Theorem diveq0d

Description: A ratio is zero iff the numerator is zero. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	divcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	divcld.3	⊢ ( 𝜑 → 𝐵 ≠ 0 )
	diveq0d.4	⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) = 0 )
Assertion	diveq0d	⊢ ( 𝜑 → 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		diveq0d.4	⊢ ( 𝜑 → ( 𝐴 / 𝐵 ) = 0 )
5		diveq0	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = 0 ) )
6	1 2 3 5	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 / 𝐵 ) = 0 ↔ 𝐴 = 0 ) )
7	4 6	mpbid	⊢ ( 𝜑 → 𝐴 = 0 )