Theorem recgt1d

Description: The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypothesis	rpred.1	|- ( ph -> A e. RR+ )
Assertion	recgt1d	|- ( ph -> ( 1 < A <-> ( 1 / A ) < 1 ) )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	|- ( ph -> A e. RR+ )
2	1	rpregt0d	|- ( ph -> ( A e. RR /\ 0 < A ) )
3		recgt1	|- ( ( A e. RR /\ 0 < A ) -> ( 1 < A <-> ( 1 / A ) < 1 ) )
4	2 3	syl	|- ( ph -> ( 1 < A <-> ( 1 / A ) < 1 ) )