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Theorem geolim2 10969
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  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2089 . . 3  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 geolim2.3 . . . 4  |-  ( ph  ->  M  e.  NN0 )
32nn0zd 8929 . . 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 271 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  A  e.  CC )
7 eluznn0 9149 . . . . 5  |-  ( ( M  e.  NN0  /\  k  e.  ( ZZ>= `  M ) )  -> 
k  e.  NN0 )
82, 7sylan 278 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  NN0 )
96, 8expcld 10149 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( A ^ k )  e.  CC )
10 eluzelz 9091 . . . . . . . . 9  |-  ( x  e.  ( ZZ>= `  M
)  ->  x  e.  ZZ )
1110adantl 272 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  ZZ )
12 0red 7552 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  0  e.  RR )
133adantr 271 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  M  e.  ZZ )
1413zred 8931 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  M  e.  RR )
1511zred 8931 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  RR )
162nn0ge0d 8792 . . . . . . . . . 10  |-  ( ph  ->  0  <_  M )
1716adantr 271 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  0  <_  M )
18 eluzle 9094 . . . . . . . . . 10  |-  ( x  e.  ( ZZ>= `  M
)  ->  M  <_  x )
1918adantl 272 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  M  <_  x )
2012, 14, 15, 17, 19letrd 7670 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  0  <_  x )
21 elnn0z 8826 . . . . . . . 8  |-  ( x  e.  NN0  <->  ( x  e.  ZZ  /\  0  <_  x ) )
2211, 20, 21sylanbrc 409 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  NN0 )
235adantr 271 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  A  e.  CC )
2423, 22expcld 10149 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( A ^ x )  e.  CC )
25 oveq2 5676 . . . . . . . 8  |-  ( n  =  x  ->  ( A ^ n )  =  ( A ^ x
) )
26 eqid 2089 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
2725, 26fvmptg 5395 . . . . . . 7  |-  ( ( x  e.  NN0  /\  ( A ^ x )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  x )  =  ( A ^ x ) )
2822, 24, 27syl2anc 404 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 x )  =  ( A ^ x
) )
2928, 24eqeltrd 2165 . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 x )  e.  CC )
30 oveq2 5676 . . . . . . . 8  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
3130, 26fvmptg 5395 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( A ^ k )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k )  =  ( A ^ k ) )
328, 9, 31syl2anc 404 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
3332, 4eqtr4d 2124 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( F `  k
) )
34 addcl 7530 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
3534adantl 272 . . . . 5  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
363, 29, 33, 35seq3feq 9960 . . . 4  |-  ( ph  ->  seq M (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  =  seq M (  +  ,  F ) )
37 seqex 9920 . . . . . 6  |-  seq 0
(  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  _V
38 ax-1cn 7501 . . . . . . . 8  |-  1  e.  CC
39 subcl 7744 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
4038, 5, 39sylancr 406 . . . . . . 7  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
41 1cnd 7567 . . . . . . . 8  |-  ( ph  ->  1  e.  CC )
42 1red 7566 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR )
43 geolim.2 . . . . . . . . . 10  |-  ( ph  ->  ( abs `  A
)  <  1 )
445, 42, 43absltap 10966 . . . . . . . . 9  |-  ( ph  ->  A #  1 )
45 apsym 8146 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A #  1  <->  1 #  A ) )
465, 41, 45syl2anc 404 . . . . . . . . 9  |-  ( ph  ->  ( A #  1  <->  1 #  A ) )
4744, 46mpbid 146 . . . . . . . 8  |-  ( ph  ->  1 #  A )
4841, 5, 47subap0d 8182 . . . . . . 7  |-  ( ph  ->  ( 1  -  A
) #  0 )
4940, 48recclapd 8311 . . . . . 6  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  e.  CC )
50 simpr 109 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  NN0 )
515adantr 271 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  A  e.  CC )
5251, 50expcld 10149 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
53 oveq2 5676 . . . . . . . . 9  |-  ( n  =  j  ->  ( A ^ n )  =  ( A ^ j
) )
5453, 26fvmptg 5395 . . . . . . . 8  |-  ( ( j  e.  NN0  /\  ( A ^ j )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  j )  =  ( A ^ j ) )
5550, 52, 54syl2anc 404 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
565, 43, 55geolim 10968 . . . . . 6  |-  ( ph  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  ~~>  ( 1  /  (
1  -  A ) ) )
57 breldmg 4657 . . . . . 6  |-  ( (  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  _V  /\  (
1  /  ( 1  -  A ) )  e.  CC  /\  seq 0 (  +  , 
( n  e.  NN0  |->  ( A ^ n ) ) )  ~~>  ( 1  /  ( 1  -  A ) ) )  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  dom  ~~>  )
5837, 49, 56, 57mp3an2i 1279 . . . . 5  |-  ( ph  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  dom  ~~>  )
59 nn0uz 9116 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
60 expcl 10036 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ j
)  e.  CC )
615, 60sylan 278 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
