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Theorem expcnv 11285
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 8992 . . . 4  |-  NN  C_  NN0
2 resmpt 4867 . . . 4  |-  ( NN  C_  NN0  ->  ( (
n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  |`  NN )  =  ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) ) )
31, 2ax-mp 5 . . 3  |-  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  |`  NN )  =  ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) )
4 expcnv.1 . . . . . 6  |-  ( ph  ->  A  e.  CC )
54abscld 10965 . . . . 5  |-  ( ph  ->  ( abs `  A
)  e.  RR )
6 expcnv.2 . . . . 5  |-  ( ph  ->  ( abs `  A
)  <  1 )
74absge0d 10968 . . . . 5  |-  ( ph  ->  0  <_  ( abs `  A ) )
85, 6, 7expcnvre 11284 . . . 4  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  ~~>  0 )
9 nnuz 9373 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
109reseq2i 4816 . . . . . 6  |-  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  |`  NN )  =  ( ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  |`  ( ZZ>= ` 
1 ) )
1110breq1i 3936 . . . . 5  |-  ( ( ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  |`  NN )  ~~>  0 
<->  ( ( n  e. 
NN0  |->  ( ( abs `  A ) ^ n
) )  |`  ( ZZ>=
`  1 ) )  ~~>  0 )
12 1z 9092 . . . . . 6  |-  1  e.  ZZ
13 nn0ex 8995 . . . . . . 7  |-  NN0  e.  _V
1413mptex 5646 . . . . . 6  |-  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  e.  _V
15 climres 11084 . . . . . 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 422 . . . . 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, 18eqbrtrrid 3964 . 2  |-  ( ph  ->  ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) )  ~~>  0 )
20 1zzd 9093 . . 3  |-  ( ph  ->  1  e.  ZZ )
2113mptex 5646 . . . 4  |-  ( n  e.  NN0  |->  ( A ^ n ) )  e.  _V
2221a1i 9 . . 3  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  e.  _V )
23 nnex 8738 . . . . 5  |-  NN  e.  _V
2423mptex 5646 . . . 4  |-  ( n  e.  NN  |->  ( ( abs `  A ) ^ n ) )  e.  _V
2524a1i 9 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) )  e.  _V )
26 nnnn0 8996 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
2726adantl 275 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  k  e. 
NN0 )
284adantr 274 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  A  e.  CC )
2928, 27expcld 10436 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( A ^ k )  e.  CC )
30 oveq2 5782 . . . . . 6  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
31 eqid 2139 . . . . . 6  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
3230, 31fvmptg 5497 . . . . 5  |-  ( ( k  e.  NN0  /\  ( A ^ k )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k )  =  ( A ^ k ) )
3327, 29, 32syl2anc 408 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
3433, 29eqeltrd 2216 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 k )  e.  CC )
35 absexp 10863 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )
364, 26, 35syl2an 287 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  ( A ^ k
) )  =  ( ( abs `  A
) ^ k ) )
3733fveq2d 5425 . . . 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 274 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  A )  e.  RR )
4039recnd 7806 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  A )  e.  CC )
4140, 27expcld 10436 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( abs `  A ) ^ k )  e.  CC )
42 oveq2 5782 . . . . . 6  |-  ( n  =  k  ->  (
( abs `  A
) ^ n )  =  ( ( abs `  A ) ^ k
) )
43 eqid 2139 . . . . . 6  |-  ( n  e.  NN  |->  ( ( abs `  A ) ^ n ) )  =  ( n  e.  NN  |->  ( ( abs `  A ) ^ n
) )
4442, 43fvmptg 5497 . . . . 5  |-  ( ( k  e.  NN  /\  ( ( abs `  A
) ^ k )  e.  CC )  -> 
( ( n  e.  NN  |->  ( ( abs `  A ) ^ n
) ) `  k
)  =  ( ( abs `  A ) ^ k ) )
4538, 41, 44syl2anc 408 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) ) `  k )  =  ( ( abs `  A ) ^ k
) )
4636, 37, 453eqtr4rd 2183 . . 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 11088 . 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 1331    e. wcel 1480   _Vcvv 2686    C_ wss 3071   class class class wbr 3929    |-> cmpt 3989    |` cres 4541   ` cfv 5123  (class class class)co 5774   CCcc 7630   RRcr 7631   0cc0 7632   1c1 7633    < clt 7812   NNcn 8732   NN0cn0 8989   ZZcz 9066   ZZ>=cuz 9338   ^cexp 10304   abscabs 10781    ~~> cli 11059
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-clim 11060
This theorem is referenced by:  explecnv  11286  geolim  11292  geo2lim  11297
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