Theorem rpreccl

Description: Closure law for reciprocation of positive reals. (Contributed by Jeff Hankins, 23-Nov-2008)

	Ref	Expression
Assertion	rpreccl	$$⊢A\in {ℝ}^{+}\to \frac{1}{A}\in {ℝ}^{+}$$

Proof

Step	Hyp	Ref	Expression
1		1rp	$$⊢1\in {ℝ}^{+}$$
2		rpdivcl	$$⊢\left(1\in {ℝ}^{+}\wedge A\in {ℝ}^{+}\right)\to \frac{1}{A}\in {ℝ}^{+}$$
3	1 2	mpan	$$⊢A\in {ℝ}^{+}\to \frac{1}{A}\in {ℝ}^{+}$$