Theorem lereci

Description: The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999)

	Ref	Expression
Hypotheses	ltplus1.1	$$⊢A\in ℝ$$
	prodgt0.2	$$⊢B\in ℝ$$
Assertion	lereci	$$⊢\left(0<A\wedge 0<B\right)\to \left(A\le B\leftrightarrow \frac{1}{B}\le \frac{1}{A}\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	$$⊢A\in ℝ$$
2		prodgt0.2	$$⊢B\in ℝ$$
3		lerec	$$⊢\left(\left(A\in ℝ\wedge 0<A\right)\wedge \left(B\in ℝ\wedge 0<B\right)\right)\to \left(A\le B\leftrightarrow \frac{1}{B}\le \frac{1}{A}\right)$$
4	2 3	mpanr1	$$⊢\left(\left(A\in ℝ\wedge 0<A\right)\wedge 0<B\right)\to \left(A\le B\leftrightarrow \frac{1}{B}\le \frac{1}{A}\right)$$
5	1 4	mpanl1	$$⊢\left(0<A\wedge 0<B\right)\to \left(A\le B\leftrightarrow \frac{1}{B}\le \frac{1}{A}\right)$$