Theorem sqrtpclii

Description: The square root of a positive real is a real. (Contributed by Mario Carneiro, 6-Sep-2013)

	Ref	Expression
Hypotheses	sqrtthi.1	$$⊢A\in ℝ$$
	sqrpclii.2	$$⊢0<A$$
Assertion	sqrtpclii	$$⊢\sqrt{A}\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	$$⊢A\in ℝ$$
2		sqrpclii.2	$$⊢0<A$$
3		0re	$$⊢0\in ℝ$$
4	3 1 2	ltleii	$$⊢0\le A$$
5	1	sqrtcli	$$⊢0\le A\to \sqrt{A}\in ℝ$$
6	4 5	ax-mp	$$⊢\sqrt{A}\in ℝ$$