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	$$⊢\phi \to A\in ℝ$$
	logge0d.2	$$⊢\phi \to 1\le A$$
Assertion	logge0d	$$⊢\phi \to 0\le \log A$$

Proof

Step	Hyp	Ref	Expression
1		relogefd.1	$$⊢\phi \to A\in ℝ$$
2		logge0d.2	$$⊢\phi \to 1\le A$$
3		logge0	$$⊢\left(A\in ℝ\wedge 1\le A\right)\to 0\le \log A$$
4	1 2 3	syl2anc	$$⊢\phi \to 0\le \log A$$