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	$$⊢A\in ℝ$$
	prodgt0.2	$$⊢B\in ℝ$$
	ltmul1.3	$$⊢C\in ℝ$$
	ltmul1i.4	$$⊢0<C$$
Assertion	ltdiv1ii	$$⊢A<B\leftrightarrow \frac{A}{C}<\frac{B}{C}$$

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	ltdiv1i	$$⊢0<C\to \left(A<B\leftrightarrow \frac{A}{C}<\frac{B}{C}\right)$$
6	4 5	ax-mp	$$⊢A<B\leftrightarrow \frac{A}{C}<\frac{B}{C}$$