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 e. CC /\ A =/= 0 ) -> ( log ` A ) e. CC )

Proof

Step Hyp Ref Expression
1 logrncl 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( log ` A ) e. ran log )
2 logrncn 
 |-  ( ( log ` A ) e. ran log -> ( log ` A ) e. CC )
3 1 2 syl 
 |-  ( ( A e. CC /\ A =/= 0 ) -> ( log ` A ) e. CC )