Theorem halfpos2

Description: A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005)

	Ref	Expression
Assertion	halfpos2	$$⊢A\in ℝ\to \left(0<A\leftrightarrow 0<\frac{A}{2}\right)$$

Proof

Step	Hyp	Ref	Expression
1		2re	$$⊢2\in ℝ$$
2		2pos	$$⊢0<2$$
3		gt0div	$$⊢\left(A\in ℝ\wedge 2\in ℝ\wedge 0<2\right)\to \left(0<A\leftrightarrow 0<\frac{A}{2}\right)$$
4	1 2 3	mp3an23	$$⊢A\in ℝ\to \left(0<A\leftrightarrow 0<\frac{A}{2}\right)$$