Theorem rplogcld

Description: Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	relogefd.1	|- ( ph -> A e. RR )
	rplogcld.2	|- ( ph -> 1 < A )
Assertion	rplogcld	|- ( ph -> ( log ` A ) e. RR+ )

Proof

Step	Hyp	Ref	Expression
1		relogefd.1	|- ( ph -> A e. RR )
2		rplogcld.2	|- ( ph -> 1 < A )
3		rplogcl	|- ( ( A e. RR /\ 1 < A ) -> ( log ` A ) e. RR+ )
4	1 2 3	syl2anc	|- ( ph -> ( log ` A ) e. RR+ )