Theorem lediv2d

Description: Division of a positive number by both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpred.1	|- ( ph -> A e. RR+ )
	rpaddcld.1	|- ( ph -> B e. RR+ )
	ltdiv2d.3	|- ( ph -> C e. RR+ )
Assertion	lediv2d	|- ( ph -> ( A <_ B <-> ( C / B ) <_ ( C / A ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	|- ( ph -> A e. RR+ )
2		rpaddcld.1	|- ( ph -> B e. RR+ )
3		ltdiv2d.3	|- ( ph -> C e. RR+ )
4	1	rpregt0d	|- ( ph -> ( A e. RR /\ 0 < A ) )
5	2	rpregt0d	|- ( ph -> ( B e. RR /\ 0 < B ) )
6	3	rpregt0d	|- ( ph -> ( C e. RR /\ 0 < C ) )
7		lediv2	|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( C / B ) <_ ( C / A ) ) )
8	4 5 6 7	syl3anc	|- ( ph -> ( A <_ B <-> ( C / B ) <_ ( C / A ) ) )