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Metamath Proof Explorer

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 , < ) )