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	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	logge0d.2	⊢ ( 𝜑 → 1 ≤ 𝐴 )
Assertion	logge0d	⊢ ( 𝜑 → 0 ≤ ( log ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		relogefd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		logge0d.2	⊢ ( 𝜑 → 1 ≤ 𝐴 )
3		logge0	⊢ ( ( 𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ) → 0 ≤ ( log ‘ 𝐴 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → 0 ≤ ( log ‘ 𝐴 ) )