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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 ⊢ ( 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 ) )