Metamath Proof Explorer

Theorem lcm4un

Description: Least common multiple of natural numbers up to 4 equals 12. (Contributed by metakunt, 25-Apr-2024)

Ref Expression
Assertion lcm4un ( lcm ‘ ( 1 ... 4 ) ) = 1 2

Proof

Step Hyp Ref Expression
1 4nn ⊢ 4 ∈ ℕ
2 id ⊢ ( 4 ∈ ℕ → 4 ∈ ℕ )
3 2 lcmfunnnd ⊢ ( 4 ∈ ℕ → ( lcm ‘ ( 1 ... 4 ) ) = ( ( lcm ‘ ( 1 ... ( 4 − 1 ) ) ) lcm 4 ) )
4 1 3 ax-mp ⊢ ( lcm ‘ ( 1 ... 4 ) ) = ( ( lcm ‘ ( 1 ... ( 4 − 1 ) ) ) lcm 4 )
5 4m1e3 ⊢ ( 4 − 1 ) = 3
6 5 oveq2i ⊢ ( 1 ... ( 4 − 1 ) ) = ( 1 ... 3 )
7 6 fveq2i ⊢ ( lcm ‘ ( 1 ... ( 4 − 1 ) ) ) = ( lcm ‘ ( 1 ... 3 ) )
8 lcm3un ⊢ ( lcm ‘ ( 1 ... 3 ) ) = 6
9 7 8 eqtri ⊢ ( lcm ‘ ( 1 ... ( 4 − 1 ) ) ) = 6
10 9 oveq1i ⊢ ( ( lcm ‘ ( 1 ... ( 4 − 1 ) ) ) lcm 4 ) = ( 6 lcm 4 )
11 6lcm4e12 ⊢ ( 6 lcm 4 ) = 1 2
12 4 10 11 3eqtri ⊢ ( lcm ‘ ( 1 ... 4 ) ) = 1 2