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Theorem geolim 12423
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 10354 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0z 10127 . . . 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 12419 . . . . 5  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
7 ax-1cn 8885 . . . . . . 7  |-  1  e.  CC
8 subcl 9141 . . . . . . 7  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
97, 4, 8sylancr 644 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
10 1re 8927 . . . . . . . . . . . 12  |-  1  e.  RR
1110ltnri 9019 . . . . . . . . . . 11  |-  -.  1  <  1
12 fveq2 5608 . . . . . . . . . . . . 13  |-  ( A  =  1  ->  ( abs `  A )  =  ( abs `  1
) )
13 abs1 11878 . . . . . . . . . . . . 13  |-  ( abs `  1 )  =  1
1412, 13syl6eq 2406 . . . . . . . . . . . 12  |-  ( A  =  1  ->  ( abs `  A )  =  1 )
1514breq1d 4114 . . . . . . . . . . 11  |-  ( A  =  1  ->  (
( abs `  A
)  <  1  <->  1  <  1 ) )
1611, 15mtbiri 294 . . . . . . . . . 10  |-  ( A  =  1  ->  -.  ( abs `  A )  <  1 )
1716necon2ai 2566 . . . . . . . . 9  |-  ( ( abs `  A )  <  1  ->  A  =/=  1 )
185, 17syl 15 . . . . . . . 8  |-  ( ph  ->  A  =/=  1 )
1918necomd 2604 . . . . . . 7  |-  ( ph  ->  1  =/=  A )
20 subeq0 9163 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
217, 4, 20sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( ( 1  -  A )  =  0  <->  1  =  A ) )
2221necon3bid 2556 . . . . . . 7  |-  ( ph  ->  ( ( 1  -  A )  =/=  0  <->  1  =/=  A ) )
2319, 22mpbird 223 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  =/=  0 )
244, 9, 23divcld 9626 . . . . 5  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  CC )
25 nn0ex 10063 . . . . . . 7  |-  NN0  e.  _V
2625mptex 5832 . . . . . 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 5953 . . . . . . . 8  |-  ( n  =  j  ->  ( A ^ n )  =  ( A ^ j
) )
29 eqid 2358 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
30 ovex 5970 . . . . . . . 8  |-  ( A ^ j )  e. 
_V
3128, 29, 30fvmpt 5685 . . . . . . 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 11214 . . . . . . 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 2432 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  e.  CC )
36 expp1 11203 . . . . . . . . . 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 8946 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ j )  x.  A )  =  ( A  x.  ( A ^ j ) ) )
4037, 39eqtrd 2390 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  =  ( A  x.  ( A ^ j ) ) )
4140oveq1d 5960 . . . . . . 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 9663 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  x.  ( A ^ j ) )  /  ( 1  -  A ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
4541, 44eqtrd 2390 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  =  ( ( A  /  (
1  -  A ) )  x.  ( A ^ j ) ) )
46 oveq1 5952 . . . . . . . . . 10  |-  ( n  =  j  ->  (
n  +  1 )  =  ( j  +  1 ) )
4746oveq2d 5961 . . . . . . . . 9  |-  ( n  =  j  ->  ( A ^ ( n  + 
1 ) )  =  ( A ^ (
j  +  1 ) ) )
4847oveq1d 5960 . . . . . . . 8  |-  ( n  =  j  ->  (
( A ^ (
n  +  1 ) )  /  ( 1  -  A ) )  =  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) ) )
49 eqid 2358 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ ( n  + 
1 ) )  / 
( 1  -  A
) ) )
50 ovex 5970 . . . . . . . 8  |-  ( ( A ^ ( j  +  1 ) )  /  ( 1  -  A ) )  e. 
_V
5148, 49, 50fvmpt 5685 . . . . . . 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 5961 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  /  ( 1  -  A ) )  x.  ( ( n  e. 
NN0  |->  ( A ^
n ) ) `  j ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
5445, 52, 533eqtr4d 2400 . . . . 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 12206 . . . 4  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  ( ( A  /  ( 1  -  A ) )  x.  0 ) )
5624mul01d 9101 . . . 4  |-  ( ph  ->  ( ( A  / 
( 1  -  A
) )  x.  0 )  =  0 )
5755, 56breqtrd 4128 . . 3  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  0 )
589, 23reccld 9619 . . 3  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  e.  CC )
59 seqex 11140 . . . 4  |-  seq  0
(  +  ,  F
)  e.  _V
6059a1i 10 . . 3  |-  ( ph  ->  seq  0 (  +  ,  F )  e. 
_V )
61 peano2nn0 10096 . . . . . 6  |-  ( j  e.  NN0  ->  ( j  +  1 )  e. 
NN0 )
62 expcl 11214 . . . . . 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 9626 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  e.  CC )
6552, 64eqeltrd 2432 . . 3  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  e.  CC )
66 nn0cn 10067 . . . . . . . 8  |-  ( j  e.  NN0  ->  j  e.  CC )
6766adantl 452 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  CC )
68 pncan 9147 . . . . . . 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 5961 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 0 ... ( ( j  +  1 )  - 
1 ) )  =  ( 0 ... j
) )
7170sumeq1d 12271 . . . 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 9665 . . . . 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 12422 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  ( ( 1  -  ( A ^ ( j  +  1 ) ) )  /  ( 1  -  A ) ) )
7752oveq2d 5961 . . . . 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 2400 . . . 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 10914 . . . . . . 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 2448 . . . . 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 11338 . . . . 5  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( 0 ... j
) )  ->  ( A ^ k )  e.  CC )
8883, 85, 87fsumser 12300 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... j
) ( A ^
k )  =  (  seq  0 (  +  ,  F ) `  j ) )
8971, 78, 883eqtr3rd 2399 . . 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 12208 . 2  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( ( 1  /  ( 1  -  A ) )  -  0 ) )
9158subid1d 9236 . 2  |-  ( ph  ->  ( ( 1  / 
( 1  -  A
) )  -  0 )  =  ( 1  /  ( 1  -  A ) ) )
9290, 91breqtrd 4128 1  |-  ( ph  ->  seq  0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864   class class class wbr 4104    e. cmpt 4158   ` cfv 5337  (class class class)co 5945   CCcc 8825   0cc0 8827   1c1 8828    + caddc 8830    x. cmul 8832    < clt 8957    - cmin 9127    / cdiv 9513   NN0cn0 10057   ZZcz 10116   ZZ>=cuz 10322   ...cfz 10874    seq cseq 11138   ^cexp 11197   abscabs 11815    ~~> cli 12054   sum_csu 12255
This theorem is referenced by:  geolim2  12424  georeclim  12425  geomulcvg  12429  geoisum  12430  cvgrat  12436  eflegeo  12498  geolim3  19823  abelthlem5  19918  logtayllem  20117  zetacvg  24048
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-pm 6863  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-fz 10875  df-fzo 10963  df-fl 11017  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-rlim 12059  df-sum 12256
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