Theorem divne0

Description: The ratio of nonzero numbers is nonzero. (Contributed by NM, 28-Dec-2007)

	Ref	Expression
Assertion	divne0	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 / 𝐵 ) ≠ 0 )

Proof

Step	Hyp	Ref	Expression
1		divne0b	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐴 ≠ 0 ↔ ( 𝐴 / 𝐵 ) ≠ 0 ) )
2	1	3expb	⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 ≠ 0 ↔ ( 𝐴 / 𝐵 ) ≠ 0 ) )
3	2	biimpa	⊢ ( ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) ∧ 𝐴 ≠ 0 ) → ( 𝐴 / 𝐵 ) ≠ 0 )
4	3	an32s	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 𝐴 / 𝐵 ) ≠ 0 )