Theorem ltdiv1ii

Description: Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999)

	Ref	Expression
Hypotheses	ltplus1.1	|- A e. RR
	prodgt0.2	|- B e. RR
	ltmul1.3	|- C e. RR
	ltmul1i.4	|- 0 < C
Assertion	ltdiv1ii	|- ( A < B <-> ( A / C ) < ( B / C ) )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	|- A e. RR
2		prodgt0.2	|- B e. RR
3		ltmul1.3	|- C e. RR
4		ltmul1i.4	|- 0 < C
5	1 2 3	ltdiv1i	|- ( 0 < C -> ( A < B <-> ( A / C ) < ( B / C ) ) )
6	4 5	ax-mp	|- ( A < B <-> ( A / C ) < ( B / C ) )