Theorem lt2mul2divd

Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	lt2mul2divd.1	|- ( ph -> A e. RR )
	lt2mul2divd.2	|- ( ph -> B e. RR+ )
	lt2mul2divd.3	|- ( ph -> C e. RR )
	lt2mul2divd.4	|- ( ph -> D e. RR+ )
Assertion	lt2mul2divd	|- ( ph -> ( ( A x. B ) < ( C x. D ) <-> ( A / D ) < ( C / B ) ) )

Proof

Step	Hyp	Ref	Expression
1		lt2mul2divd.1	|- ( ph -> A e. RR )
2		lt2mul2divd.2	|- ( ph -> B e. RR+ )
3		lt2mul2divd.3	|- ( ph -> C e. RR )
4		lt2mul2divd.4	|- ( ph -> D e. RR+ )
5	2	rpregt0d	|- ( ph -> ( B e. RR /\ 0 < B ) )
6	4	rpregt0d	|- ( ph -> ( D e. RR /\ 0 < D ) )
7		lt2mul2div	|- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( A x. B ) < ( C x. D ) <-> ( A / D ) < ( C / B ) ) )
8	1 5 3 6 7	syl22anc	|- ( ph -> ( ( A x. B ) < ( C x. D ) <-> ( A / D ) < ( C / B ) ) )