Theorem sqn0rp

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

	Ref	Expression
Assertion	sqn0rp	|- ( ( A e. RR /\ A =/= 0 ) -> ( A ^ 2 ) e. RR+ )

Proof

Step	Hyp	Ref	Expression
1		resqcl	|- ( A e. RR -> ( A ^ 2 ) e. RR )
2	1	adantr	|- ( ( A e. RR /\ A =/= 0 ) -> ( A ^ 2 ) e. RR )
3		sqgt0	|- ( ( A e. RR /\ A =/= 0 ) -> 0 < ( A ^ 2 ) )
4	2 3	elrpd	|- ( ( A e. RR /\ A =/= 0 ) -> ( A ^ 2 ) e. RR+ )