Theorem lemuldiv2

Description: 'Less than or equal' relationship between division and multiplication. (Contributed by NM, 10-Mar-2006)

	Ref	Expression
Assertion	lemuldiv2	|- ( ( 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		lemuldiv	|- ( ( 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 ) ) )