6255, 61eqeltrd 2165 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  e.  CC )
6359, 2, 62iserex 10790 . . . . 5  |-  ( ph  ->  (  seq 0 (  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  dom  ~~>  <->  seq M (  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  dom  ~~>  ) )
6458, 63mpbid 146 . . . 4  |-  ( ph  ->  seq M (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  dom  ~~>  )
6536, 64eqeltrrd 2166 . . 3  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
661, 3, 4, 9, 65isumclim2 10879 . 2  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  sum_ k  e.  ( ZZ>= `  M )
( A ^ k
) )
67 simpr 109 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
685adantr 271 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  A  e.  CC )
6968, 67expcld 10149 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  CC )
7067, 69, 31syl2anc 404 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
71 expcl 10036 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
725, 71sylan 278 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  CC )
7359, 1, 2, 70, 72, 58isumsplit 10948 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  NN0  ( A ^ k )  =  ( sum_ k  e.  ( 0 ... ( M  -  1 ) ) ( A ^
k )  +  sum_ k  e.  ( ZZ>= `  M ) ( A ^ k ) ) )
74 0zd 8825 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
7559, 74, 70, 72, 56isumclim 10878 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  NN0  ( A ^ k )  =  ( 1  / 
( 1  -  A
) ) )
7673, 75eqtr3d 2123 . . . . 5  |-  ( ph  ->  ( sum_ k  e.  ( 0 ... ( M  -  1 ) ) ( A ^ k
)  +  sum_ k  e.  ( ZZ>= `  M )
( A ^ k
) )  =  ( 1  /  ( 1  -  A ) ) )
775, 44, 2geoserap 10964 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( M  -  1 ) ) ( A ^ k
)  =  ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) ) )
7877oveq1d 5683 . . . . 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
) ) )
7976, 78eqtr3d 2123 . . . 4  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  =  ( ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) )  + 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k ) ) )
8079oveq1d 5683 . . 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 ) ) ) )
815, 2expcld 10149 . . . . . 6  |-  ( ph  ->  ( A ^ M
)  e.  CC )
82 subcl 7744 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A ^ M )  e.  CC )  -> 
( 1  -  ( A ^ M ) )  e.  CC )
8338, 81, 82sylancr 406 . . . . 5  |-  ( ph  ->  ( 1  -  ( A ^ M ) )  e.  CC )
8441, 83, 40, 48divsubdirapd 8360 . . . 4  |-  ( ph  ->  ( ( 1  -  ( 1  -  ( A ^ M ) ) )  /  ( 1  -  A ) )  =  ( ( 1  /  ( 1  -  A ) )  -  ( ( 1  -  ( A ^ M
) )  /  (
1  -  A ) ) ) )
85 nncan 7774 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A ^ M )  e.  CC )  -> 
( 1  -  (
1  -  ( A ^ M ) ) )  =  ( A ^ M ) )
8638, 81, 85sylancr 406 . . . . 5  |-  ( ph  ->  ( 1  -  (
1  -  ( A ^ M ) ) )  =  ( A ^ M ) )
8786oveq1d 5683 . . . 4  |-  ( ph  ->  ( ( 1  -  ( 1  -  ( A ^ M ) ) )  /  ( 1  -  A ) )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
8884, 87eqtr3d 2123 . . 3  |-  ( ph  ->  ( ( 1  / 
( 1  -  A
) )  -  (
( 1  -  ( A ^ M ) )  /  ( 1  -  A ) ) )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
8983, 40, 48divclapd 8320 . . . 4  |-  ( ph  ->  ( ( 1  -  ( A ^ M
) )  /  (
1  -  A ) )  e.  CC )
901, 3, 32, 9, 64isumcl 10882 . . . 4  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k )  e.  CC )
9189, 90pncan2d 7858 . . 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 ) )
9280, 88, 913eqtr3rd 2130 . 2  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
9366, 92breqtrd 3877 1  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  ( ( A ^ M )  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   _Vcvv 2622   class class class wbr 3853    |-> cmpt 3907   dom cdm 4454   ` cfv 5030  (class class class)co 5668   CCcc 7411   0cc0 7413   1c1 7414    + caddc 7416    < clt 7585    <_ cle 7586    - cmin 7716   # cap 8121    / cdiv 8202   NN0cn0 8736   ZZcz 8813   ZZ>=cuz 9082   ...cfz 9487    seqcseq 9915   ^cexp 10017   abscabs 10493    ~~> cli 10729   sum_csu 10805
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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-mulrcl 7507  ax-addcom 7508  ax-mulcom 7509  ax-addass 7510  ax-mulass 7511  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-1rid 7515  ax-0id 7516  ax-rnegex 7517  ax-precex 7518  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524  ax-pre-mulgt0 7525  ax-pre-mulext 7526  ax-arch 7527  ax-caucvg 7528
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-if 3400  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-id 4131  df-po 4134  df-iso 4135  df-iord 4204  df-on 4206  df-ilim 4207  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-isom 5039  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-irdg 6151  df-frec 6172  df-1o 6197  df-oadd 6201  df-er 6308  df-en 6514  df-dom 6515  df-fin 6516  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-reap 8115  df-ap 8122  df-div 8203  df-inn 8486  df-2 8544  df-3 8545  df-4 8546  df-n0 8737  df-z 8814  df-uz 9083  df-q 9168  df-rp 9198  df-fz 9488  df-fzo 9617  df-iseq 9916  df-seq3 9917  df-exp 10018  df-ihash 10247  df-cj 10339  df-re 10340  df-im 10341  df-rsqrt 10494  df-abs 10495  df-clim 10730  df-isum 10806
This theorem is referenced by:  geoisum1  10976  geoisum1c  10977
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