Theorem ltdiv2dd

Description: Division of a positive number by both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	ltdiv2dd.a	|- ( ph -> A e. RR+ )
	ltdiv2dd.b	|- ( ph -> B e. RR+ )
	ltdiv2dd.c	|- ( ph -> C e. RR+ )
	ltdiv2dd.altb	|- ( ph -> A < B )
Assertion	ltdiv2dd	|- ( ph -> ( C / B ) < ( C / A ) )

Proof

Step	Hyp	Ref	Expression
1		ltdiv2dd.a	|- ( ph -> A e. RR+ )
2		ltdiv2dd.b	|- ( ph -> B e. RR+ )
3		ltdiv2dd.c	|- ( ph -> C e. RR+ )
4		ltdiv2dd.altb	|- ( ph -> A < B )
5	1 2 3	ltdiv2d	|- ( ph -> ( A < B <-> ( C / B ) < ( C / A ) ) )
6	4 5	mpbid	|- ( ph -> ( C / B ) < ( C / A ) )