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Theorem geolim 11273
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 9353 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 9059 . . 3  |-  ( ph  ->  0  e.  ZZ )
3 geolim.1 . . . . . 6  |-  ( ph  ->  A  e.  CC )
4 geolim.2 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  <  1 )
53, 4expcnv 11266 . . . . 5  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
6 ax-1cn 7706 . . . . . . 7  |-  1  e.  CC
7 subcl 7954 . . . . . . 7  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
86, 3, 7sylancr 410 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
9 1cnd 7775 . . . . . . 7  |-  ( ph  ->  1  e.  CC )
10 1red 7774 . . . . . . . . 9  |-  ( ph  ->  1  e.  RR )
113, 10, 4absltap 11271 . . . . . . . 8  |-  ( ph  ->  A #  1 )
12 apsym 8361 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A #  1  <->  1 #  A ) )
133, 6, 12sylancl 409 . . . . . . . 8  |-  ( ph  ->  ( A #  1  <->  1 #  A ) )
1411, 13mpbid 146 . . . . . . 7  |-  ( ph  ->  1 #  A )
159, 3, 14subap0d 8399 . . . . . 6  |-  ( ph  ->  ( 1  -  A
) #  0 )
163, 8, 15divclapd 8543 . . . . 5  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  CC )
17 nn0ex 8976 . . . . . . 7  |-  NN0  e.  _V
1817mptex 5639 . . . . . 6  |-  ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) )  e.  _V
1918a1i 9 . . . . 5  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  e.  _V )
20 simpr 109 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  NN0 )
213adantr 274 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  A  e.  CC )
2221, 20expcld 10417 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
23 oveq2 5775 . . . . . . . 8  |-  ( n  =  j  ->  ( A ^ n )  =  ( A ^ j
) )
24 eqid 2137 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
2523, 24fvmptg 5490 . . . . . . 7  |-  ( ( j  e.  NN0  /\  ( A ^ j )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  j )  =  ( A ^ j ) )
2620, 22, 25syl2anc 408 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
27 expcl 10304 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ j
)  e.  CC )
283, 27sylan 281 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
2926, 28eqeltrd 2214 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  e.  CC )
30 expp1 10293 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ (
j  +  1 ) )  =  ( ( A ^ j )  x.  A ) )
313, 30sylan 281 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  =  ( ( A ^
j )  x.  A
) )
3228, 21mulcomd 7780 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ j )  x.  A )  =  ( A  x.  ( A ^ j ) ) )
3331, 32eqtrd 2170 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  =  ( A  x.  ( A ^ j ) ) )
3433oveq1d 5782 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  =  ( ( A  x.  ( A ^ j ) )  /  ( 1  -  A ) ) )
358adantr 274 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 1  -  A )  e.  CC )
3615adantr 274 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 1  -  A ) #  0 )
3721, 28, 35, 36div23apd 8581 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  x.  ( A ^ j ) )  /  ( 1  -  A ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
3834, 37eqtrd 2170 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  =  ( ( A  /  (
1  -  A ) )  x.  ( A ^ j ) ) )
39 peano2nn0 9010 . . . . . . . . . 10  |-  ( j  e.  NN0  ->  ( j  +  1 )  e. 
NN0 )
4039adantl 275 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( j  +  1 )  e. 
NN0 )
4121, 40expcld 10417 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  e.  CC )
4241, 35, 36divclapd 8543 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  e.  CC )
43 oveq1 5774 . . . . . . . . . 10  |-  ( n  =  j  ->  (
n  +  1 )  =  ( j  +  1 ) )
4443oveq2d 5783 . . . . . . . . 9  |-  ( n  =  j  ->  ( A ^ ( n  + 
1 ) )  =  ( A ^ (
j  +  1 ) ) )
4544oveq1d 5782 . . . . . . . 8  |-  ( n  =  j  ->  (
( A ^ (
n  +  1 ) )  /  ( 1  -  A ) )  =  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) ) )
46 eqid 2137 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ ( n  + 
1 ) )  / 
( 1  -  A
) ) )
4745, 46fvmptg 5490 . . . . . . 7  |-  ( ( j  e.  NN0  /\  ( ( A ^
( j  +  1 ) )  /  (
1  -  A ) )  e.  CC )  ->  ( ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  =  ( ( A ^
( j  +  1 ) )  /  (
1  -  A ) ) )
4820, 42, 47syl2anc 408 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  =  ( ( A ^
( j  +  1 ) )  /  (
1  -  A ) ) )
4926oveq2d 5783 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  /  ( 1  -  A ) )  x.  ( ( n  e. 
