Theorem sqgt0

Description: The square of a nonzero real is positive. (Contributed by NM, 8-Sep-2007)

	Ref	Expression
Assertion	sqgt0	$$⊢\left(A\in ℝ\wedge A\ne 0\right)\to 0<A^2$$

Proof

Step	Hyp	Ref	Expression
1		msqgt0	$$⊢\left(A\in ℝ\wedge A\ne 0\right)\to 0<AA$$
2		recn	$$⊢A\in ℝ\to A\in ℂ$$
3		sqval	$$⊢A\in ℂ\to A^2=AA$$
4	2 3	syl	$$⊢A\in ℝ\to A^2=AA$$
5	4	adantr	$$⊢\left(A\in ℝ\wedge A\ne 0\right)\to A^2=AA$$
6	1 5	breqtrrd	$$⊢\left(A\in ℝ\wedge A\ne 0\right)\to 0<A^2$$