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Theorem geolim2 11236
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 2117 . . 3  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 geolim2.3 . . . 4  |-  ( ph  ->  M  e.  NN0 )
32nn0zd 9129 . . 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 9349 . . . . 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 10379 . . 3  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( A ^ k )  e.  CC )
10 eluzelz 9291 . . . . . . . . 9  |-  ( x  e.  ( ZZ>= `  M
)  ->  x  e.  ZZ )
1110adantl 275 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  ZZ )
12 0red 7735 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  0  e.  RR )
133adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  M  e.  ZZ )
1413zred 9131 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  M  e.  RR )
1511zred 9131 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  RR )
162nn0ge0d 8991 . . . . . . . . . 10  |-  ( ph  ->  0  <_  M )
1716adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  0  <_  M )
18 eluzle 9294 . . . . . . . . . 10  |-  ( x  e.  ( ZZ>= `  M
)  ->  M  <_  x )
1918adantl 275 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  M  <_  x )
2012, 14, 15, 17, 19letrd 7854 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  0  <_  x )
21 elnn0z 9025 . . . . . . . 8  |-  ( x  e.  NN0  <->  ( x  e.  ZZ  /\  0  <_  x ) )
2211, 20, 21sylanbrc 413 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  x  e.  NN0 )
235adantr 274 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  A  e.  CC )
2423, 22expcld 10379 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( A ^ x )  e.  CC )
25 oveq2 5750 . . . . . . . 8  |-  ( n  =  x  ->  ( A ^ n )  =  ( A ^ x
) )
26 eqid 2117 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
2725, 26fvmptg 5465 . . . . . . 7  |-  ( ( x  e.  NN0  /\  ( A ^ x )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  x )  =  ( A ^ x ) )
2822, 24, 27syl2anc 408 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 x )  =  ( A ^ x
) )
2928, 24eqeltrd 2194 . . . . 5  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 x )  e.  CC )
30 oveq2 5750 . . . . . . . 8  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
3130, 26fvmptg 5465 . . . . . . 7  |-  ( ( k  e.  NN0  /\  ( A ^ k )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k )  =  ( A ^ k ) )
328, 9, 31syl2anc 408 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
3332, 4eqtr4d 2153 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( F `  k
) )
34 addcl 7713 . . . . . 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 10200 . . . 4  |-  ( ph  ->  seq M (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  =  seq M (  +  ,  F ) )
37 seqex 10175 . . . . . 6  |-  seq 0
(  +  ,  ( n  e.  NN0  |->  ( A ^ n ) ) )  e.  _V
38 ax-1cn 7681 . . . . . . . 8  |-  1  e.  CC
39 subcl 7929 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
4038, 5, 39sylancr 410 . . . . . . 7  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
41 1cnd 7750 . . . . . . . 8  |-  ( ph  ->  1  e.  CC )
42 1red 7749 . . . . . . . . . 10  |-  ( ph  ->  1  e.  RR )
43 geolim.2 . . . . . . . . . 10  |-  ( ph  ->  ( abs `  A
)  <  1 )
445, 42, 43absltap 11233 . . . . . . . . 9  |-  ( ph  ->  A #  1 )
45 apsym 8335 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A #  1  <->  1 #  A ) )
465, 41, 45syl2anc 408 . . . . . . . . 9  |-  ( ph  ->  ( A #  1  <->  1 #  A ) )
4744, 46mpbid 146 . . . . . . . 8  |-  ( ph  ->  1 #  A )
4841, 5, 47subap0d 8373 . . . . . . 7  |-  ( ph  ->  ( 1  -  A
) #  0 )
4940, 48recclapd 8508 . . . . . 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 10379 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
53 oveq2 5750 . . . . . . . . 9  |-  ( n  =  j  ->  ( A ^ n )  =  ( A ^ j
) )
5453, 26fvmptg 5465 . . . . . . . 8  |-  ( ( j  e.  NN0  /\  ( A ^ j )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  j )  =  ( A ^ j ) )
5550, 52, 54syl2anc 408 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
565, 43, 55geolim 11235 . . . . . 6  |-  ( ph  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  ~~>  ( 1  /  (
1  -  A ) ) )
57 breldmg 4715 . . . . . 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 1305 . . . . 5  |-  ( ph  ->  seq 0 (  +  ,  ( n  e. 
