Theorem lediv23d

Description: Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltdiv23d.1	|- ( ph -> A e. RR )
	ltdiv23d.2	|- ( ph -> B e. RR+ )
	ltdiv23d.3	|- ( ph -> C e. RR+ )
	lediv23d.4	|- ( ph -> ( A / B ) <_ C )
Assertion	lediv23d	|- ( ph -> ( A / C ) <_ B )

Proof

Step	Hyp	Ref	Expression
1		ltdiv23d.1	|- ( ph -> A e. RR )
2		ltdiv23d.2	|- ( ph -> B e. RR+ )
3		ltdiv23d.3	|- ( ph -> C e. RR+ )
4		lediv23d.4	|- ( ph -> ( A / B ) <_ C )
5	2	rpregt0d	|- ( ph -> ( B e. RR /\ 0 < B ) )
6	3	rpregt0d	|- ( ph -> ( C e. RR /\ 0 < C ) )
7		lediv23	|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) <_ C <-> ( A / C ) <_ B ) )
8	1 5 6 7	syl3anc	|- ( ph -> ( ( A / B ) <_ C <-> ( A / C ) <_ B ) )
9	4 8	mpbid	|- ( ph -> ( A / C ) <_ B )