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Metamath Proof Explorer

Theorem msq11

Description: The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion msq11 
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A x. A ) = ( B x. B ) <-> A = B ) )

Proof

Step Hyp Ref Expression
1 le2msq 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A <_ B <-> ( A x. A ) <_ ( B x. B ) ) )
2 le2msq 
 |-  ( ( ( B e. RR /\ 0 <_ B ) /\ ( A e. RR /\ 0 <_ A ) ) -> ( B <_ A <-> ( B x. B ) <_ ( A x. A ) ) )
3 2 ancoms 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( B <_ A <-> ( B x. B ) <_ ( A x. A ) ) )
4 1 3 anbi12d 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A <_ B /\ B <_ A ) <-> ( ( A x. A ) <_ ( B x. B ) /\ ( B x. B ) <_ ( A x. A ) ) ) )
5 simpll 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> A e. RR )
6 simprl 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> B e. RR )
7 5 6 letri3d 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
8 5 5 remulcld 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A x. A ) e. RR )
9 6 6 remulcld 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( B x. B ) e. RR )
10 8 9 letri3d 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A x. A ) = ( B x. B ) <-> ( ( A x. A ) <_ ( B x. B ) /\ ( B x. B ) <_ ( A x. A ) ) ) )
11 4 7 10 3bitr4rd 
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A x. A ) = ( B x. B ) <-> A = B ) )