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Metamath Proof Explorer

Theorem sqrtgt0

Description: The square root function is positive for positive input. (Contributed by Mario Carneiro, 10-Jul-2013) (Revised by Mario Carneiro, 6-Sep-2013)

Ref Expression
Assertion sqrtgt0 ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 0 < ( √ ‘ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 0re ⊢ 0 ∈ ℝ
2 ltle ⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 < 𝐴 → 0 ≤ 𝐴 ) )
3 1 2 mpan ⊢ ( 𝐴 ∈ ℝ → ( 0 < 𝐴 → 0 ≤ 𝐴 ) )
4 3 imp ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 0 ≤ 𝐴 )
5 resqrtcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( √ ‘ 𝐴 ) ∈ ℝ )
6 4 5 syldan ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( √ ‘ 𝐴 ) ∈ ℝ )
7 sqrtge0 ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → 0 ≤ ( √ ‘ 𝐴 ) )
8 4 7 syldan ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 0 ≤ ( √ ‘ 𝐴 ) )
9 gt0ne0 ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 𝐴 ≠ 0 )
10 sq0i ⊢ ( ( √ ‘ 𝐴 ) = 0 → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 0 )
11 resqrtth ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 )
12 4 11 syldan ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 )
13 12 eqeq1d ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = 0 ↔ 𝐴 = 0 ) )
14 10 13 imbitrid ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( ( √ ‘ 𝐴 ) = 0 → 𝐴 = 0 ) )
15 14 necon3d ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 𝐴 ≠ 0 → ( √ ‘ 𝐴 ) ≠ 0 ) )
16 9 15 mpd ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( √ ‘ 𝐴 ) ≠ 0 )
17 6 8 16 ne0gt0d ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 0 < ( √ ‘ 𝐴 ) )