Theorem nnrecl 9459

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.)

Assertion

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

Distinct variable group:   A, n

Proof of Theorem nnrecl

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> A e. RR )
2		gt0ap0 8865	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> A # 0 )
3	1, 2	rerecclapd 9073	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR )
4		arch 9458	. . 3 |- ( ( 1 / A ) e. RR -> E. n e. NN ( 1 / A ) < n )
5	3, 4	syl 14	. 2 |- ( ( A e. RR /\ 0 < A ) -> E. n e. NN ( 1 / A ) < n )
6		recgt0 9089	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )
7	3, 6	jca 306	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / A ) e. RR /\ 0 < ( 1 / A ) ) )
8		nnre 9209	. . . . . 6 |- ( n e. NN -> n e. RR )
9		nngt0 9227	. . . . . 6 |- ( n e. NN -> 0 < n )
10	8, 9	jca 306	. . . . 5 |- ( n e. NN -> ( n e. RR /\ 0 < n ) )
11		ltrec 9122	. . . . 5 |- ( ( ( ( 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 289	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ n e. NN ) -> ( ( 1 / A ) < n <-> ( 1 / n ) < ( 1 / ( 1 / A ) ) ) )
13		recn 8225	. . . . . . . 8 |- ( A e. RR -> A e. CC )
14	13	adantr 276	. . . . . . 7 |- ( ( A e. RR /\ 0 < A ) -> A e. CC )
15	14, 2	recrecapd 9024	. . . . . 6 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / ( 1 / A ) ) = A )
16	15	breq2d 4105	. . . . 5 |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / n ) < ( 1 / ( 1 / A ) ) <-> ( 1 / n ) < A ) )
17	16	adantr 276	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ n e. NN ) -> ( ( 1 / n ) < ( 1 / ( 1 / A ) ) <-> ( 1 / n ) < A ) )
18	12, 17	bitrd 188	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ n e. NN ) -> ( ( 1 / A ) < n <-> ( 1 / n ) < A ) )
19	18	rexbidva 2530	. 2 |- ( ( A e. RR /\ 0 < A ) -> ( E. n e. NN ( 1 / A ) < n <-> E. n e. NN ( 1 / n ) < A ) )
20	5, 19	mpbid 147	1 |- ( ( A e. RR /\ 0 < A ) -> E. n e. NN ( 1 / n ) < A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2202   E.wrex 2512   class class class wbr 4093  (class class class)co 6028   CCcc 8090   RRcr 8091   0cc0 8092   1c1 8093   < clt 8273   / cdiv 8911   NNcn 9202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203

This theorem is referenced by:  qbtwnre  10579  trilpolemlt1  16773