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Theorem lcmftp 14602
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 14610, 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 10945 . . . . . . 7  |-  0  e.  ZZ
2 eltpg 4013 . . . . . . 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 210 . . . . 5  |-  ( ( 0  =  A  \/  0  =  B  \/  0  =  C )  ->  0  e.  { A ,  B ,  C }
)
5 tpssi 4137 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  { A ,  B ,  C }  C_  ZZ )
64, 5anim12ci 570 . . . 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 14585 . . . 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 10946 . . . . . . . . . 10  |-  ( C  e.  ZZ  ->  0  e.  ZZ )
10 lcmcom 14550 . . . . . . . . . 10  |-  ( ( 0  e.  ZZ  /\  C  e.  ZZ )  ->  ( 0 lcm  C )  =  ( C lcm  0
) )
119, 10mpancom 674 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  (
0 lcm  C )  =  ( C lcm  0 ) )
12 lcm0val 14551 . . . . . . . . 9  |-  ( C  e.  ZZ  ->  ( C lcm  0 )  =  0 )
1311, 12eqtrd 2484 . . . . . . . 8  |-  ( C  e.  ZZ  ->  (
0 lcm  C )  =  0 )
1413eqcomd 2456 . . . . . . 7  |-  ( C  e.  ZZ  ->  0  =  ( 0 lcm  C
) )
15143ad2ant3 1030 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  0  =  ( 0 lcm  C
) )
1615adantl 468 . . . . 5  |-  ( ( 0  =  A  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  0  =  ( 0 lcm  C )
)
17 0zd 10946 . . . . . . . . . . 11  |-  ( B  e.  ZZ  ->  0  e.  ZZ )
18 lcmcom 14550 . . . . . . . . . . 11  |-  ( ( 0  e.  ZZ  /\  B  e.  ZZ )  ->  ( 0 lcm  B )  =  ( B lcm  0
) )
1917, 18mpancom 674 . . . . . . . . . 10  |-  ( B  e.  ZZ  ->  (
0 lcm  B )  =  ( B lcm  0 ) )
20 lcm0val 14551 . . . . . . . . . 10  |-  ( B  e.  ZZ  ->  ( B lcm  0 )  =  0 )
2119, 20eqtrd 2484 . . . . . . . . 9  |-  ( B  e.  ZZ  ->  (
0 lcm  B )  =  0 )
2221eqcomd 2456 . . . . . . . 8  |-  ( B  e.  ZZ  ->  0  =  ( 0 lcm  B
) )
23223ad2ant2 1029 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  0  =  ( 0 lcm  B
) )
2423adantl 468 . . . . . 6  |-  ( ( 0  =  A  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  0  =  ( 0 lcm  B )
)
2524oveq1d 6303 . . . . 5  |-  ( ( 0  =  A  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  ( 0 lcm 
C )  =  ( ( 0 lcm  B ) lcm 
C ) )
26 oveq1 6295 . . . . . . 7  |-  ( 0  =  A  ->  (
0 lcm  B )  =  ( A lcm  B ) )
2726oveq1d 6303 . . . . . 6  |-  ( 0  =  A  ->  (
( 0 lcm  B ) lcm 
C )  =  ( ( A lcm  B ) lcm 
C ) )
2827adantr 467 . . . . 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 2488 . . . 4  |-  ( ( 0  =  A  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  0  =  ( ( A lcm  B
) lcm  C ) )
30 lcm0val 14551 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( A lcm  0 )  =  0 )
3130eqcomd 2456 . . . . . . . 8  |-  ( A  e.  ZZ  ->  0  =  ( A lcm  0
) )
32313ad2ant1 1028 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  0  =  ( A lcm  0
) )
3332adantl 468 . . . . . 6  |-  ( ( 0  =  B  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  0  =  ( A lcm  0 )
)
3433oveq1d 6303 . . . . 5  |-  ( ( 0  =  B  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  ( 0 lcm 
C )  =  ( ( A lcm  0 ) lcm 
C ) )
35133ad2ant3 1030 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
0 lcm  C )  =  0 )
3635adantl 468 . . . . 5  |-  ( ( 0  =  B  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  ( 0 lcm 
C )  =  0 )
37 oveq2 6296 . . . . . . 7  |-  ( 0  =  B  ->  ( A lcm  0 )  =  ( A lcm  B ) )
3837adantr 467 . . . . . 6  |-  ( ( 0  =  B  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  ( A lcm  0 )  =  ( A lcm  B ) )
3938oveq1d 6303 . . . . 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 2492 . . . 4  |-  ( ( 0  =  B  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  0  =  ( ( A lcm  B
) lcm  C ) )
41 lcmcl 14559 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A lcm  B )  e.  NN0 )
4241nn0zd 11035 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A lcm  B )  e.  ZZ )
43 lcm0val 14551 . . . . . . . 8  |-  ( ( A lcm  B )  e.  ZZ  ->  ( ( A lcm  B ) lcm  0 )  =  0 )
4443eqcomd 2456 . . . . . . 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 1027 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  0  =  ( ( A lcm 
