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 ) )