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 ) ) ) |
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 ) ) ) |