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Theorem geo2lim 12607
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 10477 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1z 10267 . . . 4  |-  1  e.  ZZ
32a1i 11 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
4 1re 9046 . . . . . . . . 9  |-  1  e.  RR
54rehalfcli 10172 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
65recni 9058 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
76a1i 11 . . . . . 6  |-  ( A  e.  CC  ->  (
1  /  2 )  e.  CC )
8 0re 9047 . . . . . . . . . 10  |-  0  e.  RR
9 halfgt0 10144 . . . . . . . . . 10  |-  0  <  ( 1  /  2
)
108, 5, 9ltleii 9152 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
11 absid 12056 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  0  <_  ( 1  / 
2 ) )  -> 
( abs `  (
1  /  2 ) )  =  ( 1  /  2 ) )
125, 10, 11mp2an 654 . . . . . . . 8  |-  ( abs `  ( 1  /  2
) )  =  ( 1  /  2 )
13 halflt1 10145 . . . . . . . 8  |-  ( 1  /  2 )  <  1
1412, 13eqbrtri 4191 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
1514a1i 11 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( 1  / 
2 ) )  <  1 )
167, 15expcnv 12598 . . . . 5  |-  ( A  e.  CC  ->  (
k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) )  ~~>  0 )
17 id 20 . . . . 5  |-  ( A  e.  CC  ->  A  e.  CC )
18 geo2lim.1 . . . . . . 7  |-  F  =  ( k  e.  NN  |->  ( A  /  (
2 ^ k ) ) )
19 nnex 9962 . . . . . . . 8  |-  NN  e.  _V
2019mptex 5925 . . . . . . 7  |-  ( k  e.  NN  |->  ( A  /  ( 2 ^ k ) ) )  e.  _V
2118, 20eqeltri 2474 . . . . . 6  |-  F  e. 
_V
2221a1i 11 . . . . 5  |-  ( A  e.  CC  ->  F  e.  _V )
23 nnnn0 10184 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  NN0 )
2423adantl 453 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  NN0 )
25 oveq2 6048 . . . . . . . . 9  |-  ( k  =  j  ->  (
( 1  /  2
) ^ k )  =  ( ( 1  /  2 ) ^
j ) )
26 eqid 2404 . . . . . . . . 9  |-  ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) )  =  ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) )
27 ovex 6065 . . . . . . . . 9  |-  ( ( 1  /  2 ) ^ j )  e. 
_V
2825, 26, 27fvmpt 5765 . . . . . . . 8  |-  ( j  e.  NN0  ->  ( ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) ) `
 j )  =  ( ( 1  / 
2 ) ^ j
) )
2924, 28syl 16 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( ( 1  /  2
) ^ j ) )
30 nnz 10259 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  ZZ )
3130adantl 453 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ZZ )
32 2cn 10026 . . . . . . . . 9  |-  2  e.  CC
33 2ne0 10039 . . . . . . . . 9  |-  2  =/=  0
34 exprec 11376 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  2  =/=  0  /\  j  e.  ZZ )  ->  (
( 1  /  2
) ^ j )  =  ( 1  / 
( 2 ^ j
) ) )
3532, 33, 34mp3an12 1269 . . . . . . . 8  |-  ( j  e.  ZZ  ->  (
( 1  /  2
) ^ j )  =  ( 1  / 
( 2 ^ j
) ) )
3631, 35syl 16 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( 1  / 
2 ) ^ j
)  =  ( 1  /  ( 2 ^ j ) ) )
3729, 36eqtrd 2436 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( 1  /  ( 2 ^ j ) ) )
38 2nn 10089 . . . . . . . . 9  |-  2  e.  NN
39 nnexpcl 11349 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  j  e.  NN0 )  -> 
( 2 ^ j
)  e.  NN )
4038, 24, 39sylancr 645 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  NN )
4140nnrecred 10001 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 1  /  (
2 ^ j ) )  e.  RR )
4241recnd 9070 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 1  /  (
2 ^ j ) )  e.  CC )
4337, 42eqeltrd 2478 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  e.  CC )
44 simpl 444 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  A  e.  CC )
4540nncnd 9972 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  CC )
4640nnne0d 10000 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  =/=  0 )
4744, 45, 46divrecd 9749 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  =  ( A  x.  ( 1  / 
( 2 ^ j
) ) ) )
48 oveq2 6048 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
4948oveq2d 6056 . . . . . . . 8  |-  ( k  =  j  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ j ) ) )
50 ovex 6065 . . . . . . . 8  |-  ( A  /  ( 2 ^ j ) )  e. 
