Theorem logrncn

Description: The range of the natural logarithm function is a subset of the complex numbers. (Contributed by Mario Carneiro, 13-May-2014)

	Ref	Expression
Assertion	logrncn	⊢ ( 𝐴 ∈ ran log → 𝐴 ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		ellogrn	⊢ ( 𝐴 ∈ ran log ↔ ( 𝐴 ∈ ℂ ∧ - π < ( ℑ ‘ 𝐴 ) ∧ ( ℑ ‘ 𝐴 ) ≤ π ) )
2	1	simp1bi	⊢ ( 𝐴 ∈ ran log → 𝐴 ∈ ℂ )