Theorem ltmul1ii

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) (Proof shortened by Paul Chapman, 25-Jan-2008)

	Ref	Expression
Hypotheses	ltplus1.1	$$⊢A\in ℝ$$
	prodgt0.2	$$⊢B\in ℝ$$
	ltmul1.3	$$⊢C\in ℝ$$
	ltmul1i.4	$$⊢0<C$$
Assertion	ltmul1ii	$$⊢A<B\leftrightarrow AC<BC$$

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	$$⊢A\in ℝ$$
2		prodgt0.2	$$⊢B\in ℝ$$
3		ltmul1.3	$$⊢C\in ℝ$$
4		ltmul1i.4	$$⊢0<C$$
5	1 2 3	ltmul1i	$$⊢0<C\to \left(A<B\leftrightarrow AC<BC\right)$$
6	4 5	ax-mp	$$⊢A<B\leftrightarrow AC<BC$$