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 ∈ ℝ+ → ( √ ‘ 𝐴 ) ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		rpre	⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ )
2		rpge0	⊢ ( 𝐴 ∈ ℝ+ → 0 ≤ 𝐴 )
3		resqrtcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( √ ‘ 𝐴 ) ∈ ℝ )
4	1 2 3	syl2anc	⊢ ( 𝐴 ∈ ℝ+ → ( √ ‘ 𝐴 ) ∈ ℝ )
5		rpgt0	⊢ ( 𝐴 ∈ ℝ+ → 0 < 𝐴 )
6		sqrtgt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 0 < ( √ ‘ 𝐴 ) )
7	1 5 6	syl2anc	⊢ ( 𝐴 ∈ ℝ+ → 0 < ( √ ‘ 𝐴 ) )
8	4 7	elrpd	⊢ ( 𝐴 ∈ ℝ+ → ( √ ‘ 𝐴 ) ∈ ℝ+ )