Metamath Proof Explorer
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 ) ) |