Theorem sqrtge0

Description: The square root function is nonnegative for nonnegative input. (Contributed by NM, 26-May-1999) (Revised by Mario Carneiro, 9-Jul-2013)

	Ref	Expression
Assertion	sqrtge0	|- ( ( A e. RR /\ 0 <_ A ) -> 0 <_ ( sqrt ` A ) )

Proof

Step	Hyp	Ref	Expression
1		resqrtthlem	|- ( ( A e. RR /\ 0 <_ A ) -> ( ( ( sqrt ` A ) ^ 2 ) = A /\ 0 <_ ( Re ` ( sqrt ` A ) ) /\ ( _i x. ( sqrt ` A ) ) e/ RR+ ) )
2	1	simp2d	|- ( ( A e. RR /\ 0 <_ A ) -> 0 <_ ( Re ` ( sqrt ` A ) ) )
3		resqrtcl	|- ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` A ) e. RR )
4	3	rered	|- ( ( A e. RR /\ 0 <_ A ) -> ( Re ` ( sqrt ` A ) ) = ( sqrt ` A ) )
5	2 4	breqtrd	|- ( ( A e. RR /\ 0 <_ A ) -> 0 <_ ( sqrt ` A ) )