This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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