Theorem rpsqrtcl

Description: The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008)

	Ref	Expression
Assertion	rpsqrtcl	$$⊢A\in {ℝ}^{+}\to \sqrt{A}\in {ℝ}^{+}$$

Proof

Step	Hyp	Ref	Expression
1		rpre	$$⊢A\in {ℝ}^{+}\to A\in ℝ$$
2		rpge0	$$⊢A\in {ℝ}^{+}\to 0\le A$$
3		resqrtcl	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \sqrt{A}\in ℝ$$
4	1 2 3	syl2anc	$$⊢A\in {ℝ}^{+}\to \sqrt{A}\in ℝ$$
5		rpgt0	$$⊢A\in {ℝ}^{+}\to 0<A$$
6		sqrtgt0	$$⊢\left(A\in ℝ\wedge 0<A\right)\to 0<\sqrt{A}$$
7	1 5 6	syl2anc	$$⊢A\in {ℝ}^{+}\to 0<\sqrt{A}$$
8	4 7	elrpd	$$⊢A\in {ℝ}^{+}\to \sqrt{A}\in {ℝ}^{+}$$