Theorem lerecd

Description: The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpred.1	$$⊢\phi \to A\in {ℝ}^{+}$$
	rpaddcld.1	$$⊢\phi \to B\in {ℝ}^{+}$$
Assertion	lerecd	$$⊢\phi \to \left(A\le B\leftrightarrow \frac{1}{B}\le \frac{1}{A}\right)$$

Proof

Step	Hyp	Ref	Expression
1		rpred.1	$$⊢\phi \to A\in {ℝ}^{+}$$
2		rpaddcld.1	$$⊢\phi \to B\in {ℝ}^{+}$$
3	1	rpregt0d	$$⊢\phi \to \left(A\in ℝ\wedge 0<A\right)$$
4	2	rpregt0d	$$⊢\phi \to \left(B\in ℝ\wedge 0<B\right)$$
5		lerec	$$⊢\left(\left(A\in ℝ\wedge 0<A\right)\wedge \left(B\in ℝ\wedge 0<B\right)\right)\to \left(A\le B\leftrightarrow \frac{1}{B}\le \frac{1}{A}\right)$$
6	3 4 5	syl2anc	$$⊢\phi \to \left(A\le B\leftrightarrow \frac{1}{B}\le \frac{1}{A}\right)$$