Theorem ltmuldivi

Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999)

	Ref	Expression
Hypotheses	ltplus1.1	$$⊢A\in ℝ$$
	prodgt0.2	$$⊢B\in ℝ$$
	ltmul1.3	$$⊢C\in ℝ$$
Assertion	ltmuldivi	$$⊢0<C\to \left(AC<B\leftrightarrow A<\frac{B}{C}\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	$$⊢A\in ℝ$$
2		prodgt0.2	$$⊢B\in ℝ$$
3		ltmul1.3	$$⊢C\in ℝ$$
4		ltmuldiv	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge \left(C\in ℝ\wedge 0<C\right)\right)\to \left(AC<B\leftrightarrow A<\frac{B}{C}\right)$$
5	1 2 4	mp3an12	$$⊢\left(C\in ℝ\wedge 0<C\right)\to \left(AC<B\leftrightarrow A<\frac{B}{C}\right)$$
6	3 5	mpan	$$⊢0<C\to \left(AC<B\leftrightarrow A<\frac{B}{C}\right)$$