Theorem logled

Description: Natural logarithm preserves <_ . (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	relogcld.1	|- ( ph -> A e. RR+ )
	relogmuld.2	|- ( ph -> B e. RR+ )
Assertion	logled	|- ( ph -> ( A <_ B <-> ( log ` A ) <_ ( log ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		relogcld.1	|- ( ph -> A e. RR+ )
2		relogmuld.2	|- ( ph -> B e. RR+ )
3		logleb	|- ( ( A e. RR+ /\ B e. RR+ ) -> ( A <_ B <-> ( log ` A ) <_ ( log ` B ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( A <_ B <-> ( log ` A ) <_ ( log ` B ) ) )