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Theorem nnrecl 9979
Description: There exists a natural number 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
StepHypRef Expression
1 simpl 443 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  RR )
2 gt0ne0 9255 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
31, 2rereccld 9603 . . 3  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
4 arch 9978 . . 3  |-  ( ( 1  /  A )  e.  RR  ->  E. n  e.  NN  ( 1  /  A )  <  n
)
53, 4syl 15 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  ->  E. n  e.  NN  ( 1  /  A
)  <  n )
6 recgt0 9616 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
73, 6jca 518 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  A )  e.  RR  /\  0  <  ( 1  /  A ) ) )
8 nnre 9769 . . . . . 6  |-  ( n  e.  NN  ->  n  e.  RR )
9 nngt0 9791 . . . . . 6  |-  ( n  e.  NN  ->  0  <  n )
108, 9jca 518 . . . . 5  |-  ( n  e.  NN  ->  (
n  e.  RR  /\  0  <  n ) )
11 ltrec 9653 . . . . 5  |-  ( ( ( ( 1  /  A )  e.  RR  /\  0  <  ( 1  /  A ) )  /\  ( n  e.  RR  /\  0  < 
n ) )  -> 
( ( 1  /  A )  <  n  <->  ( 1  /  n )  <  ( 1  / 
( 1  /  A
) ) ) )
127, 10, 11syl2an 463 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  n  e.  NN )  ->  ( ( 1  /  A )  < 
n  <->  ( 1  /  n )  <  (
1  /  ( 1  /  A ) ) ) )
13 recn 8843 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
1413adantr 451 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  CC )
1514, 2recrecd 9549 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  (
1  /  A ) )  =  A )
1615breq2d 4051 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( ( 1  /  n )  <  (
1  /  ( 1  /  A ) )  <-> 
( 1  /  n
)  <  A )
)
1716adantr 451 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  n  e.  NN )  ->  ( ( 1  /  n )  < 
( 1  /  (
1  /  A ) )  <->  ( 1  /  n )  <  A
) )
1812, 17bitrd 244 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  n  e.  NN )  ->  ( ( 1  /  A )  < 
n  <->  ( 1  /  n )  <  A
) )
1918rexbidva 2573 . 2  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( E. n  e.  NN  ( 1  /  A )  <  n  <->  E. n  e.  NN  (
1  /  n )  <  A ) )
205, 19mpbid 201 1  |-  ( ( A  e.  RR  /\  0  <  A )  ->  E. n  e.  NN  ( 1  /  n
)  <  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   E.wrex 2557   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    < clt 8883    / cdiv 9439   NNcn 9762
This theorem is referenced by:  qbtwnre  10542  met1stc  18083  met2ndci  18084  bcthlem4  18765  ismbf3d  19025  itg2seq  19113  itg2gt0  19131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763
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