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Theorem geo2lim 10910
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 9054 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 8777 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
3 halfcn 8630 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
43a1i 9 . . . . . 6  |-  ( A  e.  CC  ->  (
1  /  2 )  e.  CC )
5 halfre 8629 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
6 halfge0 8632 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
7 absid 10504 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  0  <_  ( 1  / 
2 ) )  -> 
( abs `  (
1  /  2 ) )  =  ( 1  /  2 ) )
85, 6, 7mp2an 417 . . . . . . . 8  |-  ( abs `  ( 1  /  2
) )  =  ( 1  /  2 )
9 halflt1 8633 . . . . . . . 8  |-  ( 1  /  2 )  <  1
108, 9eqbrtri 3864 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
1110a1i 9 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( 1  / 
2 ) )  <  1 )
124, 11expcnv 10898 . . . . 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 8428 . . . . . . . 8  |-  NN  e.  _V
1615mptex 5523 . . . . . . 7  |-  ( k  e.  NN  |->  ( A  /  ( 2 ^ k ) ) )  e.  _V
1714, 16eqeltri 2160 . . . . . 6  |-  F  e. 
_V
1817a1i 9 . . . . 5  |-  ( A  e.  CC  ->  F  e.  _V )
19 nnnn0 8680 . . . . . . . 8  |-  ( j  e.  NN  ->  j  e.  NN0 )
2019adantl 271 . . . . . . 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 10086 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( 1  / 
2 ) ^ j
)  e.  CC )
23 oveq2 5660 . . . . . . . 8  |-  ( k  =  j  ->  (
( 1  /  2
) ^ k )  =  ( ( 1  /  2 ) ^
j ) )
24 eqid 2088 . . . . . . . 8  |-  ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) )  =  ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) )
2523, 24fvmptg 5380 . . . . . . 7  |-  ( ( j  e.  NN0  /\  ( ( 1  / 
2 ) ^ j
)  e.  CC )  ->  ( ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) ) `
 j )  =  ( ( 1  / 
2 ) ^ j
) )
2620, 22, 25syl2anc 403 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( ( 1  /  2
) ^ j ) )
2726, 22eqeltrd 2164 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  e.  CC )
28 simpl 107 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  A  e.  CC )
29 2nn 8577 . . . . . . . . 9  |-  2  e.  NN
30 nnexpcl 9968 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  j  e.  NN0 )  -> 
( 2 ^ j
)  e.  NN )
3129, 20, 30sylancr 405 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  NN )
3231nncnd 8436 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  CC )
3331nnap0d 8468 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
) #  0 )
3428, 32, 33divrecapd 8260 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  =  ( A  x.  ( 1  / 
( 2 ^ j
) ) ) )
35 simpr 108 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  NN )
3628, 32, 33divclapd 8257 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  e.  CC )
37 oveq2 5660 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
3837oveq2d 5668 . . . . . . . 8  |-  ( k  =  j  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ j ) ) )
3938, 14fvmptg 5380 . . . . . . 7  |-  ( ( j  e.  NN  /\  ( A  /  (
2 ^ j ) )  e.  CC )  ->  ( F `  j )  =  ( A  /  ( 2 ^ j ) ) )
4035, 36, 39syl2anc 403 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  =  ( A  /  ( 2 ^ j ) ) )
41 2cn 8493 . . . . . . . . 9  |-  2  e.  CC
42 2ap0 8515 . . . . . . . . 9  |-  2 #  0
43 nnz 8769 . . . . . . . . . 10  |-  ( j  e.  NN  ->  j  e.  ZZ )
4443adantl 271 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ZZ )
45 exprecap 9996 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  2 #  0  /\  j  e.  ZZ )  ->  (
( 1  /  2
) ^ j )  =  ( 1  / 
( 2 ^ j
) ) )
4641, 42, 44, 45mp3an12i 1277 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( 1  / 
2 ) ^ j
)  =  ( 1  /  ( 2 ^ j ) ) )
