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Metamath Proof Explorer

Theorem sqrtgt0ii

Description: The square root of a positive real is positive. (Contributed by NM, 26-May-1999) (Revised by Mario Carneiro, 6-Sep-2013)

Ref Expression
Hypotheses sqrtthi.1 ⊢ 𝐴 ∈ ℝ
sqrpclii.2 ⊢ 0 < 𝐴
Assertion sqrtgt0ii 0 < ( √ ‘ 𝐴 )

Proof

Step Hyp Ref Expression
1 sqrtthi.1 ⊢ 𝐴 ∈ ℝ
2 sqrpclii.2 ⊢ 0 < 𝐴
3 1 sqrtgt0i ⊢ ( 0 < 𝐴 → 0 < ( √ ‘ 𝐴 ) )
4 2 3 ax-mp ⊢ 0 < ( √ ‘ 𝐴 )