NN0  |->  ( A ^
n ) ) `  j ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
5038, 48, 493eqtr4d 2180 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  =  ( ( A  / 
( 1  -  A
) )  x.  (
( n  e.  NN0  |->  ( A ^ n ) ) `  j ) ) )
511, 2, 5, 16, 19, 29, 50climmulc2 11093 . . . 4  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  ( ( A  /  ( 1  -  A ) )  x.  0 ) )
5216mul01d 8148 . . . 4  |-  ( ph  ->  ( ( A  / 
( 1  -  A
) )  x.  0 )  =  0 )
5351, 52breqtrd 3949 . . 3  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  0 )
548, 15recclapd 8534 . . 3  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  e.  CC )
55 seqex 10213 . . . 4  |-  seq 0
(  +  ,  F
)  e.  _V
5655a1i 9 . . 3  |-  ( ph  ->  seq 0 (  +  ,  F )  e. 
_V )
57 expcl 10304 . . . . . 6  |-  ( ( A  e.  CC  /\  ( j  +  1 )  e.  NN0 )  ->  ( A ^ (
j  +  1 ) )  e.  CC )
583, 39, 57syl2an 287 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  e.  CC )
5958, 35, 36divclapd 8543 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  e.  CC )
6048, 59eqeltrd 2214 . . 3  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  e.  CC )
61 nn0cn 8980 . . . . . . . 8  |-  ( j  e.  NN0  ->  j  e.  CC )
6261adantl 275 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  CC )
63 pncan 7961 . . . . . . 7  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  +  1 )  -  1 )  =  j )
6462, 6, 63sylancl 409 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
j  +  1 )  -  1 )  =  j )
6564oveq2d 5783 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 0 ... ( ( j  +  1 )  - 
1 ) )  =  ( 0 ... j
) )
6665sumeq1d 11128 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  sum_ k  e.  ( 0 ... j ) ( A ^ k ) )
67 1cnd 7775 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  1  e.  CC )
6867, 58, 35, 36divsubdirapd 8583 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
1  -  ( A ^ ( j  +  1 ) ) )  /  ( 1  -  A ) )  =  ( ( 1  / 
( 1  -  A
) )  -  (
( A ^ (
j  +  1 ) )  /  ( 1  -  A ) ) ) )
6911adantr 274 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  A #  1
)
7021, 69, 40geoserap 11269 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  ( ( 1  -  ( A ^ ( j  +  1 ) ) )  /  ( 1  -  A ) ) )
7148oveq2d 5783 . . . . 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 ) ) ) )
7268, 70, 713eqtr4d 2180 . . . 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 ) ) )
73 simpll 518 . . . . . 6  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>= `  0 )
)  ->  ph )
74 elnn0uz 9356 . . . . . . . 8  |-  ( k  e.  NN0  <->  k  e.  (
ZZ>= `  0 ) )
7574biimpri 132 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  NN0 )
7675adantl 275 . . . . . 6  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>= `  0 )
)  ->  k  e.  NN0 )
77 geolim.3 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( A ^ k ) )
7873, 76, 77syl2anc 408 . . . . 5  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( F `  k )  =  ( A ^ k ) )
7920, 1eleqtrdi 2230 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  ( ZZ>= `  0 )
)
8021adantr 274 . . . . . 6  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>= `  0 )
)  ->  A  e.  CC )
8180, 76expcld 10417 . . . . 5  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( A ^ k )  e.  CC )
8278, 79, 81fsum3ser 11159 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... j
) ( A ^
k )  =  (  seq 0 (  +  ,  F ) `  j ) )
8366, 72, 823eqtr3rd 2179 . . 3  |-  ( (
ph  /\  j  e.  NN0 )  ->  (  seq 0 (  +  ,  F ) `  j
)  =  ( ( 1  /  ( 1  -  A ) )  -  ( ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j ) ) )
841, 2, 53, 54, 56, 60, 83climsubc2 11095 . 2  |-  ( ph  ->  seq 0 (  +  ,  F )  ~~>  ( ( 1  /  ( 1  -  A ) )  -  0 ) )
8554subid1d 8055 . 2  |-  ( ph  ->  ( ( 1  / 
( 1  -  A
) )  -  0 )  =  ( 1  /  ( 1  -  A ) ) )
8684, 85breqtrd 3949 1  |-  ( ph  ->  seq 0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   _Vcvv 2681   class class class wbr 3924    |-> cmpt 3984   ` cfv 5118  (class class class)co 5767   CCcc 7611   0cc0 7613   1c1 7614    + caddc 7616    x. cmul 7618    < clt 7793    - cmin 7926   # cap 8336    / cdiv 8425   NN0cn0 8970   ZZ>=cuz 9319   ...cfz 9783    seqcseq 10211   ^cexp 10285   abscabs 10762    ~~> cli 11040   sum_csu 11115
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-frec 6281  df-1o 6306  df-oadd 6310  df-er 6422  df-en 6628  df-dom 6629  df-fin 6630  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-fzo 9913  df-seqfrec 10212  df-exp 10286  df-ihash 10515  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-clim 11041  df-sumdc 11116
This theorem is referenced by:  geolim2  11274  georeclim  11275  geoisum  11279  eflegeo  11397
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