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Theorem lcmftp 14688
Description: The least common multiple of a triple of integers is the least common multiple of the third integer and the the least common multiple of the first two integers. Although there would be a shorter proof using lcmfunsn 14696, this explicit proof (not based on induction) should be kept. (Proof modification is discouraged.) (Contributed by AV, 23-Aug-2020.)
Assertion
Ref Expression
lcmftp  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (lcm `  { A ,  B ,  C } )  =  ( ( A lcm  B
) lcm  C ) )

Proof of Theorem lcmftp
Dummy variables  k  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0z 10972 . . . . . . 7  |-  0  e.  ZZ
2 eltpg 4005 . . . . . . 7  |-  ( 0  e.  ZZ  ->  (
0  e.  { A ,  B ,  C }  <->  ( 0  =  A  \/  0  =  B  \/  0  =  C )
) )
31, 2ax-mp 5 . . . . . 6  |-  ( 0  e.  { A ,  B ,  C }  <->  ( 0  =  A  \/  0  =  B  \/  0  =  C )
)
43biimpri 211 . . . . 5  |-  ( ( 0  =  A  \/  0  =  B  \/  0  =  C )  ->  0  e.  { A ,  B ,  C }
)
5 tpssi 4130 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  { A ,  B ,  C }  C_  ZZ )
64, 5anim12ci 577 . . . 4  |-  ( ( ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( { A ,  B ,  C }  C_  ZZ  /\  0  e. 
{ A ,  B ,  C } ) )
7 lcmf0val 14671 . . . 4  |-  ( ( { A ,  B ,  C }  C_  ZZ  /\  0  e.  { A ,  B ,  C }
)  ->  (lcm `  { A ,  B ,  C } )  =  0 )
86, 7syl 17 . . 3  |-  ( ( ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
(lcm `  { A ,  B ,  C }
)  =  0 )
9 0zd 10973 . . . . . . . . . 10  |-  ( C  e.  ZZ  ->  0  e.  ZZ )
10 lcmcom 14636 . . . . . . . . . 10  |-  ( ( 0  e.  ZZ  /\  C  e.  ZZ )  ->  ( 0 lcm  C )  =  ( C lcm  0
) )
119, 10mpancom 682 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  (
0 lcm  C )  =  ( C lcm  0 ) )
12 lcm0val 14637 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  ( C lcm  0 )  =  0 )
1311, 12eqtrd 2505 . . . . . . . 8  |-  ( C  e.  ZZ  ->  (
0 lcm  C )  =  0 )
1413eqcomd 2477 . . . . . . 7  |-  ( C  e.  ZZ  ->  0  =  ( 0 lcm  C
) )
15143ad2ant3 1053 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  0  =  ( 0 lcm  C
) )
1615adantl 473 . . . . 5  |-  ( ( 0  =  A  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  0  =  ( 0 lcm  C )
)
17 0zd 10973 . . . . . . . . . . 11  |-  ( B  e.  ZZ  ->  0  e.  ZZ )
18 lcmcom 14636 . . . . . . . . . . 11  |-  ( ( 0  e.  ZZ  /\  B  e.  ZZ )  ->  ( 0 lcm  B )  =  ( B lcm  0
) )
1917, 18mpancom 682 . . . . . . . . . 10  |-  ( B  e.  ZZ  ->  (
0 lcm  B )  =  ( B lcm  0 ) )
20 lcm0val 14637 . . . . . . . . . 10  |-  ( B  e.  ZZ  ->  ( B lcm  0 )  =  0 )
2119, 20eqtrd 2505 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  (
0 lcm  B )  =  0 )
2221eqcomd 2477 . . . . . . . 8  |-  ( B  e.  ZZ  ->  0  =  ( 0 lcm  B
) )
23223ad2ant2 1052 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  0  =  ( 0 lcm  B
) )
2423adantl 473 . . . . . 6  |-  ( ( 0  =  A  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  0  =  ( 0 lcm  B )
)
2524oveq1d 6323 . . . . 5  |-  ( ( 0  =  A  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  ( 0 lcm 
C )  =  ( ( 0 lcm  B ) lcm 
C ) )
26 oveq1 6315 . . . . . . 7  |-  ( 0  =  A  ->  (
0 lcm  B )  =  ( A lcm  B ) )
2726oveq1d 6323 . . . . . 6  |-  ( 0  =  A  ->  (
( 0 lcm  B ) lcm 
