ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  expcnv Unicode version

Theorem expcnv 11547
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 9210 . . . 4  |-  NN  C_  NN0
2 resmpt 4973 . . . 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 11225 . . . . 5  |-  ( ph  ->  ( abs `  A
)  e.  RR )
6 expcnv.2 . . . . 5  |-  ( ph  ->  ( abs `  A
)  <  1 )
74absge0d 11228 . . . . 5  |-  ( ph  ->  0  <_  ( abs `  A ) )
85, 6, 7expcnvre 11546 . . . 4  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  ~~>  0 )
9 nnuz 9595 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
109reseq2i 4922 . . . . . 6  |-  ( ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  |`  NN )  =  ( ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  |`  ( ZZ>= ` 
1 ) )
1110breq1i 4025 . . . . 5  |-  ( ( ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  |`  NN )  ~~>  0 
<->  ( ( n  e. 
NN0  |->  ( ( abs `  A ) ^ n
) )  |`  ( ZZ>=
`  1 ) )  ~~>  0 )
12 1z 9310 . . . . . 6  |-  1  e.  ZZ
13 nn0ex 9213 . . . . . . 7  |-  NN0  e.  _V
1413mptex 5763 . . . . . 6  |-  ( n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  e.  _V
15 climres 11346 . . . . . 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 426 . . . . 5  |-  ( ( ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  |`  ( ZZ>= ` 
1 ) )  ~~>  0  <->  (
n  e.  NN0  |->  ( ( abs `  A ) ^ n ) )  ~~>  0 )
1711, 16bitri 184 . . . 4  |-  ( ( ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  |`  NN )  ~~>  0 
<->  ( n  e.  NN0  |->  ( ( abs `  A
) ^ n ) )  ~~>  0 )
188, 17sylibr 134 . . 3  |-  ( ph  ->  ( ( n  e. 
NN0  |->  ( ( abs `  A ) ^ n
) )  |`  NN )  ~~>  0 )
193, 18eqbrtrrid 4054 . 2  |-  ( ph  ->  ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) )  ~~>  0 )
20 1zzd 9311 . . 3  |-  ( ph  ->  1  e.  ZZ )
2113mptex 5763 . . . 4  |-  ( n  e.  NN0  |->  ( A ^ n ) )  e.  _V
2221a1i 9 . . 3  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  e.  _V )
23 nnex 8956 . . . . 5  |-  NN  e.  _V
2423mptex 5763 . . . 4  |-  ( n  e.  NN  |->  ( ( abs `  A ) ^ n ) )  e.  _V
2524a1i 9 . . 3  |-  ( ph  ->  ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) )  e.  _V )
26 nnnn0 9214 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
2726adantl 277 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  k  e. 
NN0 )
284adantr 276 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  A  e.  CC )
2928, 27expcld 10688 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( A ^ k )  e.  CC )
30 oveq2 5905 . . . . . 6  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
31 eqid 2189 . . . . . 6  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
3230, 31fvmptg 5613 . . . . 5  |-  ( ( k  e.  NN0  /\  ( A ^ k )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k )  =  ( A ^ k ) )
3327, 29, 32syl2anc 411 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
3433, 29eqeltrd 2266 . . 3  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 k )  e.  CC )
35 absexp 11123 . . . . 5  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )
364, 26, 35syl2an 289 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  ( A ^ k
) )  =  ( ( abs `  A
) ^ k ) )
3733fveq2d 5538 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  ( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k ) )  =  ( abs `  ( A ^ k ) ) )
38 simpr 110 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
395adantr 276 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  A )  e.  RR )
4039recnd 8017 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  A )  e.  CC )
4140, 27expcld 10688 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( abs `  A ) ^ k )  e.  CC )
42 oveq2 5905 . . . . . 6  |-  ( n  =  k  ->  (
( abs `  A
) ^ n )  =  ( ( abs `  A ) ^ k
) )
43 eqid 2189 . . . . . 6  |-  ( n  e.  NN  |->  ( ( abs `  A ) ^ n ) )  =  ( n  e.  NN  |->  ( ( abs `  A ) ^ n
) )
4442, 43fvmptg 5613 . . . . 5  |-  ( ( k  e.  NN  /\  ( ( abs `  A
) ^ k )  e.  CC )  -> 
( ( n  e.  NN  |->  ( ( abs `  A ) ^ n
) ) `  k
)  =  ( ( abs `  A ) ^ k ) )
4538, 41, 44syl2anc 411 . . . 4  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) ) `  k )  =  ( ( abs `  A ) ^ k
) )
4636, 37, 453eqtr4rd 2233 . . 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 11350 . 2  |-  ( ph  ->  ( ( n  e. 
NN0  |->  ( A ^
n ) )  ~~>  0  <->  (
n  e.  NN  |->  ( ( abs `  A
) ^ n ) )  ~~>  0 ) )
4819, 47mpbird 167 1  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   _Vcvv 2752    C_ wss 3144   class class class wbr 4018    |-> cmpt 4079    |` cres 4646   ` cfv 5235  (class class class)co 5897   CCcc 7840   RRcr 7841   0cc0 7842   1c1 7843    < clt 8023   NNcn 8950   NN0cn0 9207   ZZcz 9284   ZZ>=cuz 9559   ^cexp 10553   abscabs 11041    ~~> cli 11321
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-frec 6417  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-seqfrec 10479  df-exp 10554  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-clim 11322
This theorem is referenced by:  explecnv  11548  geolim  11554  geo2lim  11559
  Copyright terms: Public domain W3C validator