Theorem divgt0d

Description: The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltp1d.1	$$⊢\phi \to A\in ℝ$$
	divgt0d.2	$$⊢\phi \to B\in ℝ$$
	divgt0d.3	$$⊢\phi \to 0<A$$
	divgt0d.4	$$⊢\phi \to 0<B$$
Assertion	divgt0d	$$⊢\phi \to 0<\frac{A}{B}$$

Proof

Step	Hyp	Ref	Expression
1		ltp1d.1	$$⊢\phi \to A\in ℝ$$
2		divgt0d.2	$$⊢\phi \to B\in ℝ$$
3		divgt0d.3	$$⊢\phi \to 0<A$$
4		divgt0d.4	$$⊢\phi \to 0<B$$
5		divgt0	$$⊢\left(\left(A\in ℝ\wedge 0<A\right)\wedge \left(B\in ℝ\wedge 0<B\right)\right)\to 0<\frac{A}{B}$$
6	1 3 2 4 5	syl22anc	$$⊢\phi \to 0<\frac{A}{B}$$