Theorem log11d

Description: The natural logarithm is one-to-one. (Contributed by SN, 25-Apr-2025)

	Ref	Expression
Hypotheses	log11d.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	log11d.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	log11d.1	⊢ ( 𝜑 → 𝐴 ≠ 0 )
	log11d.2	⊢ ( 𝜑 → 𝐵 ≠ 0 )
Assertion	log11d	⊢ ( 𝜑 → ( ( log ‘ 𝐴 ) = ( log ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		log11d.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		log11d.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		log11d.1	⊢ ( 𝜑 → 𝐴 ≠ 0 )
4		log11d.2	⊢ ( 𝜑 → 𝐵 ≠ 0 )
5		fveq2	⊢ ( ( log ‘ 𝐴 ) = ( log ‘ 𝐵 ) → ( exp ‘ ( log ‘ 𝐴 ) ) = ( exp ‘ ( log ‘ 𝐵 ) ) )
6		eflog	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 )
7	1 3 6	syl2anc	⊢ ( 𝜑 → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 )
8		eflog	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( exp ‘ ( log ‘ 𝐵 ) ) = 𝐵 )
9	2 4 8	syl2anc	⊢ ( 𝜑 → ( exp ‘ ( log ‘ 𝐵 ) ) = 𝐵 )
10	7 9	eqeq12d	⊢ ( 𝜑 → ( ( exp ‘ ( log ‘ 𝐴 ) ) = ( exp ‘ ( log ‘ 𝐵 ) ) ↔ 𝐴 = 𝐵 ) )
11	5 10	imbitrid	⊢ ( 𝜑 → ( ( log ‘ 𝐴 ) = ( log ‘ 𝐵 ) → 𝐴 = 𝐵 ) )
12		fveq2	⊢ ( 𝐴 = 𝐵 → ( log ‘ 𝐴 ) = ( log ‘ 𝐵 ) )
13	11 12	impbid1	⊢ ( 𝜑 → ( ( log ‘ 𝐴 ) = ( log ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )