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	|- ( ph -> A e. RR )
	recgt0d.2	|- ( ph -> 0 < A )
Assertion	recgt0d	|- ( ph -> 0 < ( 1 / A ) )

Proof

Step	Hyp	Ref	Expression
1		ltp1d.1	|- ( ph -> A e. RR )
2		recgt0d.2	|- ( ph -> 0 < A )
3		recgt0	|- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )
4	1 2 3	syl2anc	|- ( ph -> 0 < ( 1 / A ) )