ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  geo2lim Unicode version

Theorem geo2lim 11278
Description: The value of the infinite geometric series  2 ^ -u 1  +  2 ^ -u 2  +... , multiplied by a constant. (Contributed by Mario Carneiro, 15-Jun-2014.)
Hypothesis
Ref Expression
geo2lim.1  |-  F  =  ( k  e.  NN  |->  ( A  /  (
2 ^ k ) ) )
Assertion
Ref Expression
geo2lim  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  ~~>  A )
Distinct variable group:    A, k
Allowed substitution hint:    F( k)

Proof of Theorem geo2lim
Dummy variables  j  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnuz 9354 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 9074 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
3 halfcn 8927 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
43a1i 9 . . . . . 6  |-  ( A  e.  CC  ->  (
1  /  2 )  e.  CC )
5 halfre 8926 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
6 halfge0 8929 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
7 absid 10836 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  0  <_  ( 1  / 
2 ) )  -> 
( abs `  (
1  /  2 ) )  =  ( 1  /  2 ) )
85, 6, 7mp2an 422 . . . . . . . 8  |-  ( abs `  ( 1  /  2
) )  =  ( 1  /  2 )
9 halflt1 8930 . . . . . . . 8  |-  ( 1  /  2 )  <  1
108, 9eqbrtri 3944 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
1110a1i 9 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( 1  / 
2 ) )  <  1 )
124, 11expcnv 11266 . . . . 5  |-  ( A  e.  CC  ->  (
k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) )  ~~>  0 )
13 id 19 . . . . 5  |-  ( A  e.  CC  ->  A  e.  CC )
14 geo2lim.1 . . . . . . 7  |-  F  =  ( k  e.  NN  |->  ( A  /  (
2 ^ k ) ) )
15 nnex 8719 . . . . . . . 8  |-  NN  e.  _V
1615mptex 5639 . . . . . . 7  |-  ( k  e.  NN  |->  ( A  /  ( 2 ^ k ) ) )  e.  _V
1714, 16eqeltri 2210 . . . . . 6  |-  F  e. 
_V
1817a1i 9 . . . . 5  |-  ( A  e.  CC  ->  F  e.  _V )
19 nnnn0 8977 . . . . . . . 8  |-  ( j  e.  NN  ->  j  e.  NN0 )
2019adantl 275 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  NN0 )
213a1i 9 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 1  /  2
)  e.  CC )
2221, 20expcld 10417 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( 1  / 
2 ) ^ j
)  e.  CC )
23 oveq2 5775 . . . . . . . 8  |-  ( k  =  j  ->  (
( 1  /  2
) ^ k )  =  ( ( 1  /  2 ) ^
j ) )
24 eqid 2137 . . . . . . . 8  |-  ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) )  =  ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) )
2523, 24fvmptg 5490 . . . . . . 7  |-  ( ( j  e.  NN0  /\  ( ( 1  / 
2 ) ^ j
)  e.  CC )  ->  ( ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) ) `
 j )  =  ( ( 1  / 
2 ) ^ j
) )
2620, 22, 25syl2anc 408 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( ( 1  /  2
) ^ j ) )
2726, 22eqeltrd 2214 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  e.  CC )
28 simpl 108 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  A  e.  CC )
29 2nn 8874 . . . . . . . . 9  |-  2  e.  NN
30 nnexpcl 10299 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  j  e.  NN0 )  -> 
( 2 ^ j
)  e.  NN )
3129, 20, 30sylancr 410 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  NN )
3231nncnd 8727 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  CC )
3331nnap0d 8759 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
) #  0 )
3428, 32, 33divrecapd 8546 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  =  ( A  x.  ( 1  / 
( 2 ^ j
) ) ) )
35 simpr 109 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  NN )
3628, 32, 33divclapd 8543 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  e.  CC )
37 oveq2 5775 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
3837oveq2d 5783 . . . . . . . 8  |-  ( k  =  j  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ j ) ) )
3938, 14fvmptg 5490 . . . . . . 7  |-  ( ( j  e.  NN  /\  ( A  /  (
2 ^ j ) )  e.  CC )  ->  ( F `  j )  =  ( A  /  ( 2 ^ j ) ) )
4035, 36, 39syl2anc 408 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  =  ( A  /  ( 2 ^ j ) ) )
41 2cn 8784 . . . . . . . . 9  |-  2  e.  CC
