Theorem gt0divd

Description: Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpgecld.1	$$⊢\phi \to A\in ℝ$$
	rpgecld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
Assertion	gt0divd	$$⊢\phi \to \left(0<A\leftrightarrow 0<\frac{A}{B}\right)$$

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	$$⊢\phi \to A\in ℝ$$
2		rpgecld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
3	2	rpred	$$⊢\phi \to B\in ℝ$$
4	2	rpgt0d	$$⊢\phi \to 0<B$$
5		gt0div	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge 0<B\right)\to \left(0<A\leftrightarrow 0<\frac{A}{B}\right)$$
6	1 3 4 5	syl3anc	$$⊢\phi \to \left(0<A\leftrightarrow 0<\frac{A}{B}\right)$$