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Metamath Proof Explorer

Theorem rplog11d

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 ) )