Theorem ltrecii

Description: The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999)

	Ref	Expression
Hypotheses	ltplus1.1	$$⊢A\in ℝ$$
	prodgt0.2	$$⊢B\in ℝ$$
	ltreci.3	$$⊢0<A$$
	ltreci.4	$$⊢0<B$$
Assertion	ltrecii	$$⊢A<B\leftrightarrow \frac{1}{B}<\frac{1}{A}$$

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	$$⊢A\in ℝ$$
2		prodgt0.2	$$⊢B\in ℝ$$
3		ltreci.3	$$⊢0<A$$
4		ltreci.4	$$⊢0<B$$
5	1 2	ltreci	$$⊢\left(0<A\wedge 0<B\right)\to \left(A<B\leftrightarrow \frac{1}{B}<\frac{1}{A}\right)$$
6	3 4 5	mp2an	$$⊢A<B\leftrightarrow \frac{1}{B}<\frac{1}{A}$$