Theorem lcmfn0val

Description: The value of the _lcm function for a set without 0. (Contributed by AV, 21-Aug-2020) (Revised by AV, 16-Sep-2020)

	Ref	Expression
Assertion	lcmfn0val	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍 ) → ( lcm ‘ 𝑍 ) = inf ( { 𝑛 ∈ ℕ ∣ ∀ 𝑚 ∈ 𝑍 𝑚 ∥ 𝑛 } , ℝ , < ) )

Proof

Step	Hyp	Ref	Expression
1		lcmfval	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ) → ( lcm ‘ 𝑍 ) = if ( 0 ∈ 𝑍 , 0 , inf ( { 𝑛 ∈ ℕ ∣ ∀ 𝑚 ∈ 𝑍 𝑚 ∥ 𝑛 } , ℝ , < ) ) )
2	1	3adant3	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍 ) → ( lcm ‘ 𝑍 ) = if ( 0 ∈ 𝑍 , 0 , inf ( { 𝑛 ∈ ℕ ∣ ∀ 𝑚 ∈ 𝑍 𝑚 ∥ 𝑛 } , ℝ , < ) ) )
3		df-nel	⊢ ( 0 ∉ 𝑍 ↔ ¬ 0 ∈ 𝑍 )
4		iffalse	⊢ ( ¬ 0 ∈ 𝑍 → if ( 0 ∈ 𝑍 , 0 , inf ( { 𝑛 ∈ ℕ ∣ ∀ 𝑚 ∈ 𝑍 𝑚 ∥ 𝑛 } , ℝ , < ) ) = inf ( { 𝑛 ∈ ℕ ∣ ∀ 𝑚 ∈ 𝑍 𝑚 ∥ 𝑛 } , ℝ , < ) )
5	3 4	sylbi	⊢ ( 0 ∉ 𝑍 → if ( 0 ∈ 𝑍 , 0 , inf ( { 𝑛 ∈ ℕ ∣ ∀ 𝑚 ∈ 𝑍 𝑚 ∥ 𝑛 } , ℝ , < ) ) = inf ( { 𝑛 ∈ ℕ ∣ ∀ 𝑚 ∈ 𝑍 𝑚 ∥ 𝑛 } , ℝ , < ) )
6	5	3ad2ant3	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍 ) → if ( 0 ∈ 𝑍 , 0 , inf ( { 𝑛 ∈ ℕ ∣ ∀ 𝑚 ∈ 𝑍 𝑚 ∥ 𝑛 } , ℝ , < ) ) = inf ( { 𝑛 ∈ ℕ ∣ ∀ 𝑚 ∈ 𝑍 𝑚 ∥ 𝑛 } , ℝ , < ) )
7	2 6	eqtrd	⊢ ( ( 𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin ∧ 0 ∉ 𝑍 ) → ( lcm ‘ 𝑍 ) = inf ( { 𝑛 ∈ ℕ ∣ ∀ 𝑚 ∈ 𝑍 𝑚 ∥ 𝑛 } , ℝ , < ) )