Theorem sqrtlti

Description: Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005)

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

Proof

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