Description: The logarithm of a number is less than 1 iff the number is less than Euler's constant. (Contributed by AV, 30-May-2020)
Ref | Expression | ||
---|---|---|---|
Assertion | loglt1b | |- ( A e. RR+ -> ( ( log ` A ) < 1 <-> A < _e ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | epr | |- _e e. RR+ |
|
2 | logltb | |- ( ( A e. RR+ /\ _e e. RR+ ) -> ( A < _e <-> ( log ` A ) < ( log ` _e ) ) ) |
|
3 | 1 2 | mpan2 | |- ( A e. RR+ -> ( A < _e <-> ( log ` A ) < ( log ` _e ) ) ) |
4 | loge | |- ( log ` _e ) = 1 |
|
5 | 4 | a1i | |- ( A e. RR+ -> ( log ` _e ) = 1 ) |
6 | 5 | breq2d | |- ( A e. RR+ -> ( ( log ` A ) < ( log ` _e ) <-> ( log ` A ) < 1 ) ) |
7 | 3 6 | bitr2d | |- ( A e. RR+ -> ( ( log ` A ) < 1 <-> A < _e ) ) |