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Theorem divcnv 11389
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 11076 . . . . . 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 9630 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( abs `  A
)  /  x )  e.  RR )
5 arch 9082 . . . . 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 484 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  ->  A  e.  CC )
8 eluzelz 9443 . . . . . . . . . . . 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 9282 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  CC )
119zred 9281 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  RR )
12 0red 7874 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  e.  RR )
13 simpllr 524 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  NN )
1413nnred 8841 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  RR )
1513nngt0d 8872 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <  j )
16 eluzle 9446 . . . . . . . . . . . . 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 8293 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <  k )
1911, 18gt0ap0d 8499 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k #  0 )
207, 10, 19absdivapd 11090 . . . . . . . . 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 7989 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <_  k )
2211, 21absidd 11062 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  k
)  =  k )
2322oveq2d 5837 . . . . . . . . 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 2190 . . . . . . . 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 484 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  A
)  e.  RR )
263ad3antrrr 484 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  ->  x  e.  RR+ )
2711, 18elrpd 9595 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  RR+ )
284ad3antrrr 484 . . . . . . . . . 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 520 . . . . . . . . . 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 8293 . . . . . . . . 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 9659 . . . . . . . 8  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( abs `  A
)  /  k )  <  x )
3224, 31eqbrtrd 3986 . . . . . . 7  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  ( A  /  k ) )  <  x )
3332ralrimiva 2530 . . . . . 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 2559 . . . 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 2530 . 2  |-  ( A  e.  CC  ->  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  /  k ) )  <  x )
38 nnuz 9469 . . 3  |-  NN  =  ( ZZ>= `  1 )
39 1zzd 9189 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
40 nnex 8834 . . . . 5  |-  NN  e.  _V
4140mptex 5692 . . . 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 8842 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  k  e.  CC )
4643nnap0d 8874 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  k #  0 )
4744, 45, 46divclapd 8658 . . . 4  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  /  k
)  e.  CC )
48 oveq2 5829 . . . . 5  |-  ( n  =  k  ->  ( A  /  n )  =  ( A  /  k
) )
49 eqid 2157 . . . . 5  |-  ( n  e.  NN  |->  ( A  /  n ) )  =  ( n  e.  NN  |->  ( A  /  n ) )
5048, 49fvmptg 5543 . . . 4  |-  ( ( k  e.  NN  /\  ( A  /  k
)  e.  CC )  ->  ( ( n  e.  NN  |->  ( A  /  n ) ) `
 k )  =  ( A  /  k
) )
5143, 47, 50syl2anc 409 . . 3  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( A  /  n ) ) `  k )  =  ( A  /  k ) )
5238, 39, 42, 51, 47clim0c 11178 . 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 1335    e. wcel 2128   A.wral 2435   E.wrex 2436   _Vcvv 2712   class class class wbr 3965    |-> cmpt 4025   ` cfv 5169  (class class class)co 5821   CCcc 7725   RRcr 7726   0cc0 7727   1c1 7728    < clt 7907    <_ cle 7908    / cdiv 8540   NNcn 8828   ZZcz 9162   ZZ>=cuz 9434   RR+crp 9555   abscabs 10892    ~~> cli 11170
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546  ax-cnex 7818  ax-resscn 7819  ax-1cn 7820  ax-1re 7821  ax-icn 7822  ax-addcl 7823  ax-addrcl 7824  ax-mulcl 7825  ax-mulrcl 7826  ax-addcom 7827  ax-mulcom 7828  ax-addass 7829  ax-mulass 7830  ax-distr 7831  ax-i2m1 7832  ax-0lt1 7833  ax-1rid 7834  ax-0id 7835  ax-rnegex 7836  ax-precex 7837  ax-cnre 7838  ax-pre-ltirr 7839  ax-pre-ltwlin 7840  ax-pre-lttrn 7841  ax-pre-apti 7842  ax-pre-ltadd 7843  ax-pre-mulgt0 7844  ax-pre-mulext 7845  ax-arch 7846  ax-caucvg 7847
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-riota 5777  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-frec 6335  df-pnf 7909  df-mnf 7910  df-xr 7911  df-ltxr 7912  df-le 7913  df-sub 8043  df-neg 8044  df-reap 8445  df-ap 8452  df-div 8541  df-inn 8829  df-2 8887  df-3 8888  df-4 8889  df-n0 9086  df-z 9163  df-uz 9435  df-rp 9556  df-seqfrec 10340  df-exp 10414  df-cj 10737  df-re 10738  df-im 10739  df-rsqrt 10893  df-abs 10894  df-clim 11171
This theorem is referenced by:  trireciplem  11392  expcnvap0  11394
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