4726, 46eqtrd 2120 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( 1  /  ( 2 ^ j ) ) )
4847oveq2d 5668 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  x.  (
( k  e.  NN0  |->  ( ( 1  / 
2 ) ^ k
) ) `  j
) )  =  ( A  x.  ( 1  /  ( 2 ^ j ) ) ) )
4934, 40, 483eqtr4d 2130 . . . . 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 10719 . . . 4  |-  ( A  e.  CC  ->  F  ~~>  ( A  x.  0
) )
51 mul01 7867 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
5250, 51breqtrd 3869 . . 3  |-  ( A  e.  CC  ->  F  ~~>  0 )
53 seqex 9857 . . . 4  |-  seq 1
(  +  ,  F
)  e.  _V
5453a1i 9 . . 3  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  e.  _V )
5540, 36eqeltrd 2164 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  e.  CC )
5640oveq2d 5668 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  -  ( F `  j )
)  =  ( A  -  ( A  / 
( 2 ^ j
) ) ) )
57 geo2sum 10908 . . . . 5  |-  ( ( j  e.  NN  /\  A  e.  CC )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  ( A  -  ( A  /  (
2 ^ j ) ) ) )
5857ancoms 264 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  ( A  -  ( A  /  (
2 ^ j ) ) ) )
59 elnnuz 9055 . . . . . . . 8  |-  ( n  e.  NN  <->  n  e.  ( ZZ>= `  1 )
)
6059biimpri 131 . . . . . . 7  |-  ( n  e.  ( ZZ>= `  1
)  ->  n  e.  NN )
6160adantl 271 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  n  e.  NN )
62 simpll 496 . . . . . . 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 8726 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  n  e.  NN0 )
6563, 64expcld 10086 . . . . . . 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 8867 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  n  e.  ZZ )
6863, 66, 67expap0d 10092 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( 2 ^ n ) #  0 )
6962, 65, 68divclapd 8257 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( A  / 
( 2 ^ n
) )  e.  CC )
70 oveq2 5660 . . . . . . . 8  |-  ( k  =  n  ->  (
2 ^ k )  =  ( 2 ^ n ) )
7170oveq2d 5668 . . . . . . 7  |-  ( k  =  n  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ n ) ) )
7271, 14fvmptg 5380 . . . . . 6  |-  ( ( n  e.  NN  /\  ( A  /  (
2 ^ n ) )  e.  CC )  ->  ( F `  n )  =  ( A  /  ( 2 ^ n ) ) )
7361, 69, 72syl2anc 403 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( F `  n )  =  ( A  /  ( 2 ^ n ) ) )
7435, 1syl6eleq 2180 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ( ZZ>= ` 
1 ) )
7573, 74, 69fsum3ser 10791 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  (  seq 1
(  +  ,  F
) `  j )
)
7656, 58, 753eqtr2rd 2127 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 j )  =  ( A  -  ( F `  j )
) )
771, 2, 52, 13, 54, 55, 76climsubc2 10721 . 2  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  ~~>  ( A  -  0 ) )
78 subid1 7702 . 2  |-  ( A  e.  CC  ->  ( A  -  0 )  =  A )
7977, 78breqtrd 3869 1  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619   class class class wbr 3845    |-> cmpt 3899   ` cfv 5015  (class class class)co 5652   CCcc 7348   RRcr 7349   0cc0 7350   1c1 7351    + caddc 7353    x. cmul 7355    < clt 7522    <_ cle 7523    - cmin 7653   # cap 8058    / cdiv 8139   NNcn 8422   2c2 8473   NN0cn0 8673   ZZcz 8750   ZZ>=cuz 9019   ...cfz 9424    seqcseq 9852   ^cexp 9954   abscabs 10430    ~~> cli 10666   sum_csu 10742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463  ax-arch 7464  ax-caucvg 7465
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6292  df-en 6458  df-dom 6459  df-fin 6460  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-n0 8674  df-z 8751  df-uz 9020  df-q 9105  df-rp 9135  df-fz 9425  df-fzo 9554  df-iseq 9853  df-seq3 9854  df-exp 9955  df-ihash 10184  df-cj 10276  df-re 10277  df-im 10278  df-rsqrt 10431  df-abs 10432  df-clim 10667  df-isum 10743
This theorem is referenced by: (None)
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