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 ) ) |
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 ) ) |