C )  =  ( ( A lcm  B ) lcm 
C ) )
2827adantr 472 . . . . 5  |-  ( ( 0  =  A  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  ( (
0 lcm  B ) lcm  C
)  =  ( ( A lcm  B ) lcm  C
) )
2916, 25, 283eqtrd 2509 . . . 4  |-  ( ( 0  =  A  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  0  =  ( ( A lcm  B
) lcm  C ) )
30 lcm0val 14637 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( A lcm  0 )  =  0 )
3130eqcomd 2477 . . . . . . . 8  |-  ( A  e.  ZZ  ->  0  =  ( A lcm  0
) )
32313ad2ant1 1051 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  0  =  ( A lcm  0
) )
3332adantl 473 . . . . . 6  |-  ( ( 0  =  B  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  0  =  ( A lcm  0 )
)
3433oveq1d 6323 . . . . 5  |-  ( ( 0  =  B  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  ( 0 lcm 
C )  =  ( ( A lcm  0 ) lcm 
C ) )
35133ad2ant3 1053 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
0 lcm  C )  =  0 )
3635adantl 473 . . . . 5  |-  ( ( 0  =  B  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  ( 0 lcm 
C )  =  0 )
37 oveq2 6316 . . . . . . 7  |-  ( 0  =  B  ->  ( A lcm  0 )  =  ( A lcm  B ) )
3837adantr 472 . . . . . 6  |-  ( ( 0  =  B  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  ( A lcm  0 )  =  ( A lcm  B ) )
3938oveq1d 6323 . . . . 5  |-  ( ( 0  =  B  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  ( ( A lcm  0 ) lcm  C )  =  ( ( A lcm 
B ) lcm  C ) )
4034, 36, 393eqtr3d 2513 . . . 4  |-  ( ( 0  =  B  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  0  =  ( ( A lcm  B
) lcm  C ) )
41 lcmcl 14645 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A lcm  B )  e.  NN0 )
4241nn0zd 11061 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A lcm  B )  e.  ZZ )
43 lcm0val 14637 . . . . . . . 8  |-  ( ( A lcm  B )  e.  ZZ  ->  ( ( A lcm  B ) lcm  0 )  =  0 )
4443eqcomd 2477 . . . . . . 7  |-  ( ( A lcm  B )  e.  ZZ  ->  0  =  ( ( A lcm  B
) lcm  0 ) )
4542, 44syl 17 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  0  =  ( ( A lcm  B ) lcm  0 ) )
46453adant3 1050 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  0  =  ( ( A lcm 
B ) lcm  0 ) )
47 oveq2 6316 . . . . 5  |-  ( 0  =  C  ->  (
( A lcm  B ) lcm  0 )  =  ( ( A lcm  B ) lcm 
C ) )
4846, 47sylan9eqr 2527 . . . 4  |-  ( ( 0  =  C  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  0  =  ( ( A lcm  B
) lcm  C ) )
4929, 40, 483jaoian 1359 . . 3  |-  ( ( ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
0  =  ( ( A lcm  B ) lcm  C
) )
508, 49eqtrd 2505 . 2  |-  ( ( ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
(lcm `  { A ,  B ,  C }
)  =  ( ( A lcm  B ) lcm  C
) )
51423adant3 1050 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A lcm  B )  e.  ZZ )
52 simp3 1032 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  C  e.  ZZ )
5351, 52jca 541 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A lcm  B )  e.  ZZ  /\  C  e.  ZZ ) )
5453adantl 473 . . . . . . . 8  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( ( A lcm  B
)  e.  ZZ  /\  C  e.  ZZ )
)
55 dvdslcm 14642 . . . . . . . 8  |-  ( ( ( A lcm  B )  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A lcm  B ) 
||  ( ( A lcm 
B ) lcm  C )  /\  C  ||  (
( A lcm  B ) lcm 
C ) ) )
5654, 55syl 17 . . . . . . 7  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( ( A lcm  B