B ) lcm  0 ) )
47 oveq2 6296 . . . . 5  |-  ( 0  =  C  ->  (
( A lcm  B ) lcm  0 )  =  ( ( A lcm  B ) lcm 
C ) )
4846, 47sylan9eqr 2506 . . . 4  |-  ( ( 0  =  C  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  ->  0  =  ( ( A lcm  B
) lcm  C ) )
4929, 40, 483jaoian 1332 . . 3  |-  ( ( ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
0  =  ( ( A lcm  B ) lcm  C
) )
508, 49eqtrd 2484 . 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 1027 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A lcm  B )  e.  ZZ )
52 simp3 1009 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  C  e.  ZZ )
5351, 52jca 535 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A lcm  B )  e.  ZZ  /\  C  e.  ZZ ) )
5453adantl 468 . . . . . . . 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 14556 . . . . . . . 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 14556 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  ||  ( A lcm  B )  /\  B  ||  ( A lcm  B ) ) )
58573adant3 1027 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  ||  ( A lcm  B
)  /\  B  ||  ( A lcm  B ) ) )
59 simp1 1007 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  A  e.  ZZ )
60 lcmcl 14559 . . . . . . . . . . . . . . . . . . . 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 11035 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
( A lcm  B ) lcm 
C )  e.  ZZ )
6359, 51, 623jca 1187 . . . . . . . . . . . . . . . . 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 14330 . . . . . . . . . . . . . . . . 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 438 . . . . . . . . . . . . . . 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 32 . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . 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 37 . . . . . . . . . . . 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 468 . . . . . . . . . . 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 32 . . . . . . . . . 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 467 . . . . . . . . 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 432 . . . . . . . 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 463 . . . . . . . . . . . . . . 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 1027 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  B  ||  ( A lcm  B ) )
7776adantl 468 . . . . . . . . . . . 12  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  B  ||  ( A lcm  B
) )
78 simp2 1008 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  B  e.  ZZ )
7978, 51, 623jca 1187 . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . 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 14330 . . . . . . . . . . . . 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 680 . . . . . . . . . . 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 32 . . . . . . . . . 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 467 . . . . . . . . 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 432 . . . . . . . 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 463 . . . . . . . . 9  |-  ( ( ( A lcm  B ) 
||  ( ( A lcm 
B ) lcm  C )  /\  C  ||  (
( A lcm  B ) lcm 
C ) )  ->  C  ||  ( ( A lcm 
B ) lcm  C ) )
8887adantl 468 . . . . . . . 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 1187 . . . . . . 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 673 . . . . . 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 4404 . . . . . . . 8  |-  ( m  =  A  ->  (
m  ||  ( ( A lcm  B ) lcm  C )  <-> 
A  ||  ( ( A lcm  B ) lcm  C ) ) )
92 breq1 4404 . . . . . . . 8  |-  ( m  =  B  ->  (
m  ||  ( ( A lcm  B ) lcm  C )  <-> 
B  ||  ( ( A lcm  B ) lcm  C ) ) )
93 breq1 4404 . . . . . . . 8  |-  ( m  =  C  ->  (
m  ||  ( ( A lcm  B ) lcm  C )  <-> 
C  ||  ( ( A lcm  B ) lcm  C ) ) )
9491, 92, 93raltpg 4022 . . . . . . 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 468 . . . . . 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 236 . . . . 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 4404 . . . . . . . . 9  |-  ( m  =  A  ->  (
m  ||  k  <->  A  ||  k
