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Theorem geolim2 12327
Description: The partial sums in the geometric series  A ^ M  +  A ^ ( M  +  1 )... converge to  ( ( A ^ M )  / 
( 1  -  A
) ). (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.)
Hypotheses
Ref Expression
geolim.1  |-  ( ph  ->  A  e.  CC )
geolim.2  |-  ( ph  ->  ( abs `  A
)  <  1 )
geolim2.3  |-  ( ph  ->  M  e.  NN0 )
geolim2.4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( A ^ k ) )
Assertion
Ref Expression
geolim2  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  ( ( A ^ M )  /  ( 1  -  A ) ) )
Distinct variable groups:    A, k    k, F    k, M    ph, k

Proof of Theorem geolim2
Dummy variables  j  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 geolim2.3 . . . 4  |-  ( ph  ->  M  e.  NN0 )
32nn0zd 10115 . . 3  |-  ( ph  ->  M  e.  ZZ )
4 geolim2.4 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  =  ( A ^ k ) )
5 geolim.1 . . . . 5  |-  ( ph  ->  A  e.  CC )
65adantr 451 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  A  e.  CC )
7 eluznn0 10288 . . . . 5  |-  ( ( M  e.  NN0  /\  k  e.  ( ZZ>= `  M ) )  -> 
k  e.  NN0 )
82, 7sylan 457 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  NN0 )
96, 8expcld 11245 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( A ^ k )  e.  CC )
10 oveq2 5866 . . . . . . . 8  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
11 eqid 2283 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
12 ovex 5883 . . . . . . . 8  |-  ( A ^ k )  e. 
_V
1310, 11, 12fvmpt 5602 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
148, 13syl 15 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
1514, 4eqtr4d 2318 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( F `  k
) )
163, 15seqfeq 11071 . . . 4  |-  ( ph  ->  seq  M (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  =  seq  M (  +  ,  F ) )
17 geolim.2 . . . . . . 7  |-  ( ph  ->  ( abs `  A
)  <  1 )
18 oveq2 5866 . . . . . . . . 9  |-  ( n  =  j  ->  ( A ^ n )  =  ( A ^ j
) )
19 ovex 5883 . . . . . . . . 9  |-  ( A ^ j )  e. 
_V
2018, 11, 19fvmpt 5602 . . . . . . . 8  |-  ( j  e.  NN0  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
2120adantl 452 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
225, 17, 21geolim 12326 . . . . . 6  |-  ( ph  ->  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  ~~>  ( 1  /  (
1  -  A ) ) )
23 seqex 11048 . . . . . . 7  |-  seq  0
(  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  _V
24 ovex 5883 . . . . . . 7  |-  ( 1  /  ( 1  -  A ) )  e. 
_V
2523, 24breldm 4883 . . . . . 6  |-  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( A ^ n ) ) )  ~~>  ( 1  /  ( 1  -  A ) )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( A ^ n ) ) )  e.  dom  ~~>  )
2622, 25syl 15 . . . . 5  |-  ( ph  ->  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  dom  ~~>  )
27 nn0uz 10262 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
28 expcl 11121 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ j
)  e.  CC )
295, 28sylan 457 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
3021, 29eqeltrd 2357 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  e.  CC )
3127, 2, 30iserex 12130 . . . . 5  |-  ( ph  ->  (  seq  0 (  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  dom  ~~>  <->  seq  M (  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  dom  ~~>  ) )
3226, 31mpbid 201 . . . 4  |-  ( ph  ->  seq  M (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  dom  ~~>  )
3316, 32eqeltrrd 2358 . . 3  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
341, 3, 4, 9, 33isumclim2 12221 . 2  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  sum_ k  e.  ( ZZ>= `  M )
( A ^ k
) )
3513adantl 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
36 expcl 11121 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
375, 36sylan 457 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  CC )
3827, 1, 2, 35, 37, 26isumsplit 12299 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  NN0  ( A ^ k )  =  ( sum_ k  e.  ( 0 ... ( M  -  1 ) ) ( A ^
k )  +  sum_ k  e.  ( ZZ>= `  M ) ( A ^ k ) ) )
39 0z 10035 . . . . . . . 8  |-  0  e.  ZZ
4039a1i 10 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
4127, 40, 35, 37, 22isumclim 12220 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  NN0  ( A ^ k )  =  ( 1  / 
( 1  -  A
) ) )
