Theorem lt2msqi

Description: The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999)

	Ref	Expression
Hypotheses	ltplus1.1	|- A e. RR
	prodgt0.2	|- B e. RR
Assertion	lt2msqi	|- ( ( 0 <_ A /\ 0 <_ B ) -> ( A < B <-> ( A x. A ) < ( B x. B ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	|- A e. RR
2		prodgt0.2	|- B e. RR
3		lt2msq	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A < B <-> ( A x. A ) < ( B x. B ) ) )
4	2 3	mpanr1	|- ( ( ( A e. RR /\ 0 <_ A ) /\ 0 <_ B ) -> ( A < B <-> ( A x. A ) < ( B x. B ) ) )
5	1 4	mpanl1	|- ( ( 0 <_ A /\ 0 <_ B ) -> ( A < B <-> ( A x. A ) < ( B x. B ) ) )