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Metamath Proof Explorer

Theorem sqrt11

Description: The square root function is one-to-one. (Contributed by Scott Fenton, 11-Jun-2013)

Ref Expression
Assertion sqrt11 
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqrt ` A ) = ( sqrt ` B ) <-> A = B ) )

Proof

Step Hyp Ref Expression
1 resqrtcl 
 |-  ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` A ) e. RR )
2 sqrtge0 
 |-  ( ( A e. RR /\ 0 <_ A ) -> 0 <_ ( sqrt ` A ) )
3 1 2 jca 
 |-  ( ( A e. RR /\ 0 <_ A ) -> ( ( sqrt ` A ) e. RR /\ 0 <_ ( sqrt ` A ) ) )
4 resqrtcl 
 |-  ( ( B e. RR /\ 0 <_ B ) -> ( sqrt ` B ) e. RR )
5 sqrtge0 
 |-  ( ( B e. RR /\ 0 <_ B ) -> 0 <_ ( sqrt ` B ) )
6 4 5 jca 
 |-  ( ( B e. RR /\ 0 <_ B ) -> ( ( sqrt ` B ) e. RR /\ 0 <_ ( sqrt ` B ) ) )
7 sq11 
 |-  ( ( ( ( sqrt ` A ) e. RR /\ 0 <_ ( sqrt ` A ) ) /\ ( ( sqrt ` B ) e. RR /\ 0 <_ ( sqrt ` B ) ) ) -> ( ( ( sqrt ` A ) ^ 2 ) = ( ( sqrt ` B ) ^ 2 ) <-> ( sqrt ` A ) = ( sqrt ` B ) ) )
8 3 6 7 syl2an 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( ( sqrt ` A ) ^ 2 ) = ( ( sqrt ` B ) ^ 2 ) <-> ( sqrt ` A ) = ( sqrt ` B ) ) )
9 resqrtth 
 |-  ( ( A e. RR /\ 0 <_ A ) -> ( ( sqrt ` A ) ^ 2 ) = A )
10 resqrtth 
 |-  ( ( B e. RR /\ 0 <_ B ) -> ( ( sqrt ` B ) ^ 2 ) = B )
11 9 10 eqeqan12d 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( ( sqrt ` A ) ^ 2 ) = ( ( sqrt ` B ) ^ 2 ) <-> A = B ) )
12 8 11 bitr3d 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqrt ` A ) = ( sqrt ` B ) <-> A = B ) )