Theorem logge0

Description: The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Assertion	logge0	|- ( ( A e. RR /\ 1 <_ A ) -> 0 <_ ( log ` A ) )

Proof

Step	Hyp	Ref	Expression
1		log1	|- ( log ` 1 ) = 0
2		simpr	|- ( ( A e. RR /\ 1 <_ A ) -> 1 <_ A )
3		1rp	|- 1 e. RR+
4		rpgecl	|- ( ( 1 e. RR+ /\ A e. RR /\ 1 <_ A ) -> A e. RR+ )
5	3 4	mp3an1	|- ( ( A e. RR /\ 1 <_ A ) -> A e. RR+ )
6		logleb	|- ( ( 1 e. RR+ /\ A e. RR+ ) -> ( 1 <_ A <-> ( log ` 1 ) <_ ( log ` A ) ) )
7	3 5 6	sylancr	|- ( ( A e. RR /\ 1 <_ A ) -> ( 1 <_ A <-> ( log ` 1 ) <_ ( log ` A ) ) )
8	2 7	mpbid	|- ( ( A e. RR /\ 1 <_ A ) -> ( log ` 1 ) <_ ( log ` A ) )
9	1 8	eqbrtrrid	|- ( ( A e. RR /\ 1 <_ A ) -> 0 <_ ( log ` A ) )