Theorem sqrtsq

Description: Square root of square. (Contributed by NM, 14-Jan-2006) (Revised by Mario Carneiro, 29-May-2016)

	Ref	Expression
Assertion	sqrtsq	|- ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` ( A ^ 2 ) ) = A )

Proof

Step	Hyp	Ref	Expression
1		eqid	|- ( A ^ 2 ) = ( A ^ 2 )
2		resqcl	|- ( A e. RR -> ( A ^ 2 ) e. RR )
3		sqge0	|- ( A e. RR -> 0 <_ ( A ^ 2 ) )
4	2 3	jca	|- ( A e. RR -> ( ( A ^ 2 ) e. RR /\ 0 <_ ( A ^ 2 ) ) )
5	4	adantr	|- ( ( A e. RR /\ 0 <_ A ) -> ( ( A ^ 2 ) e. RR /\ 0 <_ ( A ^ 2 ) ) )
6		sqrtsq2	|- ( ( ( ( A ^ 2 ) e. RR /\ 0 <_ ( A ^ 2 ) ) /\ ( A e. RR /\ 0 <_ A ) ) -> ( ( sqrt ` ( A ^ 2 ) ) = A <-> ( A ^ 2 ) = ( A ^ 2 ) ) )
7	5 6	mpancom	|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqrt ` ( A ^ 2 ) ) = A <-> ( A ^ 2 ) = ( A ^ 2 ) ) )
8	1 7	mpbiri	|- ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` ( A ^ 2 ) ) = A )