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Theorem geolim 11070
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 9152 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 8860 . . 3  |-  ( ph  ->  0  e.  ZZ )
3 geolim.1 . . . . . 6  |-  ( ph  ->  A  e.  CC )
4 geolim.2 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  <  1 )
53, 4expcnv 11063 . . . . 5  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
6 ax-1cn 7535 . . . . . . 7  |-  1  e.  CC
7 subcl 7778 . . . . . . 7  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
86, 3, 7sylancr 406 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
9 1cnd 7601 . . . . . . 7  |-  ( ph  ->  1  e.  CC )
10 1red 7600 . . . . . . . . 9  |-  ( ph  ->  1  e.  RR )
113, 10, 4absltap 11068 . . . . . . . 8  |-  ( ph  ->  A #  1 )
12 apsym 8180 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A #  1  <->  1 #  A ) )
133, 6, 12sylancl 405 . . . . . . . 8  |-  ( ph  ->  ( A #  1  <->  1 #  A ) )
1411, 13mpbid 146 . . . . . . 7  |-  ( ph  ->  1 #  A )
159, 3, 14subap0d 8216 . . . . . 6  |-  ( ph  ->  ( 1  -  A
) #  0 )
163, 8, 15divclapd 8354 . . . . 5  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  CC )
17 nn0ex 8777 . . . . . . 7  |-  NN0  e.  _V
1817mptex 5562 . . . . . 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 271 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  A  e.  CC )
2221, 20expcld 10217 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
23 oveq2 5698 . . . . . . . 8  |-  ( n  =  j  ->  ( A ^ n )  =  ( A ^ j
) )
24 eqid 2095 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
2523, 24fvmptg 5415 . . . . . . 7  |-  ( ( j  e.  NN0  /\  ( A ^ j )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  j )  =  ( A ^ j ) )
2620, 22, 25syl2anc 404 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
27 expcl 10104 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ j
)  e.  CC )
283, 27sylan 278 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
2926, 28eqeltrd 2171 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  e.  CC )
30 expp1 10093 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  j  e.  NN0 )  -> 
( A ^ (
j  +  1 ) )  =  ( ( A ^ j )  x.  A ) )
313, 30sylan 278 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  =  ( ( A ^
j )  x.  A
) )
3228, 21mulcomd 7606 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ j )  x.  A )  =  ( A  x.  ( A ^ j ) ) )
3331, 32eqtrd 2127 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  =  ( A  x.  ( A ^ j ) ) )
3433oveq1d 5705 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  =  ( ( A  x.  ( A ^ j ) )  /  ( 1  -  A ) ) )
358adantr 271 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 1  -  A )  e.  CC )
3615adantr 271 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 1  -  A ) #  0 )
3721, 28, 35, 36div23apd 8392 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  x.  ( A ^ j ) )  /  ( 1  -  A ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
3834, 37eqtrd 2127 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  =  ( ( A  /  (
1  -  A ) )  x.  ( A ^ j ) ) )
39 peano2nn0 8811 . . . . . . . . . 10  |-  ( j  e.  NN0  ->  ( j  +  1 )  e. 
NN0 )
4039adantl 272 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( j  +  1 )  e. 
NN0 )
4121, 40expcld 10217 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  e.  CC )
4241, 35, 36divclapd 8354 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  e.  CC )
43 oveq1 5697 . . . . . . . . . 10  |-  ( n  =  j  ->  (
n  +  1 )  =  ( j  +  1 ) )
4443oveq2d 5706 . . . . . . . . 9  |-  ( n  =  j  ->  ( A ^ ( n  + 
1 ) )  =  ( A ^ (
j  +  1 ) ) )
4544oveq1d 5705 . . . . . . . 8  |-  ( n  =  j  ->  (
( A ^ (
n  +  1 ) )  /  ( 1  -  A ) )  =  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) ) )
46 eqid 2095 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ ( n  + 
1 ) )  / 
( 1  -  A
) ) )
4745, 46fvmptg 5415 . . . . . . 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 404 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  =  ( ( A ^
( j  +  1 ) )  /  (
1  -  A ) ) )
4926oveq2d 5706 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  /  ( 1  -  A ) )  x.  ( ( n  e. 
