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Theorem geolim 11461
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 9508 . . 3  |-  NN0  =  ( ZZ>= `  0 )
2 0zd 9211 . . 3  |-  ( ph  ->  0  e.  ZZ )
3 geolim.1 . . . . . 6  |-  ( ph  ->  A  e.  CC )
4 geolim.2 . . . . . 6  |-  ( ph  ->  ( abs `  A
)  <  1 )
53, 4expcnv 11454 . . . . 5  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
6 ax-1cn 7854 . . . . . . 7  |-  1  e.  CC
7 subcl 8105 . . . . . . 7  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  -  A
)  e.  CC )
86, 3, 7sylancr 412 . . . . . 6  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
9 1cnd 7923 . . . . . . 7  |-  ( ph  ->  1  e.  CC )
10 1red 7922 . . . . . . . . 9  |-  ( ph  ->  1  e.  RR )
113, 10, 4absltap 11459 . . . . . . . 8  |-  ( ph  ->  A #  1 )
12 apsym 8512 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( A #  1  <->  1 #  A ) )
133, 6, 12sylancl 411 . . . . . . . 8  |-  ( ph  ->  ( A #  1  <->  1 #  A ) )
1411, 13mpbid 146 . . . . . . 7  |-  ( ph  ->  1 #  A )
159, 3, 14subap0d 8550 . . . . . 6  |-  ( ph  ->  ( 1  -  A
) #  0 )
163, 8, 15divclapd 8694 . . . . 5  |-  ( ph  ->  ( A  /  (
1  -  A ) )  e.  CC )
17 nn0ex 9128 . . . . . . 7  |-  NN0  e.  _V
1817mptex 5719 . . . . . 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 10596 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ j )  e.  CC )
23 oveq2 5858 . . . . . . . 8  |-  ( n  =  j  ->  ( A ^ n )  =  ( A ^ j
) )
24 eqid 2170 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
2523, 24fvmptg 5570 . . . . . . 7  |-  ( ( j  e.  NN0  /\  ( A ^ j )  e.  CC )  -> 
( ( n  e. 
NN0  |->  ( A ^
n ) ) `  j )  =  ( A ^ j ) )
2620, 22, 25syl2anc 409 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  =  ( A ^ j
) )
27 expcl 10481 . . . . . . 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 2247 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( A ^ n ) ) `
 j )  e.  CC )
30 expp1 10470 . . . . . . . . . 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 7928 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ j )  x.  A )  =  ( A  x.  ( A ^ j ) ) )
3331, 32eqtrd 2203 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  =  ( A  x.  ( A ^ j ) ) )
3433oveq1d 5865 . . . . . . 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 8732 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  x.  ( A ^ j ) )  /  ( 1  -  A ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
3834, 37eqtrd 2203 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  =  ( ( A  /  (
1  -  A ) )  x.  ( A ^ j ) ) )
39 peano2nn0 9162 . . . . . . . . . 10  |-  ( j  e.  NN0  ->  ( j  +  1 )  e. 
NN0 )
4039adantl 275 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( j  +  1 )  e. 
NN0 )
4121, 40expcld 10596 . . . . . . . 8  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( A ^ ( j  +  1 ) )  e.  CC )
4241, 35, 36divclapd 8694 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  e.  CC )
43 oveq1 5857 . . . . . . . . . 10  |-  ( n  =  j  ->  (
n  +  1 )  =  ( j  +  1 ) )
4443oveq2d 5866 . . . . . . . . 9  |-  ( n  =  j  ->  ( A ^ ( n  + 
1 ) )  =  ( A ^ (
j  +  1 ) ) )
4544oveq1d 5865 . . . . . . . 8  |-  ( n  =  j  ->  (
( A ^ (
n  +  1 ) )  /  ( 1  -  A ) )  =  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) ) )
46 eqid 2170 . . . . . . . 8  |-  ( n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) )  =  ( n  e. 
NN0  |->  ( ( A ^ ( n  + 
1 ) )  / 
( 1  -  A
) ) )
4745, 46fvmptg 5570 . . . . . . 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 409 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  =  ( ( A ^
( j  +  1 ) )  /  (
1  -  A ) ) )
4926oveq2d 5866 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A  /  ( 1  -  A ) )  x.  ( ( n  e. 
