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	syl5ib	⊢ ( ( 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 ) )