Metamath Proof Explorer
Description: The logarithm isn't 0 if its argument isn't 0 or 1, deduction form.
(Contributed by SN, 25-Apr-2025)
|
|
Ref |
Expression |
|
Hypotheses |
logccne0d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
logccne0d.0 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
|
|
logccne0d.1 |
⊢ ( 𝜑 → 𝐴 ≠ 1 ) |
|
Assertion |
logccne0d |
⊢ ( 𝜑 → ( log ‘ 𝐴 ) ≠ 0 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
logccne0d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
logccne0d.0 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
3 |
|
logccne0d.1 |
⊢ ( 𝜑 → 𝐴 ≠ 1 ) |
4 |
|
logccne0 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1 ) → ( log ‘ 𝐴 ) ≠ 0 ) |
5 |
1 2 3 4
|
syl3anc |
⊢ ( 𝜑 → ( log ‘ 𝐴 ) ≠ 0 ) |