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 ∈ ℝ
	sqrpclii.2	⊢ 0 < 𝐴
Assertion	sqrtpclii	⊢ ( √ ‘ 𝐴 ) ∈ ℝ

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	⊢ 𝐴 ∈ ℝ
2		sqrpclii.2	⊢ 0 < 𝐴
3		0re	⊢ 0 ∈ ℝ
4	3 1 2	ltleii	⊢ 0 ≤ 𝐴
5	1	sqrtcli	⊢ ( 0 ≤ 𝐴 → ( √ ‘ 𝐴 ) ∈ ℝ )
6	4 5	ax-mp	⊢ ( √ ‘ 𝐴 ) ∈ ℝ