Metamath Proof Explorer


Theorem mulne0

Description: The product of two nonzero numbers is nonzero. (Contributed by NM, 30-Dec-2007)

Ref Expression
Assertion mulne0 ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 · 𝐵 ) ≠ 0 )

Proof

Step Hyp Ref Expression
1 mulne0b ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ( 𝐴 · 𝐵 ) ≠ 0 ) )
2 1 biimpa ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 · 𝐵 ) ≠ 0 )
3 2 an4s ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 · 𝐵 ) ≠ 0 )