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 ) )