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Metamath Proof Explorer

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 )