Metamath Proof Explorer


Theorem mulne0b

Description: The product of two nonzero numbers is nonzero. (Contributed by NM, 1-Aug-2004) (Proof shortened by Andrew Salmon, 19-Nov-2011)

Ref Expression
Assertion mulne0b ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ( 𝐴 · 𝐵 ) ≠ 0 ) )

Proof

Step Hyp Ref Expression
1 neanior ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ¬ ( 𝐴 = 0 ∨ 𝐵 = 0 ) )
2 mul0or ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) = 0 ↔ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) )
3 2 necon3abid ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ≠ 0 ↔ ¬ ( 𝐴 = 0 ∨ 𝐵 = 0 ) ) )
4 1 3 bitr4id ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ( 𝐴 · 𝐵 ) ≠ 0 ) )