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Theorem geolim2 12686
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 2443 . . 3  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 geolim2.3 . . . 4  |-  ( ph  ->  M  e.  NN0 )
32nn0zd 10411 . . 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 453 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  A  e.  CC )
7 eluznn0 10584 . . . . 5  |-  ( ( M  e.  NN0  /\  k  e.  ( ZZ>= `  M ) )  -> 
k  e.  NN0 )
82, 7sylan 459 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  NN0 )
96, 8expcld 11561 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( A ^ k )  e.  CC )
10 oveq2 6125 . . . . . . . 8  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
11 eqid 2443 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
12 ovex 6142 . . . . . . . 8  |-  ( A ^ k )  e. 
_V
1310, 11, 12fvmpt 5842 . . . . . . 7  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
148, 13syl 16 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
1514, 4eqtr4d 2478 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( F `  k
) )
163, 15seqfeq 11386 . . . 4  |-  ( ph  ->  seq  M (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  =  seq  M (  +  ,  F ) )
17 geolim.2 . . . . . . 7  |-  ( ph  ->  ( abs `  A
)  <  1 )
18 oveq2 6125 . . . . . . . . 9  |-  ( n  =  j  ->  ( A ^ n )  =  ( A ^ j
) )
19 ovex 6142 . . . . . . . . 9  |-  ( A ^ j )  e. 
_V
2018, 11, 19fvmpt 5842 . . . . . . . 8  |-  ( j  e.  NN0  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
2120adantl 454 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
225, 17, 21geolim 12685 . . . . . 6  |-  ( ph  ->  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  ~~>  ( 1  /  (
1  -  A ) ) )
23 seqex 11363 . . . . . . 7  |-  seq  0
(  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  _V
24 ovex 6142 . . . . . . 7  |-  ( 1  /  ( 1  -  A ) )  e. 
_V
2523, 24breldm 5109 . . . . . 6  |-  (  seq  0 (  +  , 
( n  e.  NN0  |->  ( A ^ n ) ) )  ~~>  ( 1  /  ( 1  -  A ) )  ->  seq  0 (  +  , 
( n  e.  NN0  |->  ( A ^ n ) ) )  e.  dom  ~~>  )
2622, 25syl 16 . . . . 5  |-  ( ph  ->  seq  0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  dom  ~~>  )
27 nn0uz 10558 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
28 expcl 11437 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ j
)  e.  CC )
295, 28sylan 459 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
3021, 29eqeltrd 2517 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  e.  CC )
3127, 2, 30iserex 12488 . . . . 5  |-  ( ph  ->  (  seq  0 (  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  dom  ~~>  <->  seq  M (  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  dom  ~~>  ) )
3226, 31mpbid 203 . . . 4  |-  ( ph  ->  seq  M (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  dom  ~~>  )
3316, 32eqeltrrd 2518 . . 3  |-  ( ph  ->  seq  M (  +  ,  F )  e. 
dom 
~~>  )
341, 3, 4, 9, 33isumclim2 12580 . 2  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  sum_ k  e.  ( ZZ>= `  M )
( A ^ k
) )
3513adantl 454 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
36 expcl 11437 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
375, 36sylan 459 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  CC )
3827, 1, 2, 35, 37, 26isumsplit 12658 . . . . . 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 10331 . . . . . . . 8  |-  0  e.  ZZ
4039a1i 11 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
4127, 40, 35, 37, 22isumclim 12579 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  NN0  ( A ^ k )  =  ( 1  / 
( 1  -  A
) ) )
4238, 41eqtr3d 2477 . . . . 5  |-  ( ph  ->  ( sum_ k  e.  ( 0 ... ( M  -  1 ) ) ( A ^ k
)  +  sum_ k  e.  ( ZZ>= `  M )
( A ^ k
) )  =  ( 1  /  ( 1  -  A ) ) )
43 1re 9128 . . . . . . . . . . 11  |-  1  e.  RR
4443ltnri 9220 . . . . . . . . . 10  |-  -.  1  <  1
45 fveq2 5763 . . . . . . . . . . . 12  |-  ( A  =  1  ->  ( abs `  A )  =  ( abs `  1
) )
46 abs1 12140 . . . . . . . . . . . 12  |-  ( abs `  1 )  =  1
4745, 46syl6eq 2491 . . . . . . . . . . 11  |-  ( A  =  1  ->  ( abs `  A )  =  1 )
4847breq1d 4253 . . . . . . . . . 10  |-  ( A  =  1  ->  (
( abs `  A
)  <  1  <->  1  <  1 ) )
4944, 48mtbiri 296 . . . . . . . . 9  |-  ( A  =  1  ->  -.  ( abs `  A )  <  1 )