)  ||  ( ( A lcm  B ) lcm  C )  /\  C  ||  (
( A lcm  B ) lcm 
C ) ) )
57 dvdslcm 14642 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  ||  ( A lcm  B )  /\  B  ||  ( A lcm  B ) ) )
58573adant3 1050 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  ( A lcm  B
)  /\  B  ||  ( A lcm  B ) ) )
59 simp1 1030 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  A  e.  ZZ )
60 lcmcl 14645 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A lcm  B )  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A lcm  B ) lcm 
C )  e.  NN0 )
6153, 60syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A lcm  B ) lcm 
C )  e.  NN0 )
6261nn0zd 11061 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A lcm  B ) lcm 
C )  e.  ZZ )
6359, 51, 623jca 1210 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  e.  ZZ  /\  ( A lcm  B )  e.  ZZ  /\  ( ( A lcm  B
) lcm  C )  e.  ZZ ) )
64 dvdstr 14414 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  ( A lcm  B )  e.  ZZ  /\  ( ( A lcm  B ) lcm  C
)  e.  ZZ )  ->  ( ( A 
||  ( A lcm  B
)  /\  ( A lcm  B )  ||  ( ( A lcm  B ) lcm  C
) )  ->  A  ||  ( ( A lcm  B
) lcm  C ) ) )
6563, 64syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A  ||  ( A lcm  B )  /\  ( A lcm  B )  ||  (
( A lcm  B ) lcm 
C ) )  ->  A  ||  ( ( A lcm 
B ) lcm  C ) ) )
6665expd 443 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  ( A lcm  B
)  ->  ( ( A lcm  B )  ||  (
( A lcm  B ) lcm 
C )  ->  A  ||  ( ( A lcm  B
) lcm  C ) ) ) )
6766com12 31 . . . . . . . . . . . . . 14  |-  ( A 
||  ( A lcm  B
)  ->  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A lcm  B ) 
||  ( ( A lcm 
B ) lcm  C )  ->  A  ||  (
( A lcm  B ) lcm 
C ) ) ) )
6867adantr 472 . . . . . . . . . . . . 13  |-  ( ( A  ||  ( A lcm 
B )  /\  B  ||  ( A lcm  B ) )  ->  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A lcm  B ) 
||  ( ( A lcm 
B ) lcm  C )  ->  A  ||  (
( A lcm  B ) lcm 
C ) ) ) )
6958, 68mpcom 36 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A lcm  B ) 
||  ( ( A lcm 
B ) lcm  C )  ->  A  ||  (
( A lcm  B ) lcm 
C ) ) )
7069adantl 473 . . . . . . . . . . 11  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( ( A lcm  B
)  ||  ( ( A lcm  B ) lcm  C )  ->  A  ||  (
( A lcm  B ) lcm 
C ) ) )
7170com12 31 . . . . . . . . . 10  |-  ( ( A lcm  B )  ||  ( ( A lcm  B
) lcm  C )  -> 
( ( -.  (
0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  A  ||  (
( A lcm  B ) lcm 
C ) ) )
7271adantr 472 . . . . . . . . 9  |-  ( ( ( A lcm  B ) 
||  ( ( A lcm 
B ) lcm  C )  /\  C  ||  (
( A lcm  B ) lcm 
C ) )  -> 
( ( -.  (
0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  A  ||  (
( A lcm  B ) lcm 
C ) ) )
7372impcom 437 . . . . . . . 8  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  ( ( A lcm  B )  ||  (
( A lcm  B ) lcm 
C )  /\  C  ||  ( ( A lcm  B
) lcm  C ) ) )  ->  A  ||  (
( A lcm  B ) lcm 
C ) )
74 simpr 468 . . . . . . . . . . . . . . 15  |-  ( ( A  ||  ( A lcm 
B )  /\  B  ||  ( A lcm  B ) )  ->  B  ||  ( A lcm  B ) )
7557, 74syl 17 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  B  ||  ( A lcm 
B ) )
76753adant3 1050 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  B  ||  ( A lcm  B ) )
7776adantl 473 . . . . . . . . . . . 12  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  B  ||  ( A lcm  B