NN0  |->  ( A ^
n ) ) )  e.  dom  ~~>  )
59 nn0uz 9316 . . . . . 6  |-  NN0  =  ( ZZ>= `  0 )
60 expcl 10266 . . . . . . . 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 2194 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  e.  CC )
6359, 2, 62iserex 11063 . . . . 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 2195 . . 3  |-  ( ph  ->  seq M (  +  ,  F )  e. 
dom 
~~>  )
661, 3, 4, 9, 65isumclim2 11146 . 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 10379 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( A ^ k )  e.  CC )
7067, 69, 31syl2anc 408 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
71 expcl 10266 . . . . . . . 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 11215 . . . . . 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 9024 . . . . . . 7  |-  ( ph  ->  0  e.  ZZ )
7559, 74, 70, 72, 56isumclim 11145 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  NN0  ( A ^ k )  =  ( 1  / 
( 1  -  A
) ) )
7673, 75eqtr3d 2152 . . . . 5  |-  ( ph  ->  ( sum_ k  e.  ( 0 ... ( M  -  1 ) ) ( A ^ k
)  +  sum_ k  e.  ( ZZ>= `  M )
( A ^ k
) )  =  ( 1  /  ( 1  -  A ) ) )
775, 44, 2geoserap 11231 . . . . . 6  |-  ( ph  -> 
sum_ k  e.  ( 0 ... ( M  -  1 ) ) ( A ^ k
)  =  ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) ) )
7877oveq1d 5757 . . . . 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 2152 . . . 4  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  =  ( ( ( 1  -  ( A ^ M ) )  /  ( 1  -  A ) )  + 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k ) ) )
8079oveq1d 5757 . . 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 10379 . . . . . 6  |-  ( ph  ->  ( A ^ M
)  e.  CC )
82 subcl 7929 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A ^ M )  e.  CC )  -> 
( 1  -  ( A ^ M ) )  e.  CC )
8338, 81, 82sylancr 410 . . . . 5  |-  ( ph  ->  ( 1  -  ( A ^ M ) )  e.  CC )
8441, 83, 40, 48divsubdirapd 8557 . . . 4  |-  ( ph  ->  ( ( 1  -  ( 1  -  ( A ^ M ) ) )  /  ( 1  -  A ) )  =  ( ( 1  /  ( 1  -  A ) )  -  ( ( 1  -  ( A ^ M
) )  /  (
1  -  A ) ) ) )
85 nncan 7959 . . . . . 6  |-  ( ( 1  e.  CC  /\  ( A ^ M )  e.  CC )  -> 
( 1  -  (
1  -  ( A ^ M ) ) )  =  ( A ^ M ) )
8638, 81, 85sylancr 410 . . . . 5  |-  ( ph  ->  ( 1  -  (
1  -  ( A ^ M ) ) )  =  ( A ^ M ) )
8786oveq1d 5757 . . . 4  |-  ( ph  ->  ( ( 1  -  ( 1  -  ( A ^ M ) ) )  /  ( 1  -  A ) )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
8884, 87eqtr3d 2152 . . 3  |-  ( ph  ->  ( ( 1  / 
( 1  -  A
) )  -  (
( 1  -  ( A ^ M ) )  /  ( 1  -  A ) ) )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
8983, 40, 48divclapd 8517 . . . 4  |-  ( ph  ->  ( ( 1  -  ( A ^ M
) )  /  (
1  -  A ) )  e.  CC )
901, 3, 32, 9, 64isumcl 11149 . . . 4  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k )  e.  CC )
9189, 90pncan2d 8043 . . 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 2159 . 2  |-  ( ph  -> 
sum_ k  e.  (
ZZ>= `  M ) ( A ^ k )  =  ( ( A ^ M )  / 
( 1  -  A
) ) )
9366, 92breqtrd 3924 1  |-  ( ph  ->  seq M (  +  ,  F )  ~~>  ( ( A ^ M )  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   _Vcvv 2660   class class class wbr 3899    |-> cmpt 3959   dom cdm 4509   ` cfv 5093  (class class class)co 5742   CCcc 7586   0cc0 7588   1c1 7589    + caddc 7591    < clt 7768    <_ cle 7769    - cmin 7901   # cap 8310    / cdiv 8399   NN0cn0 8935   ZZcz 9012   ZZ>=cuz 9282   ...cfz 9745    seqcseq 10173   ^cexp 10247   abscabs 10724    ~~> cli 11002   sum_csu 11077
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-isom 5102  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-frec 6256  df-1o 6281  df-oadd 6285  df-er 6397  df-en 6603  df-dom 6604  df-fin 6605  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-2 8743  df-3 8744  df-4 8745  df-n0 8936  df-z 9013  df-uz 9283  df-q 9368  df-rp 9398  df-fz 9746  df-fzo 9875  df-seqfrec 10174  df-exp 10248  df-ihash 10477  df-cj 10569  df-re 10570  df-im 10571  df-rsqrt 10725  df-abs 10726  df-clim 11003  df-sumdc 11078
This theorem is referenced by:  geoisum1  11243  geoisum1c  11244  trilpolemisumle  13127
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