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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 ) )