) )
78 simp2 1031 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  B  e.  ZZ )
7978, 51, 623jca 1210 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( B  e.  ZZ  /\  ( A lcm  B )  e.  ZZ  /\  ( ( A lcm  B
) lcm  C )  e.  ZZ ) )
8079adantl 473 . . . . . . . . . . . . 13  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( B  e.  ZZ  /\  ( A lcm  B )  e.  ZZ  /\  (
( A lcm  B ) lcm 
C )  e.  ZZ ) )
81 dvdstr 14414 . . . . . . . . . . . . 13  |-  ( ( B  e.  ZZ  /\  ( A lcm  B )  e.  ZZ  /\  ( ( A lcm  B ) lcm  C
)  e.  ZZ )  ->  ( ( B 
||  ( A lcm  B
)  /\  ( A lcm  B )  ||  ( ( A lcm  B ) lcm  C
) )  ->  B  ||  ( ( A lcm  B
) lcm  C ) ) )
8280, 81syl 17 . . . . . . . . . . . 12  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( ( B  ||  ( A lcm  B )  /\  ( A lcm  B ) 
||  ( ( A lcm 
B ) lcm  C ) )  ->  B  ||  (
( A lcm  B ) lcm 
C ) ) )
8377, 82mpand 689 . . . . . . . . . . 11  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( ( A lcm  B
)  ||  ( ( A lcm  B ) lcm  C )  ->  B  ||  (
( A lcm  B ) lcm 
C ) ) )
8483com12 31 . . . . . . . . . 10  |-  ( ( A lcm  B )  ||  ( ( A lcm  B
) lcm  C )  -> 
( ( -.  (
0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  B  ||  (
( A lcm  B ) lcm 
C ) ) )
8584adantr 472 . . . . . . . . 9  |-  ( ( ( A lcm  B ) 
||  ( ( A lcm 
B ) lcm  C )  /\  C  ||  (
( A lcm  B ) lcm 
C ) )  -> 
( ( -.  (
0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  B  ||  (
( A lcm  B ) lcm 
C ) ) )
8685impcom 437 . . . . . . . 8  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  ( ( A lcm  B )  ||  (
( A lcm  B ) lcm 
C )  /\  C  ||  ( ( A lcm  B
) lcm  C ) ) )  ->  B  ||  (
( A lcm  B ) lcm 
C ) )
87 simpr 468 . . . . . . . . 9  |-  ( ( ( A lcm  B ) 
||  ( ( A lcm 
B ) lcm  C )  /\  C  ||  (
( A lcm  B ) lcm 
C ) )  ->  C  ||  ( ( A lcm 
B ) lcm  C ) )
8887adantl 473 . . . . . . . 8  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  ( ( A lcm  B )  ||  (
( A lcm  B ) lcm 
C )  /\  C  ||  ( ( A lcm  B
) lcm  C ) ) )  ->  C  ||  (
( A lcm  B ) lcm 
C ) )
8973, 86, 883jca 1210 . . . . . . 7  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  ( ( A lcm  B )  ||  (
( A lcm  B ) lcm 
C )  /\  C  ||  ( ( A lcm  B
) lcm  C ) ) )  ->  ( A  ||  ( ( A lcm  B
) lcm  C )  /\  B  ||  ( ( A lcm 
B ) lcm  C )  /\  C  ||  (
( A lcm  B ) lcm 
C ) ) )
9056, 89mpdan 681 . . . . . 6  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( A  ||  (
( A lcm  B ) lcm 
C )  /\  B  ||  ( ( A lcm  B
) lcm  C )  /\  C  ||  ( ( A lcm 
B ) lcm  C ) ) )
91 breq1 4398 . . . . . . . 8  |-  ( m  =  A  ->  (
m  ||  ( ( A lcm  B ) lcm  C )  <-> 
A  ||  ( ( A lcm  B ) lcm  C ) ) )
92 breq1 4398 . . . . . . . 8  |-  ( m  =  B  ->  (
m  ||  ( ( A lcm  B ) lcm  C )  <-> 
B  ||  ( ( A lcm  B ) lcm  C ) ) )
93 breq1 4398 . . . . . . . 8  |-  ( m  =  C  ->  (
m  ||  ( ( A lcm  B ) lcm  C )  <-> 
C  ||  ( ( A lcm  B ) lcm  C ) ) )
9491, 92, 93raltpg 4014 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A. m  e.  { A ,  B ,  C }
m  ||  ( ( A lcm  B ) lcm  C )  <-> 
( A  ||  (
( A lcm  B ) lcm 
C )  /\  B  ||  ( ( A lcm  B
) lcm  C )  /\  C  ||  ( ( A lcm 
B ) lcm  C ) ) ) )
9594adantl 473 . . . . . 6  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( A. m  e. 
