Metamath Proof Explorer
Description: The natural logarithm is one-to-one on positive reals. (Contributed by SN, 25-Apr-2025)
|
|
Ref |
Expression |
|
Hypotheses |
rplog11d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |
|
|
rplog11d.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
|
Assertion |
rplog11d |
⊢ ( 𝜑 → ( ( log ‘ 𝐴 ) = ( log ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
rplog11d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |
2 |
|
rplog11d.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
3 |
1
|
rpcnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
4 |
2
|
rpcnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
5 |
1
|
rpne0d |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
6 |
2
|
rpne0d |
⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
7 |
3 4 5 6
|
log11d |
⊢ ( 𝜑 → ( ( log ‘ 𝐴 ) = ( log ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) ) |