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Theorem divcnv 11234
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 108 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  A  e.  CC )
21abscld 10921 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( abs `  A
)  e.  RR )
3 simpr 109 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  e.  RR+ )
42, 3rerpdivcld 9483 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( abs `  A
)  /  x )  e.  RR )
5 arch 8942 . . . . 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 483 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  ->  A  e.  CC )
8 eluzelz 9303 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ>= `  j
)  ->  k  e.  ZZ )
98adantl 275 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  ZZ )
109zcnd 9142 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  CC )
119zred 9141 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  RR )
12 0red 7735 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  e.  RR )
13 simpllr 508 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  NN )
1413nnred 8701 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  RR )
1513nngt0d 8732 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <  j )
16 eluzle 9306 . . . . . . . . . . . . 13  |-  ( k  e.  ( ZZ>= `  j
)  ->  j  <_  k )
1716adantl 275 . . . . . . . . . . . 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 8153 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <  k )
1911, 18gt0ap0d 8359 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k #  0 )
207, 10, 19absdivapd 10935 . . . . . . . . 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 7849 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <_  k )
2211, 21absidd 10907 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  k
)  =  k )
2322oveq2d 5758 . . . . . . . . 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 2150 . . . . . . . 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 483 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  A
)  e.  RR )
263ad3antrrr 483 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  ->  x  e.  RR+ )
2711, 18elrpd 9449 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  RR+ )
284ad3antrrr 483 . . . . . . . . . 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 504 . . . . . . . . . 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 8153 . . . . . . . . 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 9512 . . . . . . . 8  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( abs `  A
)  /  k )  <  x )
3224, 31eqbrtrd 3920 . . . . . . 7  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  ( A  /  k ) )  <  x )
3332ralrimiva 2482 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  ->  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( A  /  k ) )  <  x )
3433ex 114 . . . . 5  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  ->  ( ( ( abs `  A )  /  x )  < 
j  ->  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  /  k ) )  <  x ) )
3534reximdva 2511 . . . 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 2482 . 2  |-  ( A  e.  CC  ->  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  /  k ) )  <  x )
38 nnuz 9329 . . 3  |-  NN  =  ( ZZ>= `  1 )
39 1zzd 9049 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
40 nnex 8694 . . . . 5  |-  NN  e.  _V
4140mptex 5614 . . . 4  |-  ( n  e.  NN  |->  ( A  /  n ) )  e.  _V
4241a1i 9 . . 3  |-  ( A  e.  CC  ->  (
n  e.  NN  |->  ( A  /  n ) )  e.  _V )
43 simpr 109 . . . 4  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  k  e.  NN )
44 simpl 108 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  A  e.  CC )
4543nncnd 8702 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  k  e.  CC )
4643nnap0d 8734 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  k #  0 )
4744, 45, 46divclapd 8518 . . . 4  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  /  k
)  e.  CC )
48 oveq2 5750 . . . . 5  |-  ( n  =  k  ->  ( A  /  n )  =  ( A  /  k
) )
49 eqid 2117 . . . . 5  |-  ( n  e.  NN  |->  ( A  /  n ) )  =  ( n  e.  NN  |->  ( A  /  n ) )
5048, 49fvmptg 5465 . . . 4  |-  ( ( k  e.  NN  /\  ( A  /  k
)  e.  CC )  ->  ( ( n  e.  NN  |->  ( A  /  n ) ) `
 k )  =  ( A  /  k
) )
5143, 47, 50syl2anc 408 . . 3  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( A  /  n ) ) `  k )  =  ( A  /  k ) )
5238, 39, 42, 51, 47clim0c 11023 . 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 166 1  |-  ( A  e.  CC  ->  (
n  e.  NN  |->  ( A  /  n ) )  ~~>  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   _Vcvv 2660   class class class wbr 3899    |-> cmpt 3959   ` cfv 5093  (class class class)co 5742   CCcc 7586   RRcr 7587   0cc0 7588   1c1 7589    < clt 7768    <_ cle 7769    / cdiv 8400   NNcn 8688   ZZcz 9022   ZZ>=cuz 9294   RR+crp 9409   abscabs 10737    ~~> cli 11015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-rp 9410  df-seqfrec 10187  df-exp 10261  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-clim 11016
This theorem is referenced by:  trireciplem  11237  expcnvap0  11239
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