Theorem rphalflt

Description: Half of a positive real is less than the original number. (Contributed by Mario Carneiro, 21-May-2014)

	Ref	Expression
Assertion	rphalflt	$$⊢A\in {ℝ}^{+}\to \frac{A}{2}<A$$

Proof

Step	Hyp	Ref	Expression
1		elrp	$$⊢A\in {ℝ}^{+}\leftrightarrow \left(A\in ℝ\wedge 0<A\right)$$
2		halfpos	$$⊢A\in ℝ\to \left(0<A\leftrightarrow \frac{A}{2}<A\right)$$
3	2	biimpa	$$⊢\left(A\in ℝ\wedge 0<A\right)\to \frac{A}{2}<A$$
4	1 3	sylbi	$$⊢A\in {ℝ}^{+}\to \frac{A}{2}<A$$