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	$$⊢\left(A\in ℝ\wedge 0<A\right)\to 0<\sqrt{A}$$

Proof

Step	Hyp	Ref	Expression
1		0re	$$⊢0\in ℝ$$
2		ltle	$$⊢\left(0\in ℝ\wedge A\in ℝ\right)\to \left(0<A\to 0\le A\right)$$
3	1 2	mpan	$$⊢A\in ℝ\to \left(0<A\to 0\le A\right)$$
4	3	imp	$$⊢\left(A\in ℝ\wedge 0<A\right)\to 0\le A$$
5		resqrtcl	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \sqrt{A}\in ℝ$$
6	4 5	syldan	$$⊢\left(A\in ℝ\wedge 0<A\right)\to \sqrt{A}\in ℝ$$
7		sqrtge0	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to 0\le \sqrt{A}$$
8	4 7	syldan	$$⊢\left(A\in ℝ\wedge 0<A\right)\to 0\le \sqrt{A}$$
9		gt0ne0	$$⊢\left(A\in ℝ\wedge 0<A\right)\to A\ne 0$$
10		sq0i	$$⊢\sqrt{A}=0\to {\sqrt{A}}^2=0$$
11		resqrtth	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to {\sqrt{A}}^2=A$$
12	4 11	syldan	$$⊢\left(A\in ℝ\wedge 0<A\right)\to {\sqrt{A}}^2=A$$
13	12	eqeq1d	$$⊢\left(A\in ℝ\wedge 0<A\right)\to \left({\sqrt{A}}^2=0\leftrightarrow A=0\right)$$
14	10 13	imbitrid	$$⊢\left(A\in ℝ\wedge 0<A\right)\to \left(\sqrt{A}=0\to A=0\right)$$
15	14	necon3d	$$⊢\left(A\in ℝ\wedge 0<A\right)\to \left(A\ne 0\to \sqrt{A}\ne 0\right)$$
16	9 15	mpd	$$⊢\left(A\in ℝ\wedge 0<A\right)\to \sqrt{A}\ne 0$$
17	6 8 16	ne0gt0d	$$⊢\left(A\in ℝ\wedge 0<A\right)\to 0<\sqrt{A}$$