Theorem diveq0ad

Description: A fraction of complex numbers is zero iff its numerator is. Deduction form of diveq0 . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	div1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	divcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	divcld.3	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	diveq0ad	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		div1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		divcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		divcld.3	⊢ ( 𝜑 → 𝐵 ≠ 0 )
4		diveq0	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = 0 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = 0 ) )