Metamath Proof Explorer

Theorem lcm5un

Description: Least common multiple of natural numbers up to 5 equals 60. (Contributed by metakunt, 25-Apr-2024)

Ref Expression
Assertion lcm5un 
|- ( _lcm ` ( 1 ... 5 ) ) = ; 6 0

Proof

Step Hyp Ref Expression
1 5nn 
 |-  5 e. NN
2 1 a1i 
 |-  ( 5 e. NN -> 5 e. NN )
3 2 lcmfunnnd 
 |-  ( 5 e. NN -> ( _lcm ` ( 1 ... 5 ) ) = ( ( _lcm ` ( 1 ... ( 5 - 1 ) ) ) lcm 5 ) )
4 1 3 ax-mp 
 |-  ( _lcm ` ( 1 ... 5 ) ) = ( ( _lcm ` ( 1 ... ( 5 - 1 ) ) ) lcm 5 )
5 5m1e4 
 |-  ( 5 - 1 ) = 4
6 5 oveq2i 
 |-  ( 1 ... ( 5 - 1 ) ) = ( 1 ... 4 )
7 6 fveq2i 
 |-  ( _lcm ` ( 1 ... ( 5 - 1 ) ) ) = ( _lcm ` ( 1 ... 4 ) )
8 7 oveq1i 
 |-  ( ( _lcm ` ( 1 ... ( 5 - 1 ) ) ) lcm 5 ) = ( ( _lcm ` ( 1 ... 4 ) ) lcm 5 )
9 lcm4un 
 |-  ( _lcm ` ( 1 ... 4 ) ) = ; 1 2
10 9 oveq1i 
 |-  ( ( _lcm ` ( 1 ... 4 ) ) lcm 5 ) = ( ; 1 2 lcm 5 )
11 8 10 eqtri 
 |-  ( ( _lcm ` ( 1 ... ( 5 - 1 ) ) ) lcm 5 ) = ( ; 1 2 lcm 5 )
12 12lcm5e60 
 |-  ( ; 1 2 lcm 5 ) = ; 6 0
13 4 11 12 3eqtri 
 |-  ( _lcm ` ( 1 ... 5 ) ) = ; 6 0