_V
5149, 18, 50fvmpt 5765 . . . . . . 7  |-  ( j  e.  NN  ->  ( F `  j )  =  ( A  / 
( 2 ^ j
) ) )
5251adantl 453 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  =  ( A  /  ( 2 ^ j ) ) )
5337oveq2d 6056 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  x.  (
( k  e.  NN0  |->  ( ( 1  / 
2 ) ^ k
) ) `  j
) )  =  ( A  x.  ( 1  /  ( 2 ^ j ) ) ) )
5447, 52, 533eqtr4d 2446 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  =  ( A  x.  ( ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) ) `
 j ) ) )
551, 3, 16, 17, 22, 43, 54climmulc2 12385 . . . 4  |-  ( A  e.  CC  ->  F  ~~>  ( A  x.  0
) )
56 mul01 9201 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
5755, 56breqtrd 4196 . . 3  |-  ( A  e.  CC  ->  F  ~~>  0 )
58 seqex 11280 . . . 4  |-  seq  1
(  +  ,  F
)  e.  _V
5958a1i 11 . . 3  |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  e.  _V )
6044, 45, 46divcld 9746 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  e.  CC )
6152, 60eqeltrd 2478 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  e.  CC )
6252oveq2d 6056 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  -  ( F `  j )
)  =  ( A  -  ( A  / 
( 2 ^ j
) ) ) )
63 geo2sum 12605 . . . . 5  |-  ( ( j  e.  NN  /\  A  e.  CC )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  ( A  -  ( A  /  (
2 ^ j ) ) ) )
6463ancoms 440 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  ( A  -  ( A  /  (
2 ^ j ) ) ) )
65 elfznn 11036 . . . . . . 7  |-  ( n  e.  ( 1 ... j )  ->  n  e.  NN )
6665adantl 453 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  n  e.  NN )
67 oveq2 6048 . . . . . . . 8  |-  ( k  =  n  ->  (
2 ^ k )  =  ( 2 ^ n ) )
6867oveq2d 6056 . . . . . . 7  |-  ( k  =  n  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ n ) ) )
69 ovex 6065 . . . . . . 7  |-  ( A  /  ( 2 ^ n ) )  e. 
_V
7068, 18, 69fvmpt 5765 . . . . . 6  |-  ( n  e.  NN  ->  ( F `  n )  =  ( A  / 
( 2 ^ n
) ) )
7166, 70syl 16 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( F `  n )  =  ( A  /  ( 2 ^ n ) ) )
72 simpr 448 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  NN )
7372, 1syl6eleq 2494 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ( ZZ>= ` 
1 ) )
74 simpll 731 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  A  e.  CC )
75 nnnn0 10184 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  e.  NN0 )
76 nnexpcl 11349 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  n  e.  NN0 )  -> 
( 2 ^ n
)  e.  NN )
7738, 75, 76sylancr 645 . . . . . . . 8  |-  ( n  e.  NN  ->  (
2 ^ n )  e.  NN )
7866, 77syl 16 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( 2 ^ n )  e.  NN )
7978nncnd 9972 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( 2 ^ n )  e.  CC )
8078nnne0d 10000 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( 2 ^ n )  =/=  0 )
8174, 79, 80divcld 9746 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( A  /  ( 2 ^ n ) )  e.  CC )
8271, 73, 81fsumser 12479 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  (  seq  1
(  +  ,  F
) `  j )
)
8362, 64, 823eqtr2rd 2443 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  (  seq  1 (  +  ,  F ) `
 j )  =  ( A  -  ( F `  j )
) )
841, 3, 57, 17, 59, 61, 83climsubc2 12387 . 2  |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  ~~>  ( A  -  0 ) )
85 subid1 9278 . 2  |-  ( A  e.  CC  ->  ( A  -  0 )  =  A )
8684, 85breqtrd 4196 1  |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278   ^cexp 11337   abscabs 11994    ~~> cli 12233   sum_csu 12434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435
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