Theorem sqrtcli

Description: The square root of a nonnegative real is a real. (Contributed by NM, 26-May-1999) (Revised by Mario Carneiro, 6-Sep-2013)

	Ref	Expression
Hypothesis	sqrtthi.1	⊢ 𝐴 ∈ ℝ
Assertion	sqrtcli	⊢ ( 0 ≤ 𝐴 → ( √ ‘ 𝐴 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	⊢ 𝐴 ∈ ℝ
2		resqrtcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( √ ‘ 𝐴 ) ∈ ℝ )
3	1 2	mpan	⊢ ( 0 ≤ 𝐴 → ( √ ‘ 𝐴 ) ∈ ℝ )