Metamath Proof Explorer


Theorem resqrtcld

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

Ref Expression
Hypotheses resqrcld.1 ( 𝜑𝐴 ∈ ℝ )
resqrcld.2 ( 𝜑 → 0 ≤ 𝐴 )
Assertion resqrtcld ( 𝜑 → ( √ ‘ 𝐴 ) ∈ ℝ )

Proof

Step Hyp Ref Expression
1 resqrcld.1 ( 𝜑𝐴 ∈ ℝ )
2 resqrcld.2 ( 𝜑 → 0 ≤ 𝐴 )
3 resqrtcl ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( √ ‘ 𝐴 ) ∈ ℝ )
4 1 2 3 syl2anc ( 𝜑 → ( √ ‘ 𝐴 ) ∈ ℝ )