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 ) )