Metamath Proof Explorer


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 ) )