Theorem divge0

Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 27-Sep-1999)

	Ref	Expression
Assertion	divge0	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge \left(B\in ℝ\wedge 0<B\right)\right)\to 0\le \frac{A}{B}$$

Proof

Step	Hyp	Ref	Expression
1		ge0div	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge 0<B\right)\to \left(0\le A\leftrightarrow 0\le \frac{A}{B}\right)$$
2	1	biimpd	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge 0<B\right)\to \left(0\le A\to 0\le \frac{A}{B}\right)$$
3	2	3exp	$$⊢A\in ℝ\to \left(B\in ℝ\to \left(0<B\to \left(0\le A\to 0\le \frac{A}{B}\right)\right)\right)$$
4	3	com34	$$⊢A\in ℝ\to \left(B\in ℝ\to \left(0\le A\to \left(0<B\to 0\le \frac{A}{B}\right)\right)\right)$$
5	4	com23	$$⊢A\in ℝ\to \left(0\le A\to \left(B\in ℝ\to \left(0<B\to 0\le \frac{A}{B}\right)\right)\right)$$
6	5	imp43	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge \left(B\in ℝ\wedge 0<B\right)\right)\to 0\le \frac{A}{B}$$