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Theorem divcnv 11505
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 109 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  A  e.  CC )
21abscld 11190 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( abs `  A
)  e.  RR )
3 simpr 110 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  ->  x  e.  RR+ )
42, 3rerpdivcld 9728 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( abs `  A
)  /  x )  e.  RR )
5 arch 9173 . . . . 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 492 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  ->  A  e.  CC )
8 eluzelz 9537 . . . . . . . . . . . 12  |-  ( k  e.  ( ZZ>= `  j
)  ->  k  e.  ZZ )
98adantl 277 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  ZZ )
109zcnd 9376 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  CC )
119zred 9375 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  RR )
12 0red 7958 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  e.  RR )
13 simpllr 534 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  NN )
1413nnred 8932 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  RR )
1513nngt0d 8963 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <  j )
16 eluzle 9540 . . . . . . . . . . . . 13  |-  ( k  e.  ( ZZ>= `  j
)  ->  j  <_  k )
1716adantl 277 . . . . . . . . . . . 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 8380 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <  k )
1911, 18gt0ap0d 8586 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k #  0 )
207, 10, 19absdivapd 11204 . . . . . . . . 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 8076 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <_  k )
2211, 21absidd 11176 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  k
)  =  k )
2322oveq2d 5891 . . . . . . . . 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 2210 . . . . . . . 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 492 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  A
)  e.  RR )
263ad3antrrr 492 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  ->  x  e.  RR+ )
2711, 18elrpd 9693 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  RR+ )
284ad3antrrr 492 . . . . . . . . . 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 528 . . . . . . . . . 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 8380 . . . . . . . . 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 9757 . . . . . . . 8  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( abs `  A
)  /  k )  <  x )
3224, 31eqbrtrd 4026 . . . . . . 7  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  ( A  /  k ) )  <  x )
3332ralrimiva 2550 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  ->  A. k  e.  ( ZZ>=
`  j ) ( abs `  ( A  /  k ) )  <  x )
3433ex 115 . . . . 5  |-  ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  ->  ( ( ( abs `  A )  /  x )  < 
j  ->  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  /  k ) )  <  x ) )
3534reximdva 2579 . . . 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 2550 . 2  |-  ( A  e.  CC  ->  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  /  k ) )  <  x )
38 nnuz 9563 . . 3  |-  NN  =  ( ZZ>= `  1 )
39 1zzd 9280 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
40 nnex 8925 . . . . 5  |-  NN  e.  _V
4140mptex 5743 . . . 4  |-  ( n  e.  NN  |->  ( A  /  n ) )  e.  _V
4241a1i 9 . . 3  |-  ( A  e.  CC  ->  (
n  e.  NN  |->  ( A  /  n ) )  e.  _V )
43 simpr 110 . . . 4  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  k  e.  NN )
44 simpl 109 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  A  e.  CC )
4543nncnd 8933 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  k  e.  CC )
4643nnap0d 8965 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  k #  0 )
4744, 45, 46divclapd 8747 . . . 4  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  /  k
)  e.  CC )
48 oveq2 5883 . . . . 5  |-  ( n  =  k  ->  ( A  /  n )  =  ( A  /  k
) )
49 eqid 2177 . . . . 5  |-  ( n  e.  NN  |->  ( A  /  n ) )  =  ( n  e.  NN  |->  ( A  /  n ) )
5048, 49fvmptg 5593 . . . 4  |-  ( ( k  e.  NN  /\  ( A  /  k
)  e.  CC )  ->  ( ( n  e.  NN  |->  ( A  /  n ) ) `
 k )  =  ( A  /  k
) )
5143, 47, 50syl2anc 411 . . 3  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( A  /  n ) ) `  k )  =  ( A  /  k ) )
5238, 39, 42, 51, 47clim0c 11294 . 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 167 1  |-  ( A  e.  CC  ->  (
n  e.  NN  |->  ( A  /  n ) )  ~~>  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   _Vcvv 2738   class class class wbr 4004    |-> cmpt 4065   ` cfv 5217  (class class class)co 5875   CCcc 7809   RRcr 7810   0cc0 7811   1c1 7812    < clt 7992    <_ cle 7993    / cdiv 8629   NNcn 8919   ZZcz 9253   ZZ>=cuz 9528   RR+crp 9653   abscabs 11006    ~~> cli 11286
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-rp 9654  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-clim 11287
This theorem is referenced by:  trireciplem  11508  expcnvap0  11510
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