Theorem sqgt0

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

	Ref	Expression
Assertion	sqgt0	|- ( ( A e. RR /\ A =/= 0 ) -> 0 < ( A ^ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		msqgt0	|- ( ( A e. RR /\ A =/= 0 ) -> 0 < ( A x. A ) )
2		recn	|- ( A e. RR -> A e. CC )
3		sqval	|- ( A e. CC -> ( A ^ 2 ) = ( A x. A ) )
4	2 3	syl	|- ( A e. RR -> ( A ^ 2 ) = ( A x. A ) )
5	4	adantr	|- ( ( A e. RR /\ A =/= 0 ) -> ( A ^ 2 ) = ( A x. A ) )
6	1 5	breqtrrd	|- ( ( A e. RR /\ A =/= 0 ) -> 0 < ( A ^ 2 ) )