) )
98 breq1 4404 . . . . . . . . 9  |-  ( m  =  B  ->  (
m  ||  k  <->  B  ||  k
) )
99 breq1 4404 . . . . . . . . 9  |-  ( m  =  C  ->  (
m  ||  k  <->  C  ||  k
) )
10097, 98, 99raltpg 4022 . . . . . . . 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 732 . . . . . . 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 463 . . . . . . . . . . 11  |-  ( ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )
)  /\  k  e.  NN )  ->  k  e.  NN )
10351ad2antlr 732 . . . . . . . . . . 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 732 . . . . . . . . . . 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 1187 . . . . . . . . . 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 467 . . . . . . . . 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 1002 . . . . . . . . . . . . . . . . 17  |-  ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  <->  ( -.  0  =  A  /\  -.  0  =  B  /\  -.  0  =  C ) )
108 eqcom 2457 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  =  A  <->  A  = 
0 )
109108notbii 298 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  0  =  A  <->  -.  A  =  0 )
110 eqcom 2457 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0  =  B  <->  B  = 
0 )
111110notbii 298 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  0  =  B  <->  -.  B  =  0 )
112109, 111anbi12i 702 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( -.  0  =  A  /\  -.  0  =  B )  <->  ( -.  A  =  0  /\  -.  B  =  0
) )
113112biimpi 198 . . . . . . . . . . . . . . . . . . 19  |-  ( ( -.  0  =  A  /\  -.  0  =  B )  ->  ( -.  A  =  0  /\  -.  B  =  0 ) )
114 ioran 493 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  ( A  =  0  \/  B  =  0 )  <->  ( -.  A  =  0  /\  -.  B  =  0 ) )
115113, 114sylibr 216 . . . . . . . . . . . . . . . . . 18  |-  ( ( -.  0  =  A  /\  -.  0  =  B )  ->  -.  ( A  =  0  \/  B  =  0
) )
1161153adant3 1027 . . . . . . . . . . . . . . . . 17  |-  ( ( -.  0  =  A  /\  -.  0  =  B  /\  -.  0  =  C )  ->  -.  ( A  =  0  \/  B  =  0
) )
117107, 116sylbi 199 . . . . . . . . . . . . . . . 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 1027 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( A  e.  ZZ  /\  B  e.  ZZ ) )
120117, 119anim12ci 570 . . . . . . . . . . . . . . 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 14555 . . . . . . . . . . . . . . 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 10639 . . . . . . . . . . . . . . 15  |-  ( ( A lcm  B )  e.  NN  ->  ( A lcm  B )  =/=  0 )
124123neneqd 2628 . . . . . . . . . . . . . 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 2457 . . . . . . . . . . . . . . . . . 18  |-  ( 0  =  C  <->  C  = 
0 )
127126notbii 298 . . . . . . . . . . . . . . . . 17  |-  ( -.  0  =  C  <->  -.  C  =  0 )
128127biimpi 198 . . . . . . . . . . . . . . . 16  |-  ( -.  0  =  C  ->  -.  C  =  0
)
1291283ad2ant3 1030 . . . . . . . . . . . . . . 15  |-  ( ( -.  0  =  A  /\  -.  0  =  B  /\  -.  0  =  C )  ->  -.  C  =  0 )
130107, 129sylbi 199 . . . . . . . . . . . . . 14  |-  ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  ->  -.  C  =  0 )
131130adantr 467 . . . . . . . . . . . . 13  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  -.  C  =  0
)
132125, 131jca 535 . . . . . . . . . . . 12  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( -.  ( A lcm 
B )  =  0  /\  -.  C  =  0 ) )
133132adantr 467 . . . . . . . . . . 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 467 . . . . . . . . . 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 493 . . . . . . . . . 10  |-  ( -.  ( ( A lcm  B
)  =  0  \/  C  =  0 )  <-> 
( -.  ( A lcm 
B )  =  0  /\  -.  C  =  0 ) )
136134, 135sylibr 216 . . . . . . . . 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 468 . . . . . . . . . . . . . . 15  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( A  e.  ZZ  /\  B  e.  ZZ ) )
138 nnz 10956 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  k  e.  ZZ )
139137, 138anim12ci 570 . . . . . . . . . . . . . 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 988 . . . . . . . . . . . . . 14  |-  ( ( k  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  <->  ( k  e.  ZZ  /\  ( A  e.  ZZ  /\  B  e.  ZZ ) ) )