NN0  |->  ( A ^
n ) ) `  j ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
5038, 48, 493eqtr4d 2137 . . . . 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 10890 . . . 4  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  ( ( A  /  ( 1  -  A ) )  x.  0 ) )
5216mul01d 7968 . . . 4  |-  ( ph  ->  ( ( A  / 
( 1  -  A
) )  x.  0 )  =  0 )
5351, 52breqtrd 3891 . . 3  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  0 )
548, 15recclapd 8345 . . 3  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  e.  CC )
55 seqex 10005 . . . 4  |-  seq 0
(  +  ,  F
)  e.  _V
5655a1i 9 . . 3  |-  ( ph  ->  seq 0 (  +  ,  F )  e. 
_V )
57 expcl 10104 . . . . . 6  |-  ( ( A  e.  CC  /\  ( j  +  1 )  e.  NN0 )  ->  ( A ^ (
j  +  1 ) )  e.  CC )
583, 39, 57syl2an 284 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  e.  CC )
5958, 35, 36divclapd 8354 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  e.  CC )
6048, 59eqeltrd 2171 . . 3  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  e.  CC )
61 nn0cn 8781 . . . . . . . 8  |-  ( j  e.  NN0  ->  j  e.  CC )
6261adantl 272 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  CC )
63 pncan 7785 . . . . . . 7  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  +  1 )  -  1 )  =  j )
6462, 6, 63sylancl 405 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
j  +  1 )  -  1 )  =  j )
6564oveq2d 5706 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 0 ... ( ( j  +  1 )  - 
1 ) )  =  ( 0 ... j
) )
6665sumeq1d 10925 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  sum_ k  e.  ( 0 ... j ) ( A ^ k ) )
67 1cnd 7601 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  1  e.  CC )
6867, 58, 35, 36divsubdirapd 8394 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
1  -  ( A ^ ( j  +  1 ) ) )  /  ( 1  -  A ) )  =  ( ( 1  / 
( 1  -  A
) )  -  (
( A ^ (
j  +  1 ) )  /  ( 1  -  A ) ) ) )
6911adantr 271 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  A #  1
)
7021, 69, 40geoserap 11066 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  ( ( 1  -  ( A ^ ( j  +  1 ) ) )  /  ( 1  -  A ) ) )
7148oveq2d 5706 . . . . 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 2137 . . . 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 497 . . . . . 6  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>= `  0 )
)  ->  ph )
74 elnn0uz 9155 . . . . . . . 8  |-  ( k  e.  NN0  <->  k  e.  (
ZZ>= `  0 ) )
7574biimpri 132 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  0
)  ->  k  e.  NN0 )
7675adantl 272 . . . . . 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 404 . . . . 5  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( F `  k )  =  ( A ^ k ) )
7920, 1syl6eleq 2187 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  ( ZZ>= `  0 )
)
8021adantr 271 . . . . . 6  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>= `  0 )
)  ->  A  e.  CC )
8180, 76expcld 10217 . . . . 5  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( A ^ k )  e.  CC )
8278, 79, 81fsum3ser 10956 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... j
) ( A ^
k )  =  (  seq 0 (  +  ,  F ) `  j ) )
8366, 72, 823eqtr3rd 2136 . . 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 10892 . 2  |-  ( ph  ->  seq 0 (  +  ,  F )  ~~>  ( ( 1  /  ( 1  -  A ) )  -  0 ) )
8554subid1d 7879 . 2  |-  ( ph  ->  ( ( 1  / 
( 1  -  A
) )  -  0 )  =  ( 1  /  ( 1  -  A ) ) )
8684, 85breqtrd 3891 1  |-  ( ph  ->  seq 0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1296    e. wcel 1445   _Vcvv 2633   class class class wbr 3867    |-> cmpt 3921   ` cfv 5049  (class class class)co 5690   CCcc 7445   0cc0 7447   1c1 7448    + caddc 7450    x. cmul 7452    < clt 7619    - cmin 7750   # cap 8155    / cdiv 8236   NN0cn0 8771   ZZ>=cuz 9118   ...cfz 9573    seqcseq 10001   ^cexp 10085   abscabs 10561    ~~> cli 10837   sum_csu 10912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560  ax-arch 7561  ax-caucvg 7562
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-isom 5058  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-frec 6194  df-1o 6219  df-oadd 6223  df-er 6332  df-en 6538  df-dom 6539  df-fin 6540  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-n0 8772  df-z 8849  df-uz 9119  df-q 9204  df-rp 9234  df-fz 9574  df-fzo 9703  df-iseq 10002  df-seq3 10003  df-exp 10086  df-ihash 10315  df-cj 10407  df-re 10408  df-im 10409  df-rsqrt 10562  df-abs 10563  df-clim 10838  df-sumdc 10913
This theorem is referenced by:  geolim2  11071  georeclim  11072  geoisum  11076  eflegeo  11157
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