Theorem divne0i

Description: The ratio of nonzero numbers is nonzero. (Contributed by NM, 9-Feb-1995)

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
	divneq0.3	⊢ 𝐴 ≠ 0
	divneq0.4	⊢ 𝐵 ≠ 0
Assertion	divne0i	⊢ ( 𝐴 / 𝐵 ) ≠ 0

Proof

Step	Hyp	Ref	Expression
1		divclz.1	⊢ 𝐴 ∈ ℂ
2		divclz.2	⊢ 𝐵 ∈ ℂ
3		divneq0.3	⊢ 𝐴 ≠ 0
4		divneq0.4	⊢ 𝐵 ≠ 0
5		divne0	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 / 𝐵 ) ≠ 0 )
6	1 3 2 4 5	mp4an	⊢ ( 𝐴 / 𝐵 ) ≠ 0