Theorem lediv1dd

Description: Division of both sides of a less than or equal to relation 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	$$⊢\phi \to C\in {ℝ}^{+}$$
	lediv1dd.4	$$⊢\phi \to A\le B$$
Assertion	lediv1dd	$$⊢\phi \to \frac{A}{C}\le \frac{B}{C}$$

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		lediv1dd.4	$$⊢\phi \to A\le B$$
5	1 2 3	lediv1d	$$⊢\phi \to \left(A\le B\leftrightarrow \frac{A}{C}\le \frac{B}{C}\right)$$
6	4 5	mpbid	$$⊢\phi \to \frac{A}{C}\le \frac{B}{C}$$