Theorem recgt0d

Description: The reciprocal of a positive number is positive. Exercise 4 of Apostol p. 21. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltp1d.1	$$⊢\phi \to A\in ℝ$$
	recgt0d.2	$$⊢\phi \to 0<A$$
Assertion	recgt0d	$$⊢\phi \to 0<\frac{1}{A}$$

Proof

Step	Hyp	Ref	Expression
1		ltp1d.1	$$⊢\phi \to A\in ℝ$$
2		recgt0d.2	$$⊢\phi \to 0<A$$
3		recgt0	$$⊢\left(A\in ℝ\wedge 0<A\right)\to 0<\frac{1}{A}$$
4	1 2 3	syl2anc	$$⊢\phi \to 0<\frac{1}{A}$$