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Metamath Proof Explorer

Theorem sq11

Description: The square function is one-to-one for nonnegative reals. (Contributed by NM, 8-Apr-2001) (Proof shortened by Mario Carneiro, 28-May-2016)

Ref Expression
Assertion sq11 
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> A = B ) )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( A e. RR /\ 0 <_ A ) -> A e. RR )
2 1 recnd 
 |-  ( ( A e. RR /\ 0 <_ A ) -> A e. CC )
3 sqval 
 |-  ( A e. CC -> ( A ^ 2 ) = ( A x. A ) )
4 2 3 syl 
 |-  ( ( A e. RR /\ 0 <_ A ) -> ( A ^ 2 ) = ( A x. A ) )
5 simpl 
 |-  ( ( B e. RR /\ 0 <_ B ) -> B e. RR )
6 5 recnd 
 |-  ( ( B e. RR /\ 0 <_ B ) -> B e. CC )
7 sqval 
 |-  ( B e. CC -> ( B ^ 2 ) = ( B x. B ) )
8 6 7 syl 
 |-  ( ( B e. RR /\ 0 <_ B ) -> ( B ^ 2 ) = ( B x. B ) )
9 4 8 eqeqan12d 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> ( A x. A ) = ( B x. B ) ) )
10 msq11 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A x. A ) = ( B x. B ) <-> A = B ) )
11 9 10 bitrd 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> A = B ) )