Theorem sqrt11i

Description: The square root function is one-to-one. (Contributed by NM, 27-Jul-1999)

	Ref	Expression
Hypotheses	sqrtthi.1	|- A e. RR
	sqr11.1	|- B e. RR
Assertion	sqrt11i	|- ( ( 0 <_ A /\ 0 <_ B ) -> ( ( sqrt ` A ) = ( sqrt ` B ) <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	|- A e. RR
2		sqr11.1	|- B e. RR
3		sqrt11	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqrt ` A ) = ( sqrt ` B ) <-> A = B ) )
4	2 3	mpanr1	|- ( ( ( A e. RR /\ 0 <_ A ) /\ 0 <_ B ) -> ( ( sqrt ` A ) = ( sqrt ` B ) <-> A = B ) )
5	1 4	mpanl1	|- ( ( 0 <_ A /\ 0 <_ B ) -> ( ( sqrt ` A ) = ( sqrt ` B ) <-> A = B ) )