NN0  |->  ( A ^
n ) ) `  j ) )  =  ( ( A  / 
( 1  -  A
) )  x.  ( A ^ j ) ) )
5038, 48, 493eqtr4d 2213 . . . . 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 11281 . . . 4  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  ( ( A  /  ( 1  -  A ) )  x.  0 ) )
5216mul01d 8299 . . . 4  |-  ( ph  ->  ( ( A  / 
( 1  -  A
) )  x.  0 )  =  0 )
5351, 52breqtrd 4013 . . 3  |-  ( ph  ->  ( n  e.  NN0  |->  ( ( A ^
( n  +  1 ) )  /  (
1  -  A ) ) )  ~~>  0 )
548, 15recclapd 8685 . . 3  |-  ( ph  ->  ( 1  /  (
1  -  A ) )  e.  CC )
55 seqex 10390 . . . 4  |-  seq 0
(  +  ,  F
)  e.  _V
5655a1i 9 . . 3  |-  ( ph  ->  seq 0 (  +  ,  F )  e. 
_V )
57 expcl 10481 . . . . . 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 8694 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( ( A ^ ( j  +  1 ) )  / 
( 1  -  A
) )  e.  CC )
6048, 59eqeltrd 2247 . . 3  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
n  e.  NN0  |->  ( ( A ^ ( n  +  1 ) )  /  ( 1  -  A ) ) ) `
 j )  e.  CC )
61 nn0cn 9132 . . . . . . . 8  |-  ( j  e.  NN0  ->  j  e.  CC )
6261adantl 275 . . . . . . 7  |-  ( (
ph  /\  j  e.  NN0 )  ->  j  e.  CC )
63 pncan 8112 . . . . . . 7  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  +  1 )  -  1 )  =  j )
6462, 6, 63sylancl 411 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( (
j  +  1 )  -  1 )  =  j )
6564oveq2d 5866 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( 0 ... ( ( j  +  1 )  - 
1 ) )  =  ( 0 ... j
) )
6665sumeq1d 11316 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  sum_ k  e.  ( 0 ... j ) ( A ^ k ) )
67 1cnd 7923 . . . . . 6  |-  ( (
ph  /\  j  e.  NN0 )  ->  1  e.  CC )
6867, 58, 35, 36divsubdirapd 8734 . . . . 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 11457 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... (
( j  +  1 )  -  1 ) ) ( A ^
k )  =  ( ( 1  -  ( A ^ ( j  +  1 ) ) )  /  ( 1  -  A ) ) )
7148oveq2d 5866 . . . . 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 2213 . . . 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 524 . . . . . 6  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>= `  0 )
)  ->  ph )
74 elnn0uz 9511 . . . . . . . 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 409 . . . . 5  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( F `  k )  =  ( A ^ k ) )
7920, 1eleqtrdi 2263 . . . . 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 10596 . . . . 5  |-  ( ( ( ph  /\  j  e.  NN0 )  /\  k  e.  ( ZZ>= `  0 )
)  ->  ( A ^ k )  e.  CC )
8278, 79, 81fsum3ser 11347 . . . 4  |-  ( (
ph  /\  j  e.  NN0 )  ->  sum_ k  e.  ( 0 ... j
) ( A ^
k )  =  (  seq 0 (  +  ,  F ) `  j ) )
8366, 72, 823eqtr3rd 2212 . . 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 11283 . 2  |-  ( ph  ->  seq 0 (  +  ,  F )  ~~>  ( ( 1  /  ( 1  -  A ) )  -  0 ) )
8554subid1d 8206 . 2  |-  ( ph  ->  ( ( 1  / 
( 1  -  A
) )  -  0 )  =  ( 1  /  ( 1  -  A ) ) )
8684, 85breqtrd 4013 1  |-  ( ph  ->  seq 0 (  +  ,  F )  ~~>  ( 1  /  ( 1  -  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   _Vcvv 2730   class class class wbr 3987    |-> cmpt 4048   ` cfv 5196  (class class class)co 5850   CCcc 7759   0cc0 7761   1c1 7762    + caddc 7764    x. cmul 7766    < clt 7941    - cmin 8077   # cap 8487    / cdiv 8576   NN0cn0 9122   ZZ>=cuz 9474   ...cfz 9952    seqcseq 10388   ^cexp 10462   abscabs 10948    ~~> cli 11228   sum_csu 11303
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-frec 6367  df-1o 6392  df-oadd 6396  df-er 6509  df-en 6715  df-dom 6716  df-fin 6717  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-fz 9953  df-fzo 10086  df-seqfrec 10389  df-exp 10463  df-ihash 10697  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950  df-clim 11229  df-sumdc 11304
This theorem is referenced by:  geolim2  11462  georeclim  11463  geoisum  11467  eflegeo  11651
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