Theorem logcl

Description: Closure of the natural logarithm function. (Contributed by NM, 21-Apr-2008) (Revised by Mario Carneiro, 23-Sep-2014)

	Ref	Expression
Assertion	logcl	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \log A\in ℂ$$

Proof

Step	Hyp	Ref	Expression
1		logrncl	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \log A\in \mathrm{ran}log$$
2		logrncn	$$⊢\log A\in \mathrm{ran}log\to \log A\in ℂ$$
3	1 2	syl	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \log A\in ℂ$$