4238, 41eqtr3d 2317 . . . . 5  |-  ( ph  ->  ( sum_ k  e.  ( 0 ... ( M  -  1 ) ) ( A ^ k
)  +  sum_ k  e.  ( ZZ>= `  M )
( A ^ k
) )  =  ( 1  /  ( 1  -  A ) ) )
43 1re 8837 . . . . . . . . . . 11  |-  1  e.  RR
4443ltnri 8929 . . . . . . . . . 10  |-  -.  1  <  1
45 fveq2 5525 . . . . . . . . . . . 12  |-  ( A  =  1  ->  ( abs `  A )  =  ( abs `  1
) )
46 abs1 11782 . . . . . . . . . . . 12  |-  ( abs `  1 )  =  1
4745, 46syl6eq 2331 . . . . . . . . . . 11  |-  ( A  =  1  ->  ( abs `  A )  =  1 )
4847breq1d 4033 . . . . . . . . . 10  |-  ( A  =  1  ->  (
( abs `  A
)  <  1  <->  1  <  1 ) )
4944, 48mtbiri 294 . . . . . . . . 9  |-  ( A  =  1  ->  -.  ( abs `  A )  <  1 )
5049necon2ai 2491 . . . . . . . 8  |-  ( ( abs `  A )  <  1  ->  A  =/=  1 )
5117, 50syl 15 . . . . . . 7  |-  ( ph  ->  A  =/=  1 )
525, 51, 2geoser 12325 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( M  -  1 ) ) ( A ^ k
)  =  ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) ) )
5352oveq1d 5873 . . . . 5  |-  ( ph  ->  ( sum_ k  e.  ( 0 ... ( M  -  1 ) ) ( A ^ k
)  +  sum_ k  e.  ( ZZ>= `  M )
( A ^ k
) )  =  ( ( ( 1  -  ( A ^ M
) )  /  (
1  -  A ) )  +  sum_ k  e.  ( ZZ>= `  M )
( A ^ k
) ) )
5442, 53eqtr3d 2317 . . . 4  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  =  ( ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) )  + 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k ) ) )
5554oveq1d 5873 . . 3  |-  ( ph  ->  ( ( 1  / 
( 1  -  A
) )  -  (
( 1  -  ( A ^ M ) )  /  ( 1  -  A ) ) )  =  ( ( ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) )  + 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k ) )  -  ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) ) ) )
56 ax-1cn 8795 . . . . . 6  |-  1  e.  CC
5756a1i 10 . . . . 5  |-  ( ph  ->  1  e.  CC )
585, 2expcld 11245 . . . . . 6  |-  ( ph  ->  ( A ^ M
)  e.  CC )
59 subcl 9051 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A ^ M )  e.  CC )  -> 
( 1  -  ( A ^ M ) )  e.  CC )
6056, 58, 59sylancr 644 . . . . 5  |-  ( ph  ->  ( 1  -  ( A ^ M ) )  e.  CC )
61 subcl 9051 . . . . . 6  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
6256, 5, 61sylancr 644 . . . . 5  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
6351necomd 2529 . . . . . 6  |-  ( ph  ->  1  =/=  A )
64 subeq0 9073 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
6556, 5, 64sylancr 644 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
6665necon3bid 2481 . . . . . 6  |-  ( ph  ->  ( ( 1  -  A )  =/=  0  <->  1  =/=  A ) )
6763, 66mpbird 223 . . . . 5  |-  ( ph  ->  ( 1  -  A
)  =/=  0 )
6857, 60, 62, 67divsubdird 9575 . . . 4  |-  ( ph  ->  ( ( 1  -  ( 1  -  ( A ^ M ) ) )  /  ( 1  -  A ) )  =  ( ( 1  /  ( 1  -  A ) )  -  ( ( 1  -  ( A ^ M
) )  /  (
1  -  A ) ) ) )
69 nncan 9076 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A ^ M )  e.  CC )  -> 
( 1  -  (
1  -  ( A ^ M ) ) )  =  ( A ^ M ) )
7056, 58, 69sylancr 644 . . . . 5  |-  ( ph  ->  ( 1  -  (
1  -  ( A ^ M ) ) )  =  ( A ^ M ) )
7170oveq1d 5873 . . . 4  |-  ( ph  ->  ( ( 1  -  ( 1  -  ( A ^ M ) ) )  /  ( 1  -  A ) )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
7268, 71eqtr3d 2317 . . 3  |-  ( ph  ->  ( ( 1  / 
( 1  -  A
) )  -  (
( 1  -  ( A ^ M ) )  /  ( 1  -  A ) ) )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
7360, 62, 67divcld 9536 . . . 4  |-  ( ph  ->  ( ( 1  -  ( A ^ M
) )  /  (
1  -  A ) )  e.  CC )
741, 3, 14, 9, 32isumcl 12224 . . . 4  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k )  e.  CC )
7573, 74pncan2d 9159 . . 3  |-  ( ph  ->  ( ( ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) )  + 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k ) )  -  ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) ) )  =  sum_ k  e.  (
ZZ>= `  M ) ( A ^ k ) )
7655, 72, 753eqtr3rd 2324 . 2  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
7734, 76breqtrd 4047 1  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  ( ( A ^ M )  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    - cmin 9037    / cdiv 9423   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046   ^cexp 11104   abscabs 11719    ~~> cli 11958   sum_csu 12158
This theorem is referenced by:  geoisum1  12335  geoisum1c  12336  rpnnen2lem3  12495  rpnnen2lem9  12501  abelthlem7  19814  log2tlbnd  20241  cntrset  25602  geomcau  26475  stirlinglem10  27832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159
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