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Metamath Proof Explorer

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 ) ) )