Description: The lcm of 12 and 5 is 60. (Contributed by metakunt, 25-Apr-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | 12lcm5e60 | ⊢ ( ; 1 2 lcm 5 ) = ; 6 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 | ⊢ 1 ∈ ℕ0 | |
2 | 2nn | ⊢ 2 ∈ ℕ | |
3 | 1 2 | decnncl | ⊢ ; 1 2 ∈ ℕ |
4 | 5nn | ⊢ 5 ∈ ℕ | |
5 | 1nn | ⊢ 1 ∈ ℕ | |
6 | 6nn | ⊢ 6 ∈ ℕ | |
7 | 6 | decnncl2 | ⊢ ; 6 0 ∈ ℕ |
8 | 12gcd5e1 | ⊢ ( ; 1 2 gcd 5 ) = 1 | |
9 | 6nn0 | ⊢ 6 ∈ ℕ0 | |
10 | 0nn0 | ⊢ 0 ∈ ℕ0 | |
11 | 9 10 | deccl | ⊢ ; 6 0 ∈ ℕ0 |
12 | 11 | nn0cni | ⊢ ; 6 0 ∈ ℂ |
13 | 12 | mulid2i | ⊢ ( 1 · ; 6 0 ) = ; 6 0 |
14 | 5nn0 | ⊢ 5 ∈ ℕ0 | |
15 | 2nn0 | ⊢ 2 ∈ ℕ0 | |
16 | eqid | ⊢ ; 1 2 = ; 1 2 | |
17 | 5cn | ⊢ 5 ∈ ℂ | |
18 | 17 | mulid2i | ⊢ ( 1 · 5 ) = 5 |
19 | 18 | oveq1i | ⊢ ( ( 1 · 5 ) + 1 ) = ( 5 + 1 ) |
20 | 5p1e6 | ⊢ ( 5 + 1 ) = 6 | |
21 | 19 20 | eqtri | ⊢ ( ( 1 · 5 ) + 1 ) = 6 |
22 | 2cn | ⊢ 2 ∈ ℂ | |
23 | 5t2e10 | ⊢ ( 5 · 2 ) = ; 1 0 | |
24 | 17 22 23 | mulcomli | ⊢ ( 2 · 5 ) = ; 1 0 |
25 | 14 1 15 16 10 1 21 24 | decmul1c | ⊢ ( ; 1 2 · 5 ) = ; 6 0 |
26 | 3 4 5 7 8 13 25 | lcmeprodgcdi | ⊢ ( ; 1 2 lcm 5 ) = ; 6 0 |