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Theorem geolim2 11475
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 2170 . . 3  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 geolim2.3 . . . 4  |-  ( ph  ->  M  e.  NN0 )
32nn0zd 9332 . . 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 274 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  A  e.  CC )
7 eluznn0 9558 . . . . 5  |-  ( ( M  e.  NN0  /\  k  e.  ( ZZ>= `  M ) )  -> 
k  e.  NN0 )
82, 7sylan 281 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  k  e.  NN0 )
96, 8expcld 10609 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( A ^ k )  e.  CC )
10 eluzelz 9496 . . . . . . . . 9  |-  ( x  e.  ( ZZ>= `  M
)  ->  x  e.  ZZ )
1110adantl 275 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  ZZ )
12 0red 7921 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  0  e.  RR )
133adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  M  e.  ZZ )
1413zred 9334 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  M  e.  RR )
1511zred 9334 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  RR )
162nn0ge0d 9191 . . . . . . . . . 10  |-  ( ph  ->  0  <_  M )
1716adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  0  <_  M )
18 eluzle 9499 . . . . . . . . . 10  |-  ( x  e.  ( ZZ>= `  M
)  ->  M  <_  x )
1918adantl 275 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  M  <_  x )
2012, 14, 15, 17, 19letrd 8043 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  0  <_  x )
21 elnn0z 9225 . . . . . . . 8  |-  ( x  e.  NN0  <->  ( x  e.  ZZ  /\  0  <_  x ) )
2211, 20, 21sylanbrc 415 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  NN0 )
235adantr 274 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  A  e.  CC )
2423, 22expcld 10609 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( A ^ x )  e.  CC )
25 oveq2 5861 . . . . . . . 8  |-  ( n  =  x  ->  ( A ^ n )  =  ( A ^ x
) )
26 eqid 2170 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
2725, 26fvmptg 5572 . . . . . . 7  |-  ( ( x  e.  NN0  /\  ( A ^ x )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  x )  =  ( A ^ x ) )
2822, 24, 27syl2anc 409 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 x )  =  ( A ^ x
) )
2928, 24eqeltrd 2247 . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 x )  e.  CC )
30 oveq2 5861 . . . . . . . 8  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
3130, 26fvmptg 5572 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( A ^ k )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k )  =  ( A ^ k ) )
328, 9, 31syl2anc 409 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
3332, 4eqtr4d 2206 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( F `  k
) )
34 addcl 7899 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
3534adantl 275 . . . . 5  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
363, 29, 33, 35seq3feq 10428 . . . 4  |-  ( ph  ->  seq M (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  =  seq M (  +  ,  F ) )
37 seqex 10403 . . . . . 6  |-  seq 0
(  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  _V
38 ax-1cn 7867 . . . . . . . 8  |-  1  e.  CC
39 subcl 8118 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
4038, 5, 39sylancr 412 . . . . . . 7  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
41 1cnd 7936 . . . . . . . 8  |-  ( ph  ->  1  e.  CC )
42 1red 7935 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR )
43 geolim.2 . . . . . . . . . 10  |-  ( ph  ->  ( abs `  A
)  <  1 )
445, 42, 43absltap 11472 . . . . . . . . 9  |-  ( ph  ->  A #  1 )
45 apsym 8525 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A #  1  <->  1 #  A ) )
465, 41, 45syl2anc 409 . . . . . . . . 9  |-  ( ph  ->  ( A #  1  <->  1 #  A ) )
4744, 46mpbid 146 . . . . . . . 8  |-  ( ph  ->  1 #  A )
4841, 5, 47subap0d 8563 . . . . . . 7  |-  ( ph  ->  ( 1  -  A
) #  0 )
4940, 48recclapd 8698 . . . . . 6  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  e.  CC )
50 simpr 109 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  NN0 )
515adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  A  e.  CC )
5251, 50expcld 10609 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
53 oveq2 5861 . . . . . . . . 9  |-  ( n  =  j  ->  ( A ^ n )  =  ( A ^ j
) )
5453, 26fvmptg 5572 . . . . . . . 8  |-  ( ( j  e.  NN0  /\  ( A ^ j )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  j )  =  ( A ^ j ) )
