Theorem ltdiv23d

Description: Swap denominator with other side of 'less than'. (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+ )
	ltdiv23d.4	|- ( ph -> ( A / B ) < C )
Assertion	ltdiv23d	|- ( 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		ltdiv23d.4	|- ( ph -> ( A / B ) < C )
5	2	rpregt0d	|- ( ph -> ( B e. RR /\ 0 < B ) )
6	3	rpregt0d	|- ( ph -> ( C e. RR /\ 0 < C ) )
7		ltdiv23	|- ( ( 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 )