141139, 140sylibr 216 . . . . . . . . . . . . 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 14566 . . . . . . . . . . . . 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 32 . . . . . . . . . . 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 1027 . . . . . . . . . 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 432 . . . . . . . . 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 1009 . . . . . . . . . 10  |-  ( ( A  ||  k  /\  B  ||  k  /\  C  ||  k )  ->  C  ||  k )
148147adantl 468 . . . . . . . . 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 14557 . . . . . . . . . 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 431 . . . . . . . . 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 1268 . . . . . . . 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 436 . . . . . . 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 219 . . . . . 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 2801 . . . . 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 535 . . . 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 198 . . . . . . . . . . . . . . . 16  |-  ( -.  0  =  A  ->  -.  A  =  0
)
157111biimpi 198 . . . . . . . . . . . . . . . 16  |-  ( -.  0  =  B  ->  -.  B  =  0
)
158156, 157anim12i 569 . . . . . . . . . . . . . . 15  |-  ( ( -.  0  =  A  /\  -.  0  =  B )  ->  ( -.  A  =  0  /\  -.  B  =  0 ) )
159158, 114sylibr 216 . . . . . . . . . . . . . 14  |-  ( ( -.  0  =  A  /\  -.  0  =  B )  ->  -.  ( A  =  0  \/  B  =  0
) )
1601593adant3 1027 . . . . . . . . . . . . 13  |-  ( ( -.  0  =  A  /\  -.  0  =  B  /\  -.  0  =  C )  ->  -.  ( A  =  0  \/  B  =  0
) )
161107, 160sylbi 199 . . . . . . . . . . . 12  |-  ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  ->  -.  ( A  =  0  \/  B  =  0 ) )
162161, 119anim12ci 570 . . . . . . . . . . 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 535 . . . . . . . 8  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
( -.  ( A lcm 
B )  =  0  /\  -.  C  =  0 ) )
166165, 135sylibr 216 . . . . . . 7  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  -.  ( ( A lcm  B
)  =  0  \/  C  =  0 ) )
16754, 166jca 535 . . . . . 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 14555 . . . . . 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 468 . . . . 5  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  { A ,  B ,  C }  C_  ZZ )
171 tpfi 7844 . . . . . 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 211 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  (
0  e.  { A ,  B ,  C }  ->  ( 0  =  A  \/  0  =  B  \/  0  =  C ) ) )
175174con3d 139 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ )  ->  ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  ->  -.  0  e.  { A ,  B ,  C } ) )
176175impcom 432 . . . . . 6  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  ->  -.  0  e.  { A ,  B ,  C }
)
177 df-nel 2624 . . . . . 6  |-  ( 0  e/  { A ,  B ,  C }  <->  -.  0  e.  { A ,  B ,  C }
)
178176, 177sylibr 216 . . . . 5  |-  ( ( -.  ( 0  =  A  \/  0  =  B  \/  0  =  C )  /\  ( A  e.  ZZ  /\  B  e.  ZZ  /\  C  e.  ZZ ) )  -> 
0  e/  { A ,  B ,  C }
)
179 lcmf 14599 . . . . 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 1269 . . . 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 236 . . 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 2456 . 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 798 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 188    \/ wo 370    /\ wa 371    \/ w3o 983    /\ w3a 984    = wceq 1443    e. wcel 1886    e/ wnel 2622   A.wral 2736    C_ wss 3403   {ctp 3971   class class class wbr 4401   ` cfv 5581  (class class class)co 6288   Fincfn 7566   0cc0 9536    <_ cle 9673   NNcn 10606   NN0cn0 10866   ZZcz 10934    || cdvds 14298   lcm clcm 14540  lcmclcmf 14541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-prod 13953  df-dvds 14299  df-gcd 14462  df-lcm 14544  df-lcmf 14546
This theorem is referenced by:  lcmf2a3a4e12  14613
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