Theorem sqrtmsq2i

Description: Relationship between square root and squares. (Contributed by NM, 31-Jul-1999)

	Ref	Expression
Hypotheses	sqrtthi.1	|- A e. RR
	sqr11.1	|- B e. RR
Assertion	sqrtmsq2i	|- ( ( 0 <_ A /\ 0 <_ B ) -> ( ( sqrt ` A ) = B <-> A = ( B x. B ) ) )

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	|- A e. RR
2		sqr11.1	|- B e. RR
3		sqrtsq2	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqrt ` A ) = B <-> A = ( B ^ 2 ) ) )
4	2 3	mpanr1	|- ( ( ( A e. RR /\ 0 <_ A ) /\ 0 <_ B ) -> ( ( sqrt ` A ) = B <-> A = ( B ^ 2 ) ) )
5	1 4	mpanl1	|- ( ( 0 <_ A /\ 0 <_ B ) -> ( ( sqrt ` A ) = B <-> A = ( B ^ 2 ) ) )
6	2	recni	|- B e. CC
7	6	sqvali	|- ( B ^ 2 ) = ( B x. B )
8	7	eqeq2i	|- ( A = ( B ^ 2 ) <-> A = ( B x. B ) )
9	5 8	bitrdi	|- ( ( 0 <_ A /\ 0 <_ B ) -> ( ( sqrt ` A ) = B <-> A = ( B x. B ) ) )