Theorem recgt1d

Description: The reciprocal of a positive number greater than 1 is less than 1. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypothesis	rpred.1	$$⊢\phi \to A\in {ℝ}^{+}$$
Assertion	recgt1d	$$⊢\phi \to \left(1<A\leftrightarrow \frac{1}{A}<1\right)$$

Proof

Step	Hyp	Ref	Expression
1		rpred.1	$$⊢\phi \to A\in {ℝ}^{+}$$
2	1	rpregt0d	$$⊢\phi \to \left(A\in ℝ\wedge 0<A\right)$$
3		recgt1	$$⊢\left(A\in ℝ\wedge 0<A\right)\to \left(1<A\leftrightarrow \frac{1}{A}<1\right)$$
4	2 3	syl	$$⊢\phi \to \left(1<A\leftrightarrow \frac{1}{A}<1\right)$$