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Theorem divcnv 10887
Description: The sequence of reciprocals of positive integers, multiplied by the factor  A, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Jim Kingdon, 22-Oct-2022.)
Assertion
Ref Expression
divcnv  |-  ( A  e.  CC  ->  (
n  e.  NN  |->  ( A  /  n ) )  ~~>  0 )
Distinct variable group:    A, n

Proof of Theorem divcnv
Dummy variables  j  k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 107 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  A  e.  CC )
21abscld 10610 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( abs `  A
)  e.  RR )
3 simpr 108 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  e.  RR+ )
42, 3rerpdivcld 9203 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( abs `  A
)  /  x )  e.  RR )
5 arch 8668 . . . . 5  |-  ( ( ( abs `  A
)  /  x )  e.  RR  ->  E. j  e.  NN  ( ( abs `  A )  /  x
)  <  j )
64, 5syl 14 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. j  e.  NN  ( ( abs `  A
)  /  x )  <  j )
71ad3antrrr 476 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  ->  A  e.  CC )
8 eluzelz 9026 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ>= `  j
)  ->  k  e.  ZZ )
98adantl 271 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  ZZ )
109zcnd 8867 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  CC )
119zred 8866 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  RR )
12 0red 7487 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  e.  RR )
13 simpllr 501 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  NN )
1413nnred 8433 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  RR )
1513nngt0d 8464 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <  j )
16 eluzle 9029 . . . . . . . . . . . . 13  |-  ( k  e.  ( ZZ>= `  j
)  ->  j  <_  k )
1716adantl 271 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  <_  k )
1812, 14, 11, 15, 17ltletrd 7899 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <  k )
1911, 18gt0ap0d 8103 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k #  0 )
207, 10, 19absdivapd 10624 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  ( A  /  k ) )  =  ( ( abs `  A )  /  ( abs `  k ) ) )
2112, 11, 18ltled 7600 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <_  k )
2211, 21absidd 10596 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  k
)  =  k )
2322oveq2d 5668 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( abs `  A
)  /  ( abs `  k ) )  =  ( ( abs `  A
)  /  k ) )
2420, 23eqtrd 2120 . . . . . . . 8  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  ( A  /  k ) )  =  ( ( abs `  A )  /  k
) )
252ad3antrrr 476 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  A
)  e.  RR )
263ad3antrrr 476 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  ->  x  e.  RR+ )
2711, 18elrpd 9169 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  RR+ )
284ad3antrrr 476 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( abs `  A
)  /  x )  e.  RR )
29 simplr 497 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( abs `  A
)  /  x )  <  j )
3028, 14, 11, 29, 17ltletrd 7899 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( abs `  A
)  /  x )  <  k )
3125, 26, 27, 30ltdiv23d 9232 . . . . . . . 8  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( abs `  A
)  /  k )  <  x )
3224, 31eqbrtrd 3865 . . . . . . 7  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  ( A  /  k ) )  <  x )
3332ralrimiva 2446 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  ->  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( A  /  k ) )  <  x )
3433ex 113 . . . . 5  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  ->  ( ( ( abs `  A )  /  x )  < 
j  ->  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  /  k ) )  <  x ) )
3534reximdva 2475 . . . 4  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( E. j  e.  NN  ( ( abs `  A )  /  x
)  <  j  ->  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( abs `  ( A  /  k
) )  <  x
) )
366, 35mpd 13 . . 3  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( abs `  ( A  /  k
) )  <  x
)
3736ralrimiva 2446 . 2  |-  ( A  e.  CC  ->  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  /  k ) )  <  x )
38 nnuz 9052 . . 3  |-  NN  =  ( ZZ>= `  1 )
39 1zzd 8775 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
40 nnex 8426 . . . . 5  |-  NN  e.  _V
4140mptex 5523 . . . 4  |-  ( n  e.  NN  |->  ( A  /  n ) )  e.  _V
4241a1i 9 . . 3  |-  ( A  e.  CC  ->  (
n  e.  NN  |->  ( A  /  n ) )  e.  _V )
43 simpr 108 . . . 4  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  k  e.  NN )
44 simpl 107 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  A  e.  CC )
4543nncnd 8434 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  k  e.  CC )
4643nnap0d 8466 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  k #  0 )
4744, 45, 46divclapd 8255 . . . 4  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  /  k
)  e.  CC )
48 oveq2 5660 . . . . 5  |-  ( n  =  k  ->  ( A  /  n )  =  ( A  /  k
) )
49 eqid 2088 . . . . 5  |-  ( n  e.  NN  |->  ( A  /  n ) )  =  ( n  e.  NN  |->  ( A  /  n ) )
5048, 49fvmptg 5380 . . . 4  |-  ( ( k  e.  NN  /\  ( A  /  k
)  e.  CC )  ->  ( ( n  e.  NN  |->  ( A  /  n ) ) `
 k )  =  ( A  /  k
) )
5143, 47, 50syl2anc 403 . . 3  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( A  /  n ) ) `  k )  =  ( A  /  k ) )
5238, 39, 42, 51, 47clim0c 10670 . 2  |-  ( A  e.  CC  ->  (
( n  e.  NN  |->  ( A  /  n
) )  ~~>  0  <->  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  /  k ) )  <  x ) )
5337, 52mpbird 165 1  |-  ( A  e.  CC  ->  (
n  e.  NN  |->  ( A  /  n ) )  ~~>  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   _Vcvv 2619   class class class wbr 3845    |-> cmpt 3899   ` cfv 5015  (class class class)co 5652   CCcc 7346   RRcr 7347   0cc0 7348   1c1 7349    < clt 7520    <_ cle 7521    / cdiv 8137   NNcn 8420   ZZcz 8748   ZZ>=cuz 9017   RR+crp 9132   abscabs 10426    ~~> cli 10662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462  ax-caucvg 7463
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-2 8479  df-3 8480  df-4 8481  df-n0 8672  df-z 8749  df-uz 9018  df-rp 9133  df-iseq 9849  df-seq3 9850  df-exp 9951  df-cj 10272  df-re 10273  df-im 10274  df-rsqrt 10427  df-abs 10428  df-clim 10663
This theorem is referenced by:  trireciplem  10890  expcnvap0  10892
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