Theorem rpsqrtcld

Description: The square root of a positive real is positive. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	sqrgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
Assertion	rpsqrtcld	⊢ ( 𝜑 → ( √ ‘ 𝐴 ) ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		sqrgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		rpsqrtcl	⊢ ( 𝐴 ∈ ℝ+ → ( √ ‘ 𝐴 ) ∈ ℝ+ )
3	1 2	syl	⊢ ( 𝜑 → ( √ ‘ 𝐴 ) ∈ ℝ+ )