Theorem sqrtsq2d

Description: Relationship between square root and squares. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	resqrcld.1	|- ( ph -> A e. RR )
	resqrcld.2	|- ( ph -> 0 <_ A )
	sqr11d.3	|- ( ph -> B e. RR )
	sqr11d.4	|- ( ph -> 0 <_ B )
Assertion	sqrtsq2d	|- ( ph -> ( ( sqrt ` A ) = B <-> A = ( B ^ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		resqrcld.1	|- ( ph -> A e. RR )
2		resqrcld.2	|- ( ph -> 0 <_ A )
3		sqr11d.3	|- ( ph -> B e. RR )
4		sqr11d.4	|- ( ph -> 0 <_ B )
5		sqrtsq2	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqrt ` A ) = B <-> A = ( B ^ 2 ) ) )
6	1 2 3 4 5	syl22anc	|- ( ph -> ( ( sqrt ` A ) = B <-> A = ( B ^ 2 ) ) )