{ A ,  B ,  C } m  ||  ( ( A lcm  B
) lcm  C )  <->  ( A  ||  ( ( A lcm  B
) lcm  C )  /\  B  ||  ( ( A lcm 
B ) lcm  C )  /\  C  ||  (
( A lcm  B ) lcm 
C ) ) ) )
9690, 95mpbird 240 . . . . 5  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  A. m  e.  { A ,  B ,  C }
m  ||  ( ( A lcm  B ) lcm  C ) )
97 breq1 4398 . . . . . . . . 9  |-  ( m  =  A  ->  (
m  ||  k  <->  A  ||  k
) )
98 breq1 4398 . . . . . . . . 9  |-  ( m  =  B  ->  (
m  ||  k  <->  B  ||  k
) )
99 breq1 4398 . . . . . . . . 9  |-  ( m  =  C  ->  (
m  ||  k  <->  C  ||  k
) )
10097, 98, 99raltpg 4014 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A. m  e.  { A ,  B ,  C }
m  ||  k  <->  ( A  ||  k  /\  B  ||  k  /\  C  ||  k
) ) )
101100ad2antlr 741 . . . . . . 7  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  ->  ( A. m  e.  { A ,  B ,  C }
m  ||  k  <->  ( A  ||  k  /\  B  ||  k  /\  C  ||  k
) ) )
102 simpr 468 . . . . . . . . . . 11  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  ->  k  e.  NN )
10351ad2antlr 741 . . . . . . . . . . 11  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  ->  ( A lcm 
B )  e.  ZZ )
10452ad2antlr 741 . . . . . . . . . . 11  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  ->  C  e.  ZZ )
105102, 103, 1043jca 1210 . . . . . . . . . 10  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  ->  ( k  e.  NN  /\  ( A lcm  B )  e.  ZZ  /\  C  e.  ZZ ) )
106105adantr 472 . . . . . . . . 9  |-  ( ( ( ( -.  (
0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  /\  ( A  ||  k  /\  B  ||  k  /\  C  ||  k ) )  -> 
( k  e.  NN  /\  ( A lcm  B )  e.  ZZ  /\  C  e.  ZZ ) )
107 3ioran 1025 . . . . . . . . . . . . . . . . 17  |-  ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  <->  ( -.  0  =  A  /\  -.  0  =  B  /\  -.  0  =  C ) )
108 eqcom 2478 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  =  A  <->  A  = 
0 )
109108notbii 303 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  0  =  A  <->  -.  A  =  0 )
110 eqcom 2478 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  =  B  <->  B  = 
0 )
111110notbii 303 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  0  =  B  <->  -.  B  =  0 )
112109, 111anbi12i 711 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( -.  0  =  A  /\  -.  0  =  B )  <->  ( -.  A  =  0  /\  -.  B  =  0
) )
113112biimpi 199 . . . . . . . . . . . . . . . . . . 19  |-  ( ( -.  0  =  A  /\  -.  0  =  B )  ->  ( -.  A  =  0  /\  -.  B  =  0 ) )
114 ioran 498 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  ( A  =  0  \/  B  =  0 )  <->  ( -.  A  =  0  /\  -.  B  =  0 ) )
115113, 114sylibr 217 . . . . . . . . . . . . . . . . . 18  |-  ( ( -.  0  =  A  /\  -.  0  =  B )  ->  -.  ( A  =  0  \/  B  =  0
) )
1161153adant3 1050 . . . . . . . . . . . . . . . . 17  |-  ( ( -.  0  =  A  /\  -.  0  =  B  /\  -.  0  =  C )  ->  -.  ( A  =  0  \/  B  =  0
) )
117107, 116sylbi 200 . . . . . . . . . . . . . . . 16  |-  ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  ->  -.  ( A  =  0  \/  B  =  0 ) )
118 id 22 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  e.  ZZ  /\  B  e.  ZZ ) )
1191183adant3 1050 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  e.  ZZ  /\  B  e.  ZZ ) )
120117, 119anim12ci 577 . . . . . . . . . . . . . . 15  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  \/  B  =  0
