Theorem nnrecl

Description: There exists a positive integer whose reciprocal is less than a given positive real. Exercise 3 of Apostol p. 28. (Contributed by NM, 8-Nov-2004)

	Ref	Expression
Assertion	nnrecl	|- ( ( A e. RR /\ 0 < A ) -> E. n e. NN ( 1 / n ) < A )

Proof

Step	Hyp	Ref	Expression
1		simpl	|- ( ( A e. RR /\ 0 < A ) -> A e. RR )
2		gt0ne0	|- ( ( A e. RR /\ 0 < A ) -> A =/= 0 )
3	1 2	rereccld	|- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR )
4		arch	|- ( ( 1 / A ) e. RR -> E. n e. NN ( 1 / A ) < n )
5	3 4	syl	|- ( ( A e. RR /\ 0 < A ) -> E. n e. NN ( 1 / A ) < n )
6		recgt0	|- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )
7	3 6	jca	|- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / A ) e. RR /\ 0 < ( 1 / A ) ) )
8		nnre	|- ( n e. NN -> n e. RR )
9		nngt0	|- ( n e. NN -> 0 < n )
10	8 9	jca	|- ( n e. NN -> ( n e. RR /\ 0 < n ) )
11		ltrec	|- ( ( ( ( 1 / A ) e. RR /\ 0 < ( 1 / A ) ) /\ ( n e. RR /\ 0 < n ) ) -> ( ( 1 / A ) < n <-> ( 1 / n ) < ( 1 / ( 1 / A ) ) ) )
12	7 10 11	syl2an	|- ( ( ( A e. RR /\ 0 < A ) /\ n e. NN ) -> ( ( 1 / A ) < n <-> ( 1 / n ) < ( 1 / ( 1 / A ) ) ) )
13		recn	|- ( A e. RR -> A e. CC )
14	13	adantr	|- ( ( A e. RR /\ 0 < A ) -> A e. CC )
15	14 2	recrecd	|- ( ( A e. RR /\ 0 < A ) -> ( 1 / ( 1 / A ) ) = A )
16	15	breq2d	|- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / n ) < ( 1 / ( 1 / A ) ) <-> ( 1 / n ) < A ) )
17	16	adantr	|- ( ( ( A e. RR /\ 0 < A ) /\ n e. NN ) -> ( ( 1 / n ) < ( 1 / ( 1 / A ) ) <-> ( 1 / n ) < A ) )
18	12 17	bitrd	|- ( ( ( A e. RR /\ 0 < A ) /\ n e. NN ) -> ( ( 1 / A ) < n <-> ( 1 / n ) < A ) )
19	18	rexbidva	|- ( ( A e. RR /\ 0 < A ) -> ( E. n e. NN ( 1 / A ) < n <-> E. n e. NN ( 1 / n ) < A ) )
20	5 19	mpbid	|- ( ( A e. RR /\ 0 < A ) -> E. n e. NN ( 1 / n ) < A )