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Theorem geolim 12326
Description: The partial sums in the infinite series  1  +  A ^ 1  +  A ^ 2... converge to  ( 1  /  (
1  -  A ) ). (Contributed by NM, 15-May-2006.)
Hypotheses
Ref Expression
geolim.1  |-  ( ph  ->  A  e.  CC )
geolim.2  |-  ( ph  ->  ( abs `  A
)  <  1 )
geolim.3  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( A ^ k ) )
Assertion
Ref Expression
geolim  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  A ) ) )
Distinct variable groups:    A, k    k, F    ph, k

Proof of Theorem geolim
Dummy variables  j  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0uz 10262 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0z 10035 . . . 4  |-  0  e.  ZZ
32a1i 10 . . 3  |-  ( ph  ->  0  e.  ZZ )
4 geolim.1 . . . . . 6  |-  ( ph  ->  A  e.  CC )
5 geolim.2 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  <  1 )
64, 5expcnv 12322 . . . . 5  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
7 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
8 subcl 9051 . . . . . . 7  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
97, 4, 8sylancr 644 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
10 1re 8837 . . . . . . . . . . . 12  |-  1  e.  RR
1110ltnri 8929 . . . . . . . . . . 11  |-  -.  1  <  1
12 fveq2 5525 . . . . . . . . . . . . 13  |-  ( A  =  1  ->  ( abs `  A )  =  ( abs `  1
) )
13 abs1 11782 . . . . . . . . . . . . 13  |-  ( abs `  1 )  =  1
1412, 13syl6eq 2331 . . . . . . . . . . . 12  |-  ( A  =  1  ->  ( abs `  A )  =  1 )
1514breq1d 4033 . . . . . . . . . . 11  |-  ( A  =  1  ->  (
( abs `  A
)  <  1  <->  1  <  1 ) )
1611, 15mtbiri 294 . . . . . . . . . 10  |-  ( A  =  1  ->  -.  ( abs `  A )  <  1 )
1716necon2ai 2491 . . . . . . . . 9  |-  ( ( abs `  A )  <  1  ->  A  =/=  1 )
185, 17syl 15 . . . . . . . 8  |-  ( ph  ->  A  =/=  1 )
1918necomd 2529 . . . . . . 7  |-  ( ph  ->  1  =/=  A )
20 subeq0 9073 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
217, 4, 20sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
2221necon3bid 2481 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  A )  =/=  0  <->  1  =/=  A ) )
2319, 22mpbird 223 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  =/=  0 )
244, 9, 23divcld 9536 . . . . 5  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  CC )
25 nn0ex 9971 . . . . . . 7  |-  NN0  e.  _V
2625mptex 5746 . . . . . 6  |-  ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) )  e.  _V
2726a1i 10 . . . . 5  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  e.  _V )
28 oveq2 5866 . . . . . . . 8  |-  ( n  =  j  ->  ( A ^ n )  =  ( A ^ j
) )
29 eqid 2283 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
30 ovex 5883 . . . . . . . 8  |-  ( A ^ j )  e. 
_V
3128, 29, 30fvmpt 5602 . . . . . . 7  |-  ( j  e.  NN0  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
3231adantl 452 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
33 expcl 11121 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ j
)  e.  CC )
344, 33sylan 457 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
3532, 34eqeltrd 2357 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  e.  CC )
36 expp1 11110 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ (
j  +  1 ) )  =  ( ( A ^ j )  x.  A ) )
374, 36sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  =  ( ( A ^
j )  x.  A
) )
384adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  NN0 )  ->  A  e.  CC )
3934, 38mulcomd 8856 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ j )  x.  A )  =  ( A  x.  ( A ^ j ) ) )
4037, 39eqtrd 2315 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  =  ( A  x.  ( A ^ j ) ) )
4140oveq1d 5873 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  =  ( ( A  x.  ( A ^ j ) )  /  ( 1  -  A ) ) )
429adantr 451 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 1  -  A )  e.  CC )
4323adantr 451 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 1  -  A )  =/=  0 )
4438, 34, 42, 43div23d 9573 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  x.  ( A ^ j ) )  /  ( 1  -  A ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
4541, 44eqtrd 2315 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  =  ( ( A  /  (
1  -  A ) )  x.  ( A ^ j ) ) )
46 oveq1 5865 . . . . . . . . . 10  |-  ( n  =  j  ->  (
n  +  1 )  =  ( j  +  1 ) )
4746oveq2d 5874 . . . . . . . . 9  |-  ( n  =  j  ->  ( A ^ ( n  + 
1 ) )  =  ( A ^ (
j  +  1 ) ) )
4847oveq1d 5873 . . . . . . . 8  |-  ( n  =  j  ->  (
( A ^ (
n  +  1 ) )  /  ( 1  -  A ) )  =  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) ) )
49 eqid 2283 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ ( n  + 
1 ) )  / 
( 1  -  A
) ) )
50 ovex 5883 . . . . . . . 8  |-  ( ( A ^ ( j  +  1 ) )  /  ( 1  -  A ) )  e. 
