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Theorem expcnv 11062
Description: A sequence of powers of a complex number  A with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Revised by Jim Kingdon, 28-Oct-2022.)
Hypotheses
Ref Expression
expcnv.1  |-  ( ph  ->  A  e.  CC )
expcnv.2  |-  ( ph  ->  ( abs `  A
)  <  1 )
Assertion
Ref Expression
expcnv  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
Distinct variable group:    A, n
Allowed substitution hint:    ph( n)

Proof of Theorem expcnv
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 nnssnn0 8774 . . . 4  |-  NN  C_  NN0
2 resmpt 4793 . . . 4  |-  ( NN  C_  NN0  ->  ( (
n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  |`  NN )  =  ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) ) )
31, 2ax-mp 7 . . 3  |-  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  |`  NN )  =  ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) )
4 expcnv.1 . . . . . 6  |-  ( ph  ->  A  e.  CC )
54abscld 10745 . . . . 5  |-  ( ph  ->  ( abs `  A
)  e.  RR )
6 expcnv.2 . . . . 5  |-  ( ph  ->  ( abs `  A
)  <  1 )
74absge0d 10748 . . . . 5  |-  ( ph  ->  0  <_  ( abs `  A ) )
85, 6, 7expcnvre 11061 . . . 4  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  ~~>  0 )
9 nnuz 9153 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
109reseq2i 4742 . . . . . 6  |-  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  |`  NN )  =  ( ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  |`  ( ZZ>= ` 
1 ) )
1110breq1i 3874 . . . . 5  |-  ( ( ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  |`  NN )  ~~>  0 
<->  ( ( n  e. 
NN0  |->  ( ( abs `  A ) ^ n
) )  |`  ( ZZ>=
`  1 ) )  ~~>  0 )
12 1z 8874 . . . . . 6  |-  1  e.  ZZ
13 nn0ex 8777 . . . . . . 7  |-  NN0  e.  _V
1413mptex 5562 . . . . . 6  |-  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  e.  _V
15 climres 10862 . . . . . 6  |-  ( ( 1  e.  ZZ  /\  ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  e.  _V )  ->  ( ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  |`  ( ZZ>= `  1 )
)  ~~>  0  <->  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n
) )  ~~>  0 ) )
1612, 14, 15mp2an 418 . . . . 5  |-  ( ( ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  |`  ( ZZ>= ` 
1 ) )  ~~>  0  <->  (
n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  ~~>  0 )
1711, 16bitri 183 . . . 4  |-  ( ( ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  |`  NN )  ~~>  0 
<->  ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  ~~>  0 )
188, 17sylibr 133 . . 3  |-  ( ph  ->  ( ( n  e. 
NN0  |->  ( ( abs `  A ) ^ n
) )  |`  NN )  ~~>  0 )
193, 18syl5eqbrr 3901 . 2  |-  ( ph  ->  ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) )  ~~>  0 )
20 1zzd 8875 . . 3  |-  ( ph  ->  1  e.  ZZ )
2113mptex 5562 . . . 4  |-  ( n  e.  NN0  |->  ( A ^ n ) )  e.  _V
2221a1i 9 . . 3  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  e.  _V )
23 nnex 8526 . . . . 5  |-  NN  e.  _V
2423mptex 5562 . . . 4  |-  ( n  e.  NN  |->  ( ( abs `  A ) ^ n ) )  e.  _V
2524a1i 9 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) )  e.  _V )
26 nnnn0 8778 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
2726adantl 272 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  k  e. 
NN0 )
284adantr 271 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  A  e.  CC )
2928, 27expcld 10217 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( A ^ k )  e.  CC )
30 oveq2 5698 . . . . . 6  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
31 eqid 2095 . . . . . 6  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
3230, 31fvmptg 5415 . . . . 5  |-  ( ( k  e.  NN0  /\  ( A ^ k )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k )  =  ( A ^ k ) )
3327, 29, 32syl2anc 404 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
3433, 29eqeltrd 2171 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 k )  e.  CC )
35 absexp 10643 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )
364, 26, 35syl2an 284 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  ( A ^ k
) )  =  ( ( abs `  A
) ^ k ) )
3733fveq2d 5344 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  ( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k ) )  =  ( abs `  ( A ^ k ) ) )
38 simpr 109 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
395adantr 271 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  A )  e.  RR )
4039recnd 7613 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  A )  e.  CC )
4140, 27expcld 10217 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( abs `  A ) ^ k )  e.  CC )
42 oveq2 5698 . . . . . 6  |-  ( n  =  k  ->  (
( abs `  A
) ^ n )  =  ( ( abs `  A ) ^ k
) )
43 eqid 2095 . . . . . 6  |-  ( n  e.  NN  |->  ( ( abs `  A ) ^ n ) )  =  ( n  e.  NN  |->  ( ( abs `  A ) ^ n
) )
4442, 43fvmptg 5415 . . . . 5  |-  ( ( k  e.  NN  /\  ( ( abs `  A
) ^ k )  e.  CC )  -> 
( ( n  e.  NN  |->  ( ( abs `  A ) ^ n
) ) `  k
)  =  ( ( abs `  A ) ^ k ) )
4538, 41, 44syl2anc 404 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) ) `  k )  =  ( ( abs `  A ) ^ k
) )
4636, 37, 453eqtr4rd 2138 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) ) `  k )  =  ( abs `  (
( n  e.  NN0  |->  ( A ^ n ) ) `  k ) ) )
479, 20, 22, 25, 34, 46climabs0 10866 . 2  |-  ( ph  ->  ( ( n  e. 
NN0  |->  ( A ^
n ) )  ~~>  0  <->  (
n  e.  NN  |->  ( ( abs `  A
) ^ n ) )  ~~>  0 ) )
4819, 47mpbird 166 1  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1296    e. wcel 1445   _Vcvv 2633    C_ wss 3013   class class class wbr 3867    |-> cmpt 3921    |` cres 4469   ` cfv 5049  (class class class)co 5690   CCcc 7445   RRcr 7446   0cc0 7447   1c1 7448    < clt 7619   NNcn 8520   NN0cn0 8771   ZZcz 8848   ZZ>=cuz 9118   ^cexp 10085   abscabs 10561    ~~> cli 10837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560  ax-arch 7561  ax-caucvg 7562
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-n0 8772  df-z 8849  df-uz 9119  df-q 9204  df-rp 9234  df-iseq 10002  df-seq3 10003  df-exp 10086  df-cj 10407  df-re 10408  df-im 10409  df-rsqrt 10562  df-abs 10563  df-clim 10838
This theorem is referenced by:  explecnv  11063  geolim  11069  geo2lim  11074
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