This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )