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Metamath Proof Explorer

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