Theorem ltrec1d

Description: Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpred.1	$$⊢\phi \to A\in {ℝ}^{+}$$
	rpaddcld.1	$$⊢\phi \to B\in {ℝ}^{+}$$
	ltrec1d.2	$$⊢\phi \to \frac{1}{A}<B$$
Assertion	ltrec1d	$$⊢\phi \to \frac{1}{B}<A$$

Proof

Step	Hyp	Ref	Expression
1		rpred.1	$$⊢\phi \to A\in {ℝ}^{+}$$
2		rpaddcld.1	$$⊢\phi \to B\in {ℝ}^{+}$$
3		ltrec1d.2	$$⊢\phi \to \frac{1}{A}<B$$
4	1	rpregt0d	$$⊢\phi \to \left(A\in ℝ\wedge 0<A\right)$$
5	2	rpregt0d	$$⊢\phi \to \left(B\in ℝ\wedge 0<B\right)$$
6		ltrec1	$$⊢\left(\left(A\in ℝ\wedge 0<A\right)\wedge \left(B\in ℝ\wedge 0<B\right)\right)\to \left(\frac{1}{A}<B\leftrightarrow \frac{1}{B}<A\right)$$
7	4 5 6	syl2anc	$$⊢\phi \to \left(\frac{1}{A}<B\leftrightarrow \frac{1}{B}<A\right)$$
8	3 7	mpbid	$$⊢\phi \to \frac{1}{B}<A$$