) ) )
121 lcmn0cl 14641 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  \/  B  =  0 ) )  ->  ( A lcm  B
)  e.  NN )
122120, 121syl 17 . . . . . . . . . . . . . 14  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( A lcm  B )  e.  NN )
123 nnne0 10664 . . . . . . . . . . . . . . 15  |-  ( ( A lcm  B )  e.  NN  ->  ( A lcm  B )  =/=  0 )
124123neneqd 2648 . . . . . . . . . . . . . 14  |-  ( ( A lcm  B )  e.  NN  ->  -.  ( A lcm  B )  =  0 )
125122, 124syl 17 . . . . . . . . . . . . 13  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  -.  ( A lcm  B )  =  0 )
126 eqcom 2478 . . . . . . . . . . . . . . . . . 18  |-  ( 0  =  C  <->  C  = 
0 )
127126notbii 303 . . . . . . . . . . . . . . . . 17  |-  ( -.  0  =  C  <->  -.  C  =  0 )
128127biimpi 199 . . . . . . . . . . . . . . . 16  |-  ( -.  0  =  C  ->  -.  C  =  0
)
1291283ad2ant3 1053 . . . . . . . . . . . . . . 15  |-  ( ( -.  0  =  A  /\  -.  0  =  B  /\  -.  0  =  C )  ->  -.  C  =  0 )
130107, 129sylbi 200 . . . . . . . . . . . . . 14  |-  ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  ->  -.  C  =  0 )
131130adantr 472 . . . . . . . . . . . . 13  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  -.  C  =  0
)
132125, 131jca 541 . . . . . . . . . . . 12  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( -.  ( A lcm 
B )  =  0  /\  -.  C  =  0 ) )
133132adantr 472 . . . . . . . . . . 11  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  ->  ( -.  ( A lcm  B )  =  0  /\  -.  C  =  0 ) )
134133adantr 472 . . . . . . . . . 10  |-  ( ( ( ( -.  (
0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  /\  ( A  ||  k  /\  B  ||  k  /\  C  ||  k ) )  -> 
( -.  ( A lcm 
B )  =  0  /\  -.  C  =  0 ) )
135 ioran 498 . . . . . . . . . 10  |-  ( -.  ( ( A lcm  B
)  =  0  \/  C  =  0 )  <-> 
( -.  ( A lcm 
B )  =  0  /\  -.  C  =  0 ) )
136134, 135sylibr 217 . . . . . . . . 9  |-  ( ( ( ( -.  (
0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  /\  ( A  ||  k  /\  B  ||  k  /\  C  ||  k ) )  ->  -.  ( ( A lcm  B
)  =  0  \/  C  =  0 ) )
137119adantl 473 . . . . . . . . . . . . . . 15  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( A  e.  ZZ  /\  B  e.  ZZ ) )
138 nnz 10983 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  k  e.  ZZ )
139137, 138anim12ci 577 . . . . . . . . . . . . . 14  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  ->  ( k  e.  ZZ  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) ) )
140 3anass 1011 . . . . . . . . . . . . . 14  |-  ( ( k  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  <->  ( k  e.  ZZ  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) ) )
141139, 140sylibr 217 . . . . . . . . . . . . 13  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  ->  ( k  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ ) )
142 lcmdvds 14652 . . . . . . . . . . . . 13  |-  ( ( k  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A  ||  k  /\  B  ||  k )  ->  ( A lcm  B
)  ||  k )
)
143141, 142syl 17 . . . . . . . . . . . 12  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  ->  ( ( A  ||  k  /\  B  ||  k )  -> 
( A lcm  B ) 
||  k ) )
144143com12 31 . . . . . . . . . . 11  |-  ( ( A  ||  k  /\  B  ||  k )  -> 
( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  /\  k  e.  NN )  ->  ( A lcm  B ) 
||  k ) )
1451443adant3 1050 . . . . . . . . . 10  |-  ( ( A  ||  k  /\  B  ||  k  /\  C  ||  k )  ->  (
( ( -.  (
0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  ->  ( A lcm 
B )  ||  k
) )
146145impcom 437 . . . . . . . . 9  |-  ( ( ( ( -.  (
0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  /\  ( A  ||  k  /\  B  ||  k  /\  C  ||  k ) )  -> 
( A lcm  B ) 
||  k )
147 simp3 1032 . . . . . . . . . 10  |-  ( ( A  ||  k  /\  B  ||  k  /\  C  ||  k )  ->  C  ||  k )
148147adantl 473 . . . . . . . . 9  |-  ( ( ( ( -.  (
0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  /\  ( A  ||  k  /\  B  ||  k  /\  C  ||  k ) )  ->  C  ||  k )
149 lcmledvds 14643 . . . . . . . . . 10  |-  ( ( ( k  e.  NN  /\  ( A lcm  B )  e.  ZZ  /\  C  e.  ZZ )  /\  -.  ( ( A lcm  B
)  =  0  \/  C  =  0 ) )  ->  ( (
( A lcm  B ) 
||  k  /\  C  ||  k )  ->  (
( A lcm  B ) lcm 
C )  <_  k
) )
150149imp 436 . . . . . . . . 9  |-  ( ( ( ( k  e.  NN  /\  ( A lcm 
B )  e.  ZZ  /\  C  e.  ZZ )  /\  -.  ( ( A lcm  B )  =  0  \/  C  =  0 ) )  /\  ( ( A lcm  B
)  ||  k  /\  C  ||  k ) )  ->  ( ( A lcm 
B ) lcm  C )  <_  k )
151106, 136, 146, 148, 150syl22anc 1293 . . . . . . . 8  |-  ( ( ( ( -.  (
0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  /\  ( A  ||  k  /\  B  ||  k  /\  C  ||  k ) )  -> 
( ( A lcm  B
) lcm  C )  <_ 
k )
152151ex 441 . . . . . . 7  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  ->  ( ( A  ||  k  /\  B  ||  k  /\  C  ||  k )  ->  (
( A lcm  B ) lcm 
C )  <_  k
) )
153101, 152sylbid 223 . . . . . 6  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  ->  ( A. m  e.  { A ,  B ,  C }
m  ||  k  ->  ( ( A lcm  B ) lcm 
C )  <_  k
) )
154153ralrimiva 2809 . . . . 5  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  A. k  e.  NN  ( A. m  e.  { A ,  B ,  C } m  ||  k  ->  ( ( A lcm  B
) lcm  C )  <_ 
k ) )
15596, 154jca 541 . . . 4  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( A. m  e. 
{ A ,  B ,  C } m  ||  ( ( A lcm  B
) lcm  C )  /\  A. k  e.  NN  ( A. m  e.  { A ,  B ,  C }
m  ||  k  ->  ( ( A lcm  B ) lcm 
C )  <_  k
) ) )
156109biimpi 199 . . . . . . . . . . . . . . . 16  |-  ( -.  0  =  A  ->  -.  A  =  0
)
157111biimpi 199 . . . . . . . . . . . . . . . 16  |-  ( -.  0  =  B  ->  -.  B  =  0
)
158156, 157anim12i 576 . . . . . . . . . . . . . . 15  |-  ( ( -.  0  =  A  /\  -.  0  =  B )  ->  ( -.  A  =  0  /\  -.  B  =  0 ) )
159158, 114sylibr 217 . . . . . . . . . . . . . 14  |-  ( ( -.  0  =  A  /\  -.  0  =  B )  ->  -.  ( A  =  0  \/  B  =  0
) )
1601593adant3 1050 . . . . . . . . . . . . 13  |-  ( ( -.  0  =  A  /\  -.  0  =  B  /\  -.  0  =  C )  ->  -.  ( A  =  0  \/  B  =  0
) )
161107, 160sylbi 200 . . . . . . . . . . . 12  |-  ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  ->  -.  ( A  =  0  \/  B  =  0 ) )
162161, 119anim12ci 577 . . . . . . . . . . 11  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  \/  B  =  0
) ) )
163162, 121syl 17 . . . . . . . . . 10  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( A lcm  B )  e.  NN )
164163, 124syl 17 . . . . . . . . 9  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  -.  ( A lcm  B )  =  0 )
165164, 131jca 541 . . . . . . . 8  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( -.  ( A lcm 