_V
5148, 49, 50fvmpt 5602 . . . . . . 7  |-  ( j  e.  NN0  ->  ( ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  =  ( ( A ^
( j  +  1 ) )  /  (
1  -  A ) ) )
5251adantl 452 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  =  ( ( A ^
( j  +  1 ) )  /  (
1  -  A ) ) )
5332oveq2d 5874 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  /  ( 1  -  A ) )  x.  ( ( n  e. 
NN0  |->  ( A ^
n ) ) `  j ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
5445, 52, 533eqtr4d 2325 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  =  ( ( A  / 
( 1  -  A
) )  x.  (
( n  e.  NN0  |->  ( A ^ n ) ) `  j ) ) )
551, 3, 6, 24, 27, 35, 54climmulc2 12110 . . . 4  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  ( ( A  /  ( 1  -  A ) )  x.  0 ) )
5624mul01d 9011 . . . 4  |-  ( ph  ->  ( ( A  / 
( 1  -  A
) )  x.  0 )  =  0 )
5755, 56breqtrd 4047 . . 3  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  0 )
589, 23reccld 9529 . . 3  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  e.  CC )
59 seqex 11048 . . . 4  |-  seq  0
(  +  ,  F
)  e.  _V
6059a1i 10 . . 3  |-  ( ph  ->  seq  0 (  +  ,  F )  e. 
_V )
61 peano2nn0 10004 . . . . . 6  |-  ( j  e.  NN0  ->  ( j  +  1 )  e. 
NN0 )
62 expcl 11121 . . . . . 6  |-  ( ( A  e.  CC  /\  ( j  +  1 )  e.  NN0 )  ->  ( A ^ (
j  +  1 ) )  e.  CC )
634, 61, 62syl2an 463 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  e.  CC )
6463, 42, 43divcld 9536 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  e.  CC )
6552, 64eqeltrd 2357 . . 3  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  e.  CC )
66 nn0cn 9975 . . . . . . . 8  |-  ( j  e.  NN0  ->  j  e.  CC )
6766adantl 452 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  CC )
68 pncan 9057 . . . . . . 7  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  +  1 )  -  1 )  =  j )
6967, 7, 68sylancl 643 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
j  +  1 )  -  1 )  =  j )
7069oveq2d 5874 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 0 ... ( ( j  +  1 )  - 
1 ) )  =  ( 0 ... j
) )
7170sumeq1d 12174 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  sum_ k  e.  ( 0 ... j ) ( A ^ k ) )
727a1i 10 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  1  e.  CC )
7372, 63, 42, 43divsubdird 9575 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
1  -  ( A ^ ( j  +  1 ) ) )  /  ( 1  -  A ) )  =  ( ( 1  / 
( 1  -  A
) )  -  (
( A ^ (
j  +  1 ) )  /  ( 1  -  A ) ) ) )
7418adantr 451 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  A  =/=  1 )
7561adantl 452 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( j  +  1 )  e. 
NN0 )
7638, 74, 75geoser 12325 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  ( ( 1  -  ( A ^ ( j  +  1 ) ) )  /  ( 1  -  A ) ) )
7752oveq2d 5874 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
1  /  ( 1  -  A ) )  -  ( ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j ) )  =  ( ( 1  /  ( 1  -  A ) )  -  ( ( A ^
( j  +  1 ) )  /  (
1  -  A ) ) ) )
7873, 76, 773eqtr4d 2325 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  ( ( 1  /  (
1  -  A ) )  -  ( ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j ) ) )
79 simpll 730 . . . . . 6  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  ph )
80 elfznn0 10822 . . . . . . 7  |-  ( k  e.  ( 0 ... j )  ->  k  e.  NN0 )
8180adantl 452 . . . . . 6  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  k  e.  NN0 )
82 geolim.3 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( A ^ k ) )
8379, 81, 82syl2anc 642 . . . . 5  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  ( F `  k )  =  ( A ^
k ) )
84 simpr 447 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  NN0 )
8584, 1syl6eleq 2373 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  ( ZZ>= `  0 )
)
8679, 4syl 15 . . . . . 6  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  A  e.  CC )
8786, 81expcld 11245 . . . . 5  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  ( A ^ k )  e.  CC )
8883, 85, 87fsumser 12203 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... j
) ( A ^
k )  =  (  seq  0 (  +  ,  F ) `  j ) )
8971, 78, 883eqtr3rd 2324 . . 3  |-  ( (
ph  /\  j  e.  NN0 )  ->  (  seq  0 (  +  ,  F ) `  j
)  =  ( ( 1  /  ( 1  -  A ) )  -  ( ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j ) ) )
901, 3, 57, 58, 60, 65, 89climsubc2 12112 . 2  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( ( 1  /  ( 1  -  A ) )  -  0 ) )
9158subid1d 9146 . 2  |-  ( ph  ->  ( ( 1  / 
( 1  -  A
) )  -  0 )  =  ( 1  /  ( 1  -  A ) ) )
9290, 91breqtrd 4047 1  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < 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:  geolim2  12327  georeclim  12328  geomulcvg  12332  geoisum  12333  cvgrat  12339  eflegeo  12401  geolim3  19719  abelthlem5  19811  logtayllem  20006  zetacvg  23689
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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