Metamath Proof Explorer

Theorem lcm6un

Description: Least common multiple of natural numbers up to 6 equals 60. (Contributed by metakunt, 25-Apr-2024)

Ref Expression
Assertion lcm6un 
|- ( _lcm ` ( 1 ... 6 ) ) = ; 6 0

Proof

Step Hyp Ref Expression
1 6nn 
 |-  6 e. NN
2 1 a1i 
 |-  ( 6 e. NN -> 6 e. NN )
3 2 lcmfunnnd 
 |-  ( 6 e. NN -> ( _lcm ` ( 1 ... 6 ) ) = ( ( _lcm ` ( 1 ... ( 6 - 1 ) ) ) lcm 6 ) )
4 1 3 ax-mp 
 |-  ( _lcm ` ( 1 ... 6 ) ) = ( ( _lcm ` ( 1 ... ( 6 - 1 ) ) ) lcm 6 )
5 6m1e5 
 |-  ( 6 - 1 ) = 5
6 5 oveq2i 
 |-  ( 1 ... ( 6 - 1 ) ) = ( 1 ... 5 )
7 6 fveq2i 
 |-  ( _lcm ` ( 1 ... ( 6 - 1 ) ) ) = ( _lcm ` ( 1 ... 5 ) )
8 7 oveq1i 
 |-  ( ( _lcm ` ( 1 ... ( 6 - 1 ) ) ) lcm 6 ) = ( ( _lcm ` ( 1 ... 5 ) ) lcm 6 )
9 lcm5un 
 |-  ( _lcm ` ( 1 ... 5 ) ) = ; 6 0
10 9 oveq1i 
 |-  ( ( _lcm ` ( 1 ... 5 ) ) lcm 6 ) = ( ; 6 0 lcm 6 )
11 8 10 eqtri 
 |-  ( ( _lcm ` ( 1 ... ( 6 - 1 ) ) ) lcm 6 ) = ( ; 6 0 lcm 6 )
12 60lcm6e60 
 |-  ( ; 6 0 lcm 6 ) = ; 6 0
13 4 11 12 3eqtri 
 |-  ( _lcm ` ( 1 ... 6 ) ) = ; 6 0