Metamath Proof Explorer

Theorem lcm8un

Description: Least common multiple of natural numbers up to 8 equals 840. (Contributed by metakunt, 25-Apr-2024)

Ref Expression
Assertion lcm8un 
|- ( _lcm ` ( 1 ... 8 ) ) = ; ; 8 4 0

Proof

Step Hyp Ref Expression
1 8nn 
 |-  8 e. NN
2 id 
 |-  ( 8 e. NN -> 8 e. NN )
3 2 lcmfunnnd 
 |-  ( 8 e. NN -> ( _lcm ` ( 1 ... 8 ) ) = ( ( _lcm ` ( 1 ... ( 8 - 1 ) ) ) lcm 8 ) )
4 1 3 ax-mp 
 |-  ( _lcm ` ( 1 ... 8 ) ) = ( ( _lcm ` ( 1 ... ( 8 - 1 ) ) ) lcm 8 )
5 8m1e7 
 |-  ( 8 - 1 ) = 7
6 5 oveq2i 
 |-  ( 1 ... ( 8 - 1 ) ) = ( 1 ... 7 )
7 6 fveq2i 
 |-  ( _lcm ` ( 1 ... ( 8 - 1 ) ) ) = ( _lcm ` ( 1 ... 7 ) )
8 7 oveq1i 
 |-  ( ( _lcm ` ( 1 ... ( 8 - 1 ) ) ) lcm 8 ) = ( ( _lcm ` ( 1 ... 7 ) ) lcm 8 )
9 lcm7un 
 |-  ( _lcm ` ( 1 ... 7 ) ) = ; ; 4 2 0
10 9 oveq1i 
 |-  ( ( _lcm ` ( 1 ... 7 ) ) lcm 8 ) = ( ; ; 4 2 0 lcm 8 )
11 8 10 eqtri 
 |-  ( ( _lcm ` ( 1 ... ( 8 - 1 ) ) ) lcm 8 ) = ( ; ; 4 2 0 lcm 8 )
12 420lcm8e840 
 |-  ( ; ; 4 2 0 lcm 8 ) = ; ; 8 4 0
13 4 11 12 3eqtri 
 |-  ( _lcm ` ( 1 ... 8 ) ) = ; ; 8 4 0