Metamath Proof Explorer

Theorem reclt1

Description: The reciprocal of a positive number less than 1 is greater than 1. (Contributed by NM, 23-Feb-2005)

Ref Expression
Assertion reclt1 
|- ( ( A e. RR /\ 0 < A ) -> ( A < 1 <-> 1 < ( 1 / A ) ) )

Proof

Step Hyp Ref Expression
1 1re 
 |-  1 e. RR
2 0lt1 
 |-  0 < 1
3 ltrec 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( 1 e. RR /\ 0 < 1 ) ) -> ( A < 1 <-> ( 1 / 1 ) < ( 1 / A ) ) )
4 1 2 3 mpanr12 
 |-  ( ( A e. RR /\ 0 < A ) -> ( A < 1 <-> ( 1 / 1 ) < ( 1 / A ) ) )
5 1div1e1 
 |-  ( 1 / 1 ) = 1
6 5 breq1i 
 |-  ( ( 1 / 1 ) < ( 1 / A ) <-> 1 < ( 1 / A ) )
7 4 6 bitrdi 
 |-  ( ( A e. RR /\ 0 < A ) -> ( A < 1 <-> 1 < ( 1 / A ) ) )