Theorem lcmfeq0b

Description: The least common multiple of a set of integers is 0 iff at least one of its element is 0. (Contributed by AV, 21-Aug-2020)

	Ref	Expression
Assertion	lcmfeq0b	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ) → ( ( lcm ‘ 𝑍 ) = 0 ↔ 0 ∈ 𝑍 ) )

Proof

Step	Hyp	Ref	Expression
1		df-nel	⊢ ( 0 ∉ 𝑍 ↔ ¬ 0 ∈ 𝑍 )
2		lcmfn0cl	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍 ) → ( lcm ‘ 𝑍 ) ∈ ℕ )
3	2	nnne0d	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍 ) → ( lcm ‘ 𝑍 ) ≠ 0 )
4	3	3expia	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ) → ( 0 ∉ 𝑍 → ( lcm ‘ 𝑍 ) ≠ 0 ) )
5	1 4	syl5bir	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ) → ( ¬ 0 ∈ 𝑍 → ( lcm ‘ 𝑍 ) ≠ 0 ) )
6	5	necon4bd	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ) → ( ( lcm ‘ 𝑍 ) = 0 → 0 ∈ 𝑍 ) )
7		lcmf0val	⊢ ( ( 𝑍 ⊆ ℤ ∧ 0 ∈ 𝑍 ) → ( lcm ‘ 𝑍 ) = 0 )
8	7	ex	⊢ ( 𝑍 ⊆ ℤ → ( 0 ∈ 𝑍 → ( lcm ‘ 𝑍 ) = 0 ) )
9	8	adantr	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ) → ( 0 ∈ 𝑍 → ( lcm ‘ 𝑍 ) = 0 ) )
10	6 9	impbid	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ) → ( ( lcm ‘ 𝑍 ) = 0 ↔ 0 ∈ 𝑍 ) )