Theorem diveq0d

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

	Ref	Expression
Hypotheses	div1d.1	$$⊢\phi \to A\in ℂ$$
	divcld.2	$$⊢\phi \to B\in ℂ$$
	divcld.3	$$⊢\phi \to B\ne 0$$
	diveq0d.4	$$⊢\phi \to \frac{A}{B}=0$$
Assertion	diveq0d	$$⊢\phi \to A=0$$

Proof

Step	Hyp	Ref	Expression
1		div1d.1	$$⊢\phi \to A\in ℂ$$
2		divcld.2	$$⊢\phi \to B\in ℂ$$
3		divcld.3	$$⊢\phi \to B\ne 0$$
4		diveq0d.4	$$⊢\phi \to \frac{A}{B}=0$$
5		diveq0	$$⊢\left(A\in ℂ\wedge B\in ℂ\wedge B\ne 0\right)\to \left(\frac{A}{B}=0\leftrightarrow A=0\right)$$
6	1 2 3 5	syl3anc	$$⊢\phi \to \left(\frac{A}{B}=0\leftrightarrow A=0\right)$$
7	4 6	mpbid	$$⊢\phi \to A=0$$