5550, 52, 54syl2anc 409 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
565, 43, 55geolim 11474 . . . . . 6  |-  ( ph  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  ~~>  ( 1  /  (
1  -  A ) ) )
57 breldmg 4817 . . . . . 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 1337 . . . . 5  |-  ( ph  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  dom  ~~>  )
59 nn0uz 9521 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
60 expcl 10494 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ j
)  e.  CC )
615, 60sylan 281 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
6255, 61eqeltrd 2247 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  e.  CC )
6359, 2, 62iserex 11302 . . . . 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 2248 . . 3  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
661, 3, 4, 9, 65isumclim2 11385 . 2  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  sum_ k  e.  ( ZZ>= `  M )
( A ^ k
) )
67 simpr 109 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
685adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN0 )  ->  A  e.  CC )
6968, 67expcld 10609 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  CC )
7067, 69, 31syl2anc 409 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
71 expcl 10494 . . . . . . . 8  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
725, 71sylan 281 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  CC )
7359, 1, 2, 70, 72, 58isumsplit 11454 . . . . . 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 9224 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
7559, 74, 70, 72, 56isumclim 11384 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  NN0  ( A ^ k )  =  ( 1  / 
( 1  -  A
) ) )
7673, 75eqtr3d 2205 . . . . 5  |-  ( ph  ->  ( sum_ k  e.  ( 0 ... ( M  -  1 ) ) ( A ^ k
)  +  sum_ k  e.  ( ZZ>= `  M )
( A ^ k
) )  =  ( 1  /  ( 1  -  A ) ) )
775, 44, 2geoserap 11470 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( M  -  1 ) ) ( A ^ k
)  =  ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) ) )
7877oveq1d 5868 . . . . 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 2205 . . . 4  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  =  ( ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) )  + 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k ) ) )
8079oveq1d 5868 . . 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 10609 . . . . . 6  |-  ( ph  ->  ( A ^ M
)  e.  CC )
82 subcl 8118 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A ^ M )  e.  CC )  -> 
( 1  -  ( A ^ M ) )  e.  CC )
8338, 81, 82sylancr 412 . . . . 5  |-  ( ph  ->  ( 1  -  ( A ^ M ) )  e.  CC )
8441, 83, 40, 48divsubdirapd 8747 . . . 4  |-  ( ph  ->  ( ( 1  -  ( 1  -  ( A ^ M ) ) )  /  ( 1  -  A ) )  =  ( ( 1  /  ( 1  -  A ) )  -  ( ( 1  -  ( A ^ M
) )  /  (
1  -  A ) ) ) )
85 nncan 8148 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A ^ M )  e.  CC )  -> 
( 1  -  (
1  -  ( A ^ M ) ) )  =  ( A ^ M ) )
8638, 81, 85sylancr 412 . . . . 5  |-  ( ph  ->  ( 1  -  (
1  -  ( A ^ M ) ) )  =  ( A ^ M ) )
8786oveq1d 5868 . . . 4  |-  ( ph  ->  ( ( 1  -  ( 1  -  ( A ^ M ) ) )  /  ( 1  -  A ) )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
8884, 87eqtr3d 2205 . . 3  |-  ( ph  ->  ( ( 1  / 
( 1  -  A
) )  -  (
( 1  -  ( A ^ M ) )  /  ( 1  -  A ) ) )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
8983, 40, 48divclapd 8707 . . . 4  |-  ( ph  ->  ( ( 1  -  ( A ^ M
) )  /  (
1  -  A ) )  e.  CC )
901, 3, 32, 9, 64isumcl 11388 . . . 4  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k )  e.  CC )
9189, 90pncan2d 8232 . . 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 2212 . 2  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
9366, 92breqtrd 4015 1  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  ( ( A ^ M )  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   _Vcvv 2730   class class class wbr 3989    |-> cmpt 4050   dom cdm 4611   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775    + caddc 7777    < clt 7954    <_ cle 7955    - cmin 8090   # cap 8500    / cdiv 8589   NN0cn0 9135   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965    seqcseq 10401   ^cexp 10475   abscabs 10961    ~~> cli 11241   sum_csu 11316
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317
This theorem is referenced by:  geoisum1  11482  geoisum1c  11483  trilpolemisumle  14070
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