Theorem ltrecii

Description: The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999)

	Ref	Expression
Hypotheses	ltplus1.1	|- A e. RR
	prodgt0.2	|- B e. RR
	ltreci.3	|- 0 < A
	ltreci.4	|- 0 < B
Assertion	ltrecii	|- ( A < B <-> ( 1 / B ) < ( 1 / A ) )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	|- A e. RR
2		prodgt0.2	|- B e. RR
3		ltreci.3	|- 0 < A
4		ltreci.4	|- 0 < B
5	1 2	ltreci	|- ( ( 0 < A /\ 0 < B ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )
6	3 4 5	mp2an	|- ( A < B <-> ( 1 / B ) < ( 1 / A ) )