B )  =  0  /\  -.  C  =  0 ) )
166165, 135sylibr 217 . . . . . . 7  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  -.  ( ( A lcm  B
)  =  0  \/  C  =  0 ) )
16754, 166jca 541 . . . . . 6  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( ( ( A lcm 
B )  e.  ZZ  /\  C  e.  ZZ )  /\  -.  ( ( A lcm  B )  =  0  \/  C  =  0 ) ) )
168 lcmn0cl 14641 . . . . . 6  |-  ( ( ( ( A lcm  B
)  e.  ZZ  /\  C  e.  ZZ )  /\  -.  ( ( A lcm 
B )  =  0  \/  C  =  0 ) )  ->  (
( A lcm  B ) lcm 
C )  e.  NN )
169167, 168syl 17 . . . . 5  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( ( A lcm  B
) lcm  C )  e.  NN )
1705adantl 473 . . . . 5  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  { A ,  B ,  C }  C_  ZZ )
171 tpfi 7865 . . . . . 6  |-  { A ,  B ,  C }  e.  Fin
172171a1i 11 . . . . 5  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  { A ,  B ,  C }  e.  Fin )
1733a1i 11 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
0  e.  { A ,  B ,  C }  <->  ( 0  =  A  \/  0  =  B  \/  0  =  C )
) )
174173biimpd 212 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
0  e.  { A ,  B ,  C }  ->  ( 0  =  A  \/  0  =  B  \/  0  =  C ) ) )
175174con3d 140 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  ->  -.  0  e.  { A ,  B ,  C } ) )
176175impcom 437 . . . . . 6  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  -.  0  e.  { A ,  B ,  C }
)
177 df-nel 2644 . . . . . 6  |-  ( 0  e/  { A ,  B ,  C }  <->  -.  0  e.  { A ,  B ,  C }
)
178176, 177sylibr 217 . . . . 5  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
0  e/  { A ,  B ,  C }
)
179 lcmf 14685 . . . . 5  |-  ( ( ( ( A lcm  B
) lcm  C )  e.  NN  /\  ( { A ,  B ,  C }  C_  ZZ  /\  { A ,  B ,  C }  e.  Fin  /\  0  e/  { A ,  B ,  C }
) )  ->  (
( ( A lcm  B
) lcm  C )  =  (lcm `  { A ,  B ,  C }
)  <->  ( A. m  e.  { A ,  B ,  C } m  ||  ( ( A lcm  B
) lcm  C )  /\  A. k  e.  NN  ( A. m  e.  { A ,  B ,  C }
m  ||  k  ->  ( ( A lcm  B ) lcm 
C )  <_  k
) ) ) )
180169, 170, 172, 178, 179syl13anc 1294 . . . 4  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( ( ( A lcm 
B ) lcm  C )  =  (lcm `  { A ,  B ,  C }
)  <->  ( A. m  e.  { A ,  B ,  C } m  ||  ( ( A lcm  B
) lcm  C )  /\  A. k  e.  NN  ( A. m  e.  { A ,  B ,  C }
m  ||  k  ->  ( ( A lcm  B ) lcm 
C )  <_  k
) ) ) )
181155, 180mpbird 240 . . 3  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( ( A lcm  B
) lcm  C )  =  (lcm `  { A ,  B ,  C }
) )
182181eqcomd 2477 . 2  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
(lcm `  { A ,  B ,  C }
)  =  ( ( A lcm  B ) lcm  C
) )
18350, 182pm2.61ian 807 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (lcm `  { A ,  B ,  C } )  =  ( ( A lcm  B
) lcm  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    \/ w3o 1006    /\ w3a 1007    = wceq 1452    e. wcel 1904    e/ wnel 2642   A.wral 2756    C_ wss 3390   {ctp 3963   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Fincfn 7587   0cc0 9557    <_ cle 9694   NNcn 10631   NN0cn0 10893   ZZcz 10961    || cdvds 14382   lcm clcm 14626  lcmclcmf 14627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-prod 14037  df-dvds 14383  df-gcd 14548  df-lcm 14630  df-lcmf 14632
This theorem is referenced by:  lcmf2a3a4e12  14699
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