Metamath Proof Explorer


Theorem logge0d

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

Ref Expression
Hypotheses relogefd.1
|- ( ph -> A e. RR )
logge0d.2
|- ( ph -> 1 <_ A )
Assertion logge0d
|- ( ph -> 0 <_ ( log ` A ) )

Proof

Step Hyp Ref Expression
1 relogefd.1
 |-  ( ph -> A e. RR )
2 logge0d.2
 |-  ( ph -> 1 <_ A )
3 logge0
 |-  ( ( A e. RR /\ 1 <_ A ) -> 0 <_ ( log ` A ) )
4 1 2 3 syl2anc
 |-  ( ph -> 0 <_ ( log ` A ) )