Metamath Proof Explorer

Theorem recgt1

Description: The reciprocal of a positive number greater than 1 is less than 1. (Contributed by NM, 28-Dec-2005)

Ref Expression
Assertion recgt1 
|- ( ( A e. RR /\ 0 < A ) -> ( 1 < A <-> ( 1 / A ) < 1 ) )

Proof

Step Hyp Ref Expression
1 1re 
 |-  1 e. RR
2 0lt1 
 |-  0 < 1
3 ltrec 
 |-  ( ( ( 1 e. RR /\ 0 < 1 ) /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < A <-> ( 1 / A ) < ( 1 / 1 ) ) )
4 1 2 3 mpanl12 
 |-  ( ( A e. RR /\ 0 < A ) -> ( 1 < A <-> ( 1 / A ) < ( 1 / 1 ) ) )
5 1div1e1 
 |-  ( 1 / 1 ) = 1
6 5 breq2i 
 |-  ( ( 1 / A ) < ( 1 / 1 ) <-> ( 1 / A ) < 1 )
7 4 6 bitrdi 
 |-  ( ( A e. RR /\ 0 < A ) -> ( 1 < A <-> ( 1 / A ) < 1 ) )