Theorem sqn0rp

Description: The square of a nonzero real is a positive real. (Contributed by AV, 5-Mar-2023)

	Ref	Expression
Assertion	sqn0rp	$$⊢\left(A\in ℝ\wedge A\ne 0\right)\to A^2\in {ℝ}^{+}$$

Proof

Step	Hyp	Ref	Expression
1		resqcl	$$⊢A\in ℝ\to A^2\in ℝ$$
2	1	adantr	$$⊢\left(A\in ℝ\wedge A\ne 0\right)\to A^2\in ℝ$$
3		sqgt0	$$⊢\left(A\in ℝ\wedge A\ne 0\right)\to 0<A^2$$
4	2 3	elrpd	$$⊢\left(A\in ℝ\wedge A\ne 0\right)\to A^2\in {ℝ}^{+}$$