Theorem lcmfcl

Description: Closure of the _lcm function. (Contributed by AV, 21-Aug-2020)

	Ref	Expression
Assertion	lcmfcl	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ) → ( lcm ‘ 𝑍 ) ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		lcmf0val	⊢ ( ( 𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍 ) → ( lcm ‘ 𝑍 ) = 0 )
2		0nn0	⊢ 0 ∈ ℕ0
3	1 2	eqeltrdi	⊢ ( ( 𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍 ) → ( lcm ‘ 𝑍 ) ∈ ℕ0 )
4	3	adantlr	⊢ ( ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ) ∧ 0 ∈ 𝑍 ) → ( lcm ‘ 𝑍 ) ∈ ℕ0 )
5		df-nel	⊢ ( 0 ∉ 𝑍 ↔ ¬ 0 ∈ 𝑍 )
6		lcmfn0cl	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍 ) → ( lcm ‘ 𝑍 ) ∈ ℕ )
7	6	3expa	⊢ ( ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ) ∧ 0 ∉ 𝑍 ) → ( lcm ‘ 𝑍 ) ∈ ℕ )
8	5 7	sylan2br	⊢ ( ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ) ∧ ¬ 0 ∈ 𝑍 ) → ( lcm ‘ 𝑍 ) ∈ ℕ )
9	8	nnnn0d	⊢ ( ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ) ∧ ¬ 0 ∈ 𝑍 ) → ( lcm ‘ 𝑍 ) ∈ ℕ0 )
10	4 9	pm2.61dan	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ) → ( lcm ‘ 𝑍 ) ∈ ℕ0 )