Theorem mulne0d

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

	Ref	Expression
Hypotheses	msq0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	mul0ord.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	mulne0d.3	⊢ ( 𝜑 → 𝐴 ≠ 0 )
	mulne0d.4	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	mulne0d	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ≠ 0 )

Proof

Step	Hyp	Ref	Expression
1		msq0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		mul0ord.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		mulne0d.3	⊢ ( 𝜑 → 𝐴 ≠ 0 )
4		mulne0d.4	⊢ ( 𝜑 → 𝐵 ≠ 0 )
5	1 2	mulne0bd	⊢ ( 𝜑 → ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ↔ ( 𝐴 · 𝐵 ) ≠ 0 ) )
6	3 4 5	mpbi2and	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ≠ 0 )