Theorem divgt0i

Description: The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999)

	Ref	Expression
Hypotheses	ltplus1.1	$$⊢A\in ℝ$$
	prodgt0.2	$$⊢B\in ℝ$$
Assertion	divgt0i	$$⊢\left(0<A\wedge 0<B\right)\to 0<\frac{A}{B}$$

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	$$⊢A\in ℝ$$
2		prodgt0.2	$$⊢B\in ℝ$$
3		divgt0	$$⊢\left(\left(A\in ℝ\wedge 0<A\right)\wedge \left(B\in ℝ\wedge 0<B\right)\right)\to 0<\frac{A}{B}$$
4	2 3	mpanr1	$$⊢\left(\left(A\in ℝ\wedge 0<A\right)\wedge 0<B\right)\to 0<\frac{A}{B}$$
5	1 4	mpanl1	$$⊢\left(0<A\wedge 0<B\right)\to 0<\frac{A}{B}$$