Theorem mulne0bd

Description: The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	msq0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	mulne0bd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	mulne0bd	⊢ ( 𝜑 → ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ( 𝐴 · 𝐵 ) ≠ 0 ) )

Proof

Step	Hyp	Ref	Expression
1		msq0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		mulne0bd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		mulne0b	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ( 𝐴 · 𝐵 ) ≠ 0 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ( 𝐴 · 𝐵 ) ≠ 0 ) )