Theorem remsqsqrt

Description: Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013)

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

Proof

Step	Hyp	Ref	Expression
1		resqrtcl	|- ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` A ) e. RR )
2	1	recnd	|- ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` A ) e. CC )
3	2	sqvald	|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqrt ` A ) ^ 2 ) = ( ( sqrt ` A ) x. ( sqrt ` A ) ) )
4		resqrtth	|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqrt ` A ) ^ 2 ) = A )
5	3 4	eqtr3d	|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqrt ` A ) x. ( sqrt ` A ) ) = A )