5049necon2ai 2656 . . . . . . . 8  |-  ( ( abs `  A )  <  1  ->  A  =/=  1 )
5117, 50syl 16 . . . . . . 7  |-  ( ph  ->  A  =/=  1 )
525, 51, 2geoser 12684 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( M  -  1 ) ) ( A ^ k
)  =  ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) ) )
5352oveq1d 6132 . . . . 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 2477 . . . 4  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  =  ( ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) )  + 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k ) ) )
5554oveq1d 6132 . . 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 9086 . . . . . 6  |-  1  e.  CC
5756a1i 11 . . . . 5  |-  ( ph  ->  1  e.  CC )
585, 2expcld 11561 . . . . . 6  |-  ( ph  ->  ( A ^ M
)  e.  CC )
59 subcl 9343 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A ^ M )  e.  CC )  -> 
( 1  -  ( A ^ M ) )  e.  CC )
6056, 58, 59sylancr 646 . . . . 5  |-  ( ph  ->  ( 1  -  ( A ^ M ) )  e.  CC )
61 subcl 9343 . . . . . 6  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
6256, 5, 61sylancr 646 . . . . 5  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
6351necomd 2694 . . . . . 6  |-  ( ph  ->  1  =/=  A )
64 subeq0 9365 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
6556, 5, 64sylancr 646 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
6665necon3bid 2643 . . . . . 6  |-  ( ph  ->  ( ( 1  -  A )  =/=  0  <->  1  =/=  A ) )
6763, 66mpbird 225 . . . . 5  |-  ( ph  ->  ( 1  -  A
)  =/=  0 )
6857, 60, 62, 67divsubdird 9867 . . . 4  |-  ( ph  ->  ( ( 1  -  ( 1  -  ( A ^ M ) ) )  /  ( 1  -  A ) )  =  ( ( 1  /  ( 1  -  A ) )  -  ( ( 1  -  ( A ^ M
) )  /  (
1  -  A ) ) ) )
69 nncan 9368 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A ^ M )  e.  CC )  -> 
( 1  -  (
1  -  ( A ^ M ) ) )  =  ( A ^ M ) )
7056, 58, 69sylancr 646 . . . . 5  |-  ( ph  ->  ( 1  -  (
1  -  ( A ^ M ) ) )  =  ( A ^ M ) )
7170oveq1d 6132 . . . 4  |-  ( ph  ->  ( ( 1  -  ( 1  -  ( A ^ M ) ) )  /  ( 1  -  A ) )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
7268, 71eqtr3d 2477 . . 3  |-  ( ph  ->  ( ( 1  / 
( 1  -  A
) )  -  (
( 1  -  ( A ^ M ) )  /  ( 1  -  A ) ) )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
7360, 62, 67divcld 9828 . . . 4  |-  ( ph  ->  ( ( 1  -  ( A ^ M
) )  /  (
1  -  A ) )  e.  CC )
741, 3, 14, 9, 32isumcl 12583 . . . 4  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k )  e.  CC )
7573, 74pncan2d 9451 . . 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 2484 . 2  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
7734, 76breqtrd 4267 1  |-  ( ph  ->  seq  M (  +  ,  F )  ~~>  ( ( A ^ M )  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1654    e. wcel 1728    =/= wne 2606   class class class wbr 4243    e. cmpt 4297   dom cdm 4913   ` cfv 5489  (class class class)co 6117   CCcc 9026   0cc0 9028   1c1 9029    + caddc 9031    < clt 9158    - cmin 9329    / cdiv 9715   NN0cn0 10259   ZZcz 10320   ZZ>=cuz 10526   ...cfz 11081    seq cseq 11361   ^cexp 11420   abscabs 12077    ~~> cli 12316   sum_csu 12517
This theorem is referenced by:  geoisum1  12694  geoisum1c  12695  rpnnen2lem3  12854  rpnnen2lem9  12860  abelthlem7  20392  log2tlbnd  20823  geomcau  26507  stirlinglem10  27920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-inf2 7632  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105  ax-pre-sup 9106
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-int 4080  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-se 4577  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-isom 5498  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-riota 6585  df-recs 6669  df-rdg 6704  df-1o 6760  df-oadd 6764  df-er 6941  df-pm 7057  df-en 7146  df-dom 7147  df-sdom 7148  df-fin 7149  df-sup 7482  df-oi 7515  df-card 7864  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-div 9716  df-nn 10039  df-2 10096  df-3 10097  df-n0 10260  df-z 10321  df-uz 10527  df-rp 10651  df-fz 11082  df-fzo 11174  df-fl 11240  df-seq 11362  df-exp 11421  df-hash 11657  df-cj 11942  df-re 11943  df-im 11944  df-sqr 12078  df-abs 12079  df-clim 12320  df-rlim 12321  df-sum 12518
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