Theorem ltmuldiv2

Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004)

	Ref	Expression
Assertion	ltmuldiv2	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. A ) < B <-> A < ( B / C ) ) )

Proof

Step	Hyp	Ref	Expression
1		recn	|- ( A e. RR -> A e. CC )
2		recn	|- ( C e. RR -> C e. CC )
3		mulcom	|- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
4	1 2 3	syl2an	|- ( ( A e. RR /\ C e. RR ) -> ( A x. C ) = ( C x. A ) )
5	4	adantrr	|- ( ( A e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A x. C ) = ( C x. A ) )
6	5	3adant2	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A x. C ) = ( C x. A ) )
7	6	breq1d	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) < B <-> ( C x. A ) < B ) )
8		ltmuldiv	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) < B <-> A < ( B / C ) ) )
9	7 8	bitr3d	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. A ) < B <-> A < ( B / C ) ) )