Theorem divgt0ii

Description: The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999)

	Ref	Expression
Hypotheses	ltplus1.1	$$⊢A\in ℝ$$
	prodgt0.2	$$⊢B\in ℝ$$
	ltreci.3	$$⊢0<A$$
	ltreci.4	$$⊢0<B$$
Assertion	divgt0ii	$$⊢0<\frac{A}{B}$$

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	$$⊢A\in ℝ$$
2		prodgt0.2	$$⊢B\in ℝ$$
3		ltreci.3	$$⊢0<A$$
4		ltreci.4	$$⊢0<B$$
5	1 2 4	divgt0i2i	$$⊢0<A\to 0<\frac{A}{B}$$
6	3 5	ax-mp	$$⊢0<\frac{A}{B}$$