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 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		logrncl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ran log )
2		logrncn	⊢ ( ( log ‘ 𝐴 ) ∈ ran log → ( log ‘ 𝐴 ) ∈ ℂ )
3	1 2	syl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ℂ )