Metamath Proof Explorer

Theorem lcm7un

Description: Least common multiple of natural numbers up to 7 equals 420. (Contributed by metakunt, 25-Apr-2024)

Ref Expression
Assertion lcm7un ( lcm ‘ ( 1 ... 7 ) ) = 4 2 0

Proof

Step Hyp Ref Expression
1 7nn ⊢ 7 ∈ ℕ
2 id ⊢ ( 7 ∈ ℕ → 7 ∈ ℕ )
3 2 lcmfunnnd ⊢ ( 7 ∈ ℕ → ( lcm ‘ ( 1 ... 7 ) ) = ( ( lcm ‘ ( 1 ... ( 7 − 1 ) ) ) lcm 7 ) )
4 1 3 ax-mp ⊢ ( lcm ‘ ( 1 ... 7 ) ) = ( ( lcm ‘ ( 1 ... ( 7 − 1 ) ) ) lcm 7 )
5 7m1e6 ⊢ ( 7 − 1 ) = 6
6 5 oveq2i ⊢ ( 1 ... ( 7 − 1 ) ) = ( 1 ... 6 )
7 6 fveq2i ⊢ ( lcm ‘ ( 1 ... ( 7 − 1 ) ) ) = ( lcm ‘ ( 1 ... 6 ) )
8 7 oveq1i ⊢ ( ( lcm ‘ ( 1 ... ( 7 − 1 ) ) ) lcm 7 ) = ( ( lcm ‘ ( 1 ... 6 ) ) lcm 7 )
9 lcm6un ⊢ ( lcm ‘ ( 1 ... 6 ) ) = 6 0
10 9 oveq1i ⊢ ( ( lcm ‘ ( 1 ... 6 ) ) lcm 7 ) = ( 6 0 lcm 7 )
11 8 10 eqtri ⊢ ( ( lcm ‘ ( 1 ... ( 7 − 1 ) ) ) lcm 7 ) = ( 6 0 lcm 7 )
12 60lcm7e420 ⊢ ( 6 0 lcm 7 ) = 4 2 0
13 4 11 12 3eqtri ⊢ ( lcm ‘ ( 1 ... 7 ) ) = 4 2 0