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	$$⊢A\in \mathrm{ran}log\to A\in ℂ$$

Proof

Step	Hyp	Ref	Expression
1		ellogrn	$$⊢A\in \mathrm{ran}log\leftrightarrow \left(A\in ℂ\wedge -\mathrm{\pi }<\Im \left(A\right)\wedge \Im \left(A\right)\le \mathrm{\pi }\right)$$
2	1	simp1bi	$$⊢A\in \mathrm{ran}log\to A\in ℂ$$