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Theorem geo2lim 12331
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 10263 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1z 10053 . . . 4  |-  1  e.  ZZ
32a1i 10 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
4 1re 8837 . . . . . . . . 9  |-  1  e.  RR
5 rehalfcl 9938 . . . . . . . . 9  |-  ( 1  e.  RR  ->  (
1  /  2 )  e.  RR )
64, 5ax-mp 8 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
76recni 8849 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
87a1i 10 . . . . . 6  |-  ( A  e.  CC  ->  (
1  /  2 )  e.  CC )
9 0re 8838 . . . . . . . . . 10  |-  0  e.  RR
10 halfgt0 9932 . . . . . . . . . 10  |-  0  <  ( 1  /  2
)
119, 6, 10ltleii 8941 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
12 absid 11781 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  0  <_  ( 1  / 
2 ) )  -> 
( abs `  (
1  /  2 ) )  =  ( 1  /  2 ) )
136, 11, 12mp2an 653 . . . . . . . 8  |-  ( abs `  ( 1  /  2
) )  =  ( 1  /  2 )
14 halflt1 9933 . . . . . . . 8  |-  ( 1  /  2 )  <  1
1513, 14eqbrtri 4042 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
1615a1i 10 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( 1  / 
2 ) )  <  1 )
178, 16expcnv 12322 . . . . 5  |-  ( A  e.  CC  ->  (
k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) )  ~~>  0 )
18 id 19 . . . . 5  |-  ( A  e.  CC  ->  A  e.  CC )
19 geo2lim.1 . . . . . . 7  |-  F  =  ( k  e.  NN  |->  ( A  /  (
2 ^ k ) ) )
20 nnex 9752 . . . . . . . 8  |-  NN  e.  _V
2120mptex 5746 . . . . . . 7  |-  ( k  e.  NN  |->  ( A  /  ( 2 ^ k ) ) )  e.  _V
2219, 21eqeltri 2353 . . . . . 6  |-  F  e. 
_V
2322a1i 10 . . . . 5  |-  ( A  e.  CC  ->  F  e.  _V )
24 nnnn0 9972 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  NN0 )
2524adantl 452 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  NN0 )
26 oveq2 5866 . . . . . . . . 9  |-  ( k  =  j  ->  (
( 1  /  2
) ^ k )  =  ( ( 1  /  2 ) ^
j ) )
27 eqid 2283 . . . . . . . . 9  |-  ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) )  =  ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) )
28 ovex 5883 . . . . . . . . 9  |-  ( ( 1  /  2 ) ^ j )  e. 
_V
2926, 27, 28fvmpt 5602 . . . . . . . 8  |-  ( j  e.  NN0  ->  ( ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) ) `
 j )  =  ( ( 1  / 
2 ) ^ j
) )
3025, 29syl 15 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( ( 1  /  2
) ^ j ) )
31 nnz 10045 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  ZZ )
3231adantl 452 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ZZ )
33 2cn 9816 . . . . . . . . 9  |-  2  e.  CC
34 2ne0 9829 . . . . . . . . 9  |-  2  =/=  0
35 exprec 11143 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  2  =/=  0  /\  j  e.  ZZ )  ->  (
( 1  /  2
) ^ j )  =  ( 1  / 
( 2 ^ j
) ) )
3633, 34, 35mp3an12 1267 . . . . . . . 8  |-  ( j  e.  ZZ  ->  (
( 1  /  2
) ^ j )  =  ( 1  / 
( 2 ^ j
) ) )
3732, 36syl 15 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( 1  / 
2 ) ^ j
)  =  ( 1  /  ( 2 ^ j ) ) )
3830, 37eqtrd 2315 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( 1  /  ( 2 ^ j ) ) )
39 2nn 9877 . . . . . . . . 9  |-  2  e.  NN
40 nnexpcl 11116 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  j  e.  NN0 )  -> 
( 2 ^ j
)  e.  NN )
4139, 25, 40sylancr 644 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  NN )
4241nnrecred 9791 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 1  /  (
2 ^ j ) )  e.  RR )
4342recnd 8861 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 1  /  (
2 ^ j ) )  e.  CC )
4438, 43eqeltrd 2357 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  e.  CC )
45 simpl 443 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  A  e.  CC )
4641nncnd 9762 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  CC )
4741nnne0d 9790 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  =/=  0 )
4845, 46, 47divrecd 9539 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  =  ( A  x.  ( 1  / 
( 2 ^ j
) ) ) )
49 oveq2 5866 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
5049oveq2d 5874 . . . . . . . 8  |-  ( k  =  j  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ j ) ) )
51 ovex 5883 . . . . . . . 8  |-  ( A  /  ( 2 ^ j ) )  e. 
