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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