Theorem divge0d

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

	Ref	Expression
Hypotheses	rpgecld.1	$$⊢\phi \to A\in ℝ$$
	rpgecld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
	divge0d.3	$$⊢\phi \to 0\le A$$
Assertion	divge0d	$$⊢\phi \to 0\le \frac{A}{B}$$

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	$$⊢\phi \to A\in ℝ$$
2		rpgecld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
3		divge0d.3	$$⊢\phi \to 0\le A$$
4	2	rpregt0d	$$⊢\phi \to \left(B\in ℝ\wedge 0<B\right)$$
5		divge0	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge \left(B\in ℝ\wedge 0<B\right)\right)\to 0\le \frac{A}{B}$$
6	1 3 4 5	syl21anc	$$⊢\phi \to 0\le \frac{A}{B}$$