom 260 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( ( M 
||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
93923impa 1195 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  ->  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
94933comr 1212 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( -.  ( M  =  0  \/  N  =  0 )  ->  ( ( M  ||  K  /\  N  ||  K )  ->  ( M lcm  N )  ||  K
) ) )
95 lcmmndc 12076 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  ( M  =  0  \/  N  =  0 ) )
96 exmiddc 837 . . . 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 1016 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( M  =  0  \/  N  =  0 )  \/  -.  ( M  =  0  \/  N  =  0 ) ) )
9928, 94, 98mpjaod 719 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 104    <-> wb 105    \/ wo 709  DECID wdc 835    /\ w3a 979    = wceq 1363    e. wcel 2158    =/= wne 2357   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   0cc0 7825    x. cmul 7830   NNcn 8933   ZZcz 9267   abscabs 11020    || cdvds 11808    gcd cgcd 11957   lcm clcm 12074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-sup 6997  df-inf 6998  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-fz 10023  df-fzo 10157  df-fl 10284  df-mod 10337  df-seqfrec 10460  df-exp 10534  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-dvds 11809  df-gcd 11958  df-lcm 12075
This theorem is referenced by:  lcmdvdsb  12098
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