Theorem relogcl

Description: Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007)

	Ref	Expression
Assertion	relogcl	|- ( A e. RR+ -> ( log ` A ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		fvres	|- ( A e. RR+ -> ( ( log |` RR+ ) ` A ) = ( log ` A ) )
2		relogf1o	|- ( log |` RR+ ) : RR+ -1-1-onto-> RR
3		f1of	|- ( ( log |` RR+ ) : RR+ -1-1-onto-> RR -> ( log |` RR+ ) : RR+ --> RR )
4	2 3	ax-mp	|- ( log |` RR+ ) : RR+ --> RR
5	4	ffvelcdmi	|- ( A e. RR+ -> ( ( log |` RR+ ) ` A ) e. RR )
6	1 5	eqeltrrd	|- ( A e. RR+ -> ( log ` A ) e. RR )