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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