Theorem ltdiv1dd

Description: Division of both sides of 'less than' by a positive number. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	ltmul1d.1	$$⊢\phi \to A\in ℝ$$
	ltmul1d.2	$$⊢\phi \to B\in ℝ$$
	ltmul1d.3	$$⊢\phi \to C\in {ℝ}^{+}$$
	ltdiv1dd.4	$$⊢\phi \to A<B$$
Assertion	ltdiv1dd	$$⊢\phi \to \frac{A}{C}<\frac{B}{C}$$

Proof

Step	Hyp	Ref	Expression
1		ltmul1d.1	$$⊢\phi \to A\in ℝ$$
2		ltmul1d.2	$$⊢\phi \to B\in ℝ$$
3		ltmul1d.3	$$⊢\phi \to C\in {ℝ}^{+}$$
4		ltdiv1dd.4	$$⊢\phi \to A<B$$
5	1 2 3	ltdiv1d	$$⊢\phi \to \left(A<B\leftrightarrow \frac{A}{C}<\frac{B}{C}\right)$$
6	4 5	mpbid	$$⊢\phi \to \frac{A}{C}<\frac{B}{C}$$