Theorem ltdiv2dd

Description: Division of a positive number by both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	ltdiv2dd.a	$$⊢\phi \to A\in {ℝ}^{+}$$
	ltdiv2dd.b	$$⊢\phi \to B\in {ℝ}^{+}$$
	ltdiv2dd.c	$$⊢\phi \to C\in {ℝ}^{+}$$
	ltdiv2dd.altb	$$⊢\phi \to A<B$$
Assertion	ltdiv2dd	$$⊢\phi \to \frac{C}{B}<\frac{C}{A}$$

Proof

Step	Hyp	Ref	Expression
1		ltdiv2dd.a	$$⊢\phi \to A\in {ℝ}^{+}$$
2		ltdiv2dd.b	$$⊢\phi \to B\in {ℝ}^{+}$$
3		ltdiv2dd.c	$$⊢\phi \to C\in {ℝ}^{+}$$
4		ltdiv2dd.altb	$$⊢\phi \to A<B$$
5	1 2 3	ltdiv2d	$$⊢\phi \to \left(A<B\leftrightarrow \frac{C}{B}<\frac{C}{A}\right)$$
6	4 5	mpbid	$$⊢\phi \to \frac{C}{B}<\frac{C}{A}$$