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Metamath Proof Explorer

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	$⊢ (A \in ℂ \land A \neq 0) \to \log A \in ℂ$

Step	Hyp	Ref	Expression
1		logrncl	$⊢ (A \in ℂ \land A \neq 0) \to \log A \in ran log$
2		logrncn	$⊢ \log A \in ran log \to \log A \in ℂ$
3	1 2	syl	$⊢ (A \in ℂ \land A \neq 0) \to \log A \in ℂ$