_V
5250, 19, 51fvmpt 5602 . . . . . . 7  |-  ( j  e.  NN  ->  ( F `  j )  =  ( A  / 
( 2 ^ j
) ) )
5352adantl 452 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  =  ( A  /  ( 2 ^ j ) ) )
5438oveq2d 5874 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  x.  (
( k  e.  NN0  |->  ( ( 1  / 
2 ) ^ k
) ) `  j
) )  =  ( A  x.  ( 1  /  ( 2 ^ j ) ) ) )
5548, 53, 543eqtr4d 2325 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  =  ( A  x.  ( ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) ) `
 j ) ) )
561, 3, 17, 18, 23, 44, 55climmulc2 12110 . . . 4  |-  ( A  e.  CC  ->  F  ~~>  ( A  x.  0
) )
57 mul01 8991 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
5856, 57breqtrd 4047 . . 3  |-  ( A  e.  CC  ->  F  ~~>  0 )
59 seqex 11048 . . . 4  |-  seq  1
(  +  ,  F
)  e.  _V
6059a1i 10 . . 3  |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  e.  _V )
6145, 46, 47divcld 9536 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  e.  CC )
6253, 61eqeltrd 2357 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  e.  CC )
6353oveq2d 5874 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  -  ( F `  j )
)  =  ( A  -  ( A  / 
( 2 ^ j
) ) ) )
64 geo2sum 12329 . . . . 5  |-  ( ( j  e.  NN  /\  A  e.  CC )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  ( A  -  ( A  /  (
2 ^ j ) ) ) )
6564ancoms 439 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  ( A  -  ( A  /  (
2 ^ j ) ) ) )
66 elfznn 10819 . . . . . . 7  |-  ( n  e.  ( 1 ... j )  ->  n  e.  NN )
6766adantl 452 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  n  e.  NN )
68 oveq2 5866 . . . . . . . 8  |-  ( k  =  n  ->  (
2 ^ k )  =  ( 2 ^ n ) )
6968oveq2d 5874 . . . . . . 7  |-  ( k  =  n  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ n ) ) )
70 ovex 5883 . . . . . . 7  |-  ( A  /  ( 2 ^ n ) )  e. 
_V
7169, 19, 70fvmpt 5602 . . . . . 6  |-  ( n  e.  NN  ->  ( F `  n )  =  ( A  / 
( 2 ^ n
) ) )
7267, 71syl 15 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( F `  n )  =  ( A  /  ( 2 ^ n ) ) )
73 simpr 447 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  NN )
7473, 1syl6eleq 2373 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ( ZZ>= ` 
1 ) )
75 simpll 730 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  A  e.  CC )
76 nnnn0 9972 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  e.  NN0 )
77 nnexpcl 11116 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  n  e.  NN0 )  -> 
( 2 ^ n
)  e.  NN )
7839, 76, 77sylancr 644 . . . . . . . 8  |-  ( n  e.  NN  ->  (
2 ^ n )  e.  NN )
7967, 78syl 15 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( 2 ^ n )  e.  NN )
8079nncnd 9762 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( 2 ^ n )  e.  CC )
8179nnne0d 9790 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( 2 ^ n )  =/=  0 )
8275, 80, 81divcld 9536 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( A  /  ( 2 ^ n ) )  e.  CC )
8372, 74, 82fsumser 12203 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  (  seq  1
(  +  ,  F
) `  j )
)
8463, 65, 833eqtr2rd 2322 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  (  seq  1 (  +  ,  F ) `
 j )  =  ( A  -  ( F `  j )
) )
851, 3, 58, 18, 60, 62, 84climsubc2 12112 . 2  |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  ~~>  ( A  -  0 ) )
86 subid1 9068 . 2  |-  ( A  e.  CC  ->  ( A  -  0 )  =  A )
8785, 86breqtrd 4047 1  |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046   ^cexp 11104   abscabs 11719    ~~> cli 11958   sum_csu 12158
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159
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