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 ) )