42 2ap0 8806 . . . . . . . . 9  |-  2 #  0
43 nnz 9066 . . . . . . . . . 10  |-  ( j  e.  NN  ->  j  e.  ZZ )
4443adantl 275 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ZZ )
45 exprecap 10327 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  2 #  0  /\  j  e.  ZZ )  ->  (
( 1  /  2
) ^ j )  =  ( 1  / 
( 2 ^ j
) ) )
4641, 42, 44, 45mp3an12i 1319 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( 1  / 
2 ) ^ j
)  =  ( 1  /  ( 2 ^ j ) ) )
4726, 46eqtrd 2170 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( 1  /  ( 2 ^ j ) ) )
4847oveq2d 5783 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  x.  (
( k  e.  NN0  |->  ( ( 1  / 
2 ) ^ k
) ) `  j
) )  =  ( A  x.  ( 1  /  ( 2 ^ j ) ) ) )
4934, 40, 483eqtr4d 2180 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  =  ( A  x.  ( ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) ) `
 j ) ) )
501, 2, 12, 13, 18, 27, 49climmulc2 11093 . . . 4  |-  ( A  e.  CC  ->  F  ~~>  ( A  x.  0
) )
51 mul01 8144 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
5250, 51breqtrd 3949 . . 3  |-  ( A  e.  CC  ->  F  ~~>  0 )
53 seqex 10213 . . . 4  |-  seq 1
(  +  ,  F
)  e.  _V
5453a1i 9 . . 3  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  e.  _V )
5540, 36eqeltrd 2214 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  e.  CC )
5640oveq2d 5783 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  -  ( F `  j )
)  =  ( A  -  ( A  / 
( 2 ^ j
) ) ) )
57 geo2sum 11276 . . . . 5  |-  ( ( j  e.  NN  /\  A  e.  CC )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  ( A  -  ( A  /  (
2 ^ j ) ) ) )
5857ancoms 266 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  ( A  -  ( A  /  (
2 ^ j ) ) ) )
59 elnnuz 9355 . . . . . . . 8  |-  ( n  e.  NN  <->  n  e.  ( ZZ>= `  1 )
)
6059biimpri 132 . . . . . . 7  |-  ( n  e.  ( ZZ>= `  1
)  ->  n  e.  NN )
6160adantl 275 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  n  e.  NN )
62 simpll 518 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  A  e.  CC )
6341a1i 9 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  2  e.  CC )
6461nnnn0d 9023 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  n  e.  NN0 )
6563, 64expcld 10417 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( 2 ^ n )  e.  CC )
6642a1i 9 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  2 #  0 )
6761nnzd 9165 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  n  e.  ZZ )
6863, 66, 67expap0d 10423 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( 2 ^ n ) #  0 )
6962, 65, 68divclapd 8543 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( A  / 
( 2 ^ n
) )  e.  CC )
70 oveq2 5775 . . . . . . . 8  |-  ( k  =  n  ->  (
2 ^ k )  =  ( 2 ^ n ) )
7170oveq2d 5783 . . . . . . 7  |-  ( k  =  n  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ n ) ) )
7271, 14fvmptg 5490 . . . . . 6  |-  ( ( n  e.  NN  /\  ( A  /  (
2 ^ n ) )  e.  CC )  ->  ( F `  n )  =  ( A  /  ( 2 ^ n ) ) )
7361, 69, 72syl2anc 408 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( F `  n )  =  ( A  /  ( 2 ^ n ) ) )
7435, 1eleqtrdi 2230 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ( ZZ>= ` 
1 ) )
7573, 74, 69fsum3ser 11159 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  (  seq 1
(  +  ,  F
) `  j )
)
7656, 58, 753eqtr2rd 2177 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 j )  =  ( A  -  ( F `  j )
) )
771, 2, 52, 13, 54, 55, 76climsubc2 11095 . 2  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  ~~>  ( A  -  0 ) )
78 subid1 7975 . 2  |-  ( A  e.  CC  ->  ( A  -  0 )  =  A )
7977, 78breqtrd 3949 1  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2681   class class class wbr 3924    |-> cmpt 3984   ` cfv 5118  (class class class)co 5767   CCcc 7611   RRcr 7612   0cc0 7613   1c1 7614    + caddc 7616    x. cmul 7618    < clt 7793    <_ cle 7794    - cmin 7926   # cap 8336    / cdiv 8425   NNcn 8713   2c2 8764   NN0cn0 8970   ZZcz 9047   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:  trilpolemeq1  13222
  Copyright terms: Public domain W3C validator