Metamath Proof Explorer
Description: Deduction form of logne0 . See logccne0d for a more general
version. (Contributed by SN, 25-Apr-2025)
|
|
Ref |
Expression |
|
Hypotheses |
logne0d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |
|
|
logne0d.1 |
⊢ ( 𝜑 → 𝐴 ≠ 1 ) |
|
Assertion |
logne0d |
⊢ ( 𝜑 → ( log ‘ 𝐴 ) ≠ 0 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
logne0d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |
2 |
|
logne0d.1 |
⊢ ( 𝜑 → 𝐴 ≠ 1 ) |
3 |
|
logne0 |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≠ 1 ) → ( log ‘ 𝐴 ) ≠ 0 ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( log ‘ 𝐴 ) ≠ 0 ) |