Theorem mulne0i

Description: The product of two nonzero numbers is nonzero. (Contributed by NM, 15-Feb-1995)

	Ref	Expression
Hypotheses	muln0.1	⊢ 𝐴 ∈ ℂ
	muln0.2	⊢ 𝐵 ∈ ℂ
	muln0.3	⊢ 𝐴 ≠ 0
	muln0.4	⊢ 𝐵 ≠ 0
Assertion	mulne0i	⊢ ( 𝐴 · 𝐵 ) ≠ 0

Proof

Step	Hyp	Ref	Expression
1		muln0.1	⊢ 𝐴 ∈ ℂ
2		muln0.2	⊢ 𝐵 ∈ ℂ
3		muln0.3	⊢ 𝐴 ≠ 0
4		muln0.4	⊢ 𝐵 ≠ 0
5		mulne0	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 · 𝐵 ) ≠ 0 )
6	1 3 2 4 5	mp4an	⊢ ( 𝐴 · 𝐵 ) ≠ 0