Theorem loglt1b

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 ) )

Proof

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 ) )