Theorem ltmul1i

Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of Apostol p. 20. (Contributed by NM, 16-May-1999)

	Ref	Expression
Hypotheses	ltplus1.1	|- A e. RR
	prodgt0.2	|- B e. RR
	ltmul1.3	|- C e. RR
Assertion	ltmul1i	|- ( 0 < C -> ( A < B <-> ( A x. C ) < ( B x. 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		ltmul1	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( A x. C ) < ( B x. C ) ) )
5	1 2 4	mp3an12	|- ( ( C e. RR /\ 0 < C ) -> ( A < B <-> ( A x. C ) < ( B x. C ) ) )
6	3 5	mpan	|- ( 0 < C -> ( A < B <-> ( A x. C ) < ( B x. C ) ) )