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Theorem divcnv 11460
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 11145 . . . . . 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 9685 . . . . 5  |-  ( ( A  e.  CC  /\  x  e.  RR+ )  -> 
( ( abs `  A
)  /  x )  e.  RR )
5 arch 9132 . . . . 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 489 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  ->  A  e.  CC )
8 eluzelz 9496 . . . . . . . . . . . 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 9335 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  CC )
119zred 9334 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  RR )
12 0red 7921 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  e.  RR )
13 simpllr 529 . . . . . . . . . . . . 13  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  NN )
1413nnred 8891 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
j  e.  RR )
1513nngt0d 8922 . . . . . . . . . . . 12  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <  j )
16 eluzle 9499 . . . . . . . . . . . . 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 8342 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <  k )
1911, 18gt0ap0d 8548 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k #  0 )
207, 10, 19absdivapd 11159 . . . . . . . . 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 8038 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
0  <_  k )
2211, 21absidd 11131 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  k
)  =  k )
2322oveq2d 5869 . . . . . . . . 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 2203 . . . . . . . 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 489 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  A
)  e.  RR )
263ad3antrrr 489 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  ->  x  e.  RR+ )
2711, 18elrpd 9650 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  RR+ )
284ad3antrrr 489 . . . . . . . . . 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 525 . . . . . . . . . 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 8342 . . . . . . . . 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 9714 . . . . . . . 8  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( ( abs `  A
)  /  k )  <  x )
3224, 31eqbrtrd 4011 . . . . . . 7  |-  ( ( ( ( ( A  e.  CC  /\  x  e.  RR+ )  /\  j  e.  NN )  /\  (
( abs `  A
)  /  x )  <  j )  /\  k  e.  ( ZZ>= `  j ) )  -> 
( abs `  ( A  /  k ) )  <  x )
3332ralrimiva 2543 . . . . . 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 2572 . . . 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 2543 . 2  |-  ( A  e.  CC  ->  A. x  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( abs `  ( A  /  k ) )  <  x )
38 nnuz 9522 . . 3  |-  NN  =  ( ZZ>= `  1 )
39 1zzd 9239 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
40 nnex 8884 . . . . 5  |-  NN  e.  _V
4140mptex 5722 . . . 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 8892 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  k  e.  CC )
4643nnap0d 8924 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  k #  0 )
4744, 45, 46divclapd 8707 . . . 4  |-  ( ( A  e.  CC  /\  k  e.  NN )  ->  ( A  /  k
)  e.  CC )
48 oveq2 5861 . . . . 5  |-  ( n  =  k  ->  ( A  /  n )  =  ( A  /  k
) )
49 eqid 2170 . . . . 5  |-  ( n  e.  NN  |->  ( A  /  n ) )  =  ( n  e.  NN  |->  ( A  /  n ) )
5048, 49fvmptg 5572 . . . 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 11249 . 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 1348    e. wcel 2141   A.wral 2448   E.wrex 2449   _Vcvv 2730   class class class wbr 3989    |-> cmpt 4050   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775    < clt 7954    <_ cle 7955    / cdiv 8589   NNcn 8878   ZZcz 9212   ZZ>=cuz 9487   RR+crp 9610   abscabs 10961    ~~> cli 11241
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242
This theorem is referenced by:  trireciplem  11463  expcnvap0  11465
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