MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  geo2lim Structured version   Unicode version

Theorem geo2lim 12652
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 10521 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1z 10311 . . . 4  |-  1  e.  ZZ
32a1i 11 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
4 1re 9090 . . . . . . . . 9  |-  1  e.  RR
54rehalfcli 10216 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
65recni 9102 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
76a1i 11 . . . . . 6  |-  ( A  e.  CC  ->  (
1  /  2 )  e.  CC )
8 0re 9091 . . . . . . . . . 10  |-  0  e.  RR
9 halfgt0 10188 . . . . . . . . . 10  |-  0  <  ( 1  /  2
)
108, 5, 9ltleii 9196 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
11 absid 12101 . . . . . . . . 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 10189 . . . . . . . 8  |-  ( 1  /  2 )  <  1
1412, 13eqbrtri 4231 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
1514a1i 11 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( 1  / 
2 ) )  <  1 )
167, 15expcnv 12643 . . . . 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 10006 . . . . . . . 8  |-  NN  e.  _V
2019mptex 5966 . . . . . . 7  |-  ( k  e.  NN  |->  ( A  /  ( 2 ^ k ) ) )  e.  _V
2118, 20eqeltri 2506 . . . . . 6  |-  F  e. 
_V
2221a1i 11 . . . . 5  |-  ( A  e.  CC  ->  F  e.  _V )
23 nnnn0 10228 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  NN0 )
2423adantl 453 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  NN0 )
25 oveq2 6089 . . . . . . . . 9  |-  ( k  =  j  ->  (
( 1  /  2
) ^ k )  =  ( ( 1  /  2 ) ^
j ) )
26 eqid 2436 . . . . . . . . 9  |-  ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) )  =  ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) )
27 ovex 6106 . . . . . . . . 9  |-  ( ( 1  /  2 ) ^ j )  e. 
_V
2825, 26, 27fvmpt 5806 . . . . . . . 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 10303 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  ZZ )
3130adantl 453 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ZZ )
32 2cn 10070 . . . . . . . . 9  |-  2  e.  CC
33 2ne0 10083 . . . . . . . . 9  |-  2  =/=  0
34 exprec 11421 . . . . . . . . 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 2468 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( 1  /  ( 2 ^ j ) ) )
38 2nn 10133 . . . . . . . . 9  |-  2  e.  NN
39 nnexpcl 11394 . . . . . . . . 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 10045 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 1  /  (
2 ^ j ) )  e.  RR )
4241recnd 9114 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 1  /  (
2 ^ j ) )  e.  CC )
4337, 42eqeltrd 2510 . . . . 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 10016 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  CC )
4640nnne0d 10044 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  =/=  0 )
4744, 45, 46divrecd 9793 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  =  ( A  x.  ( 1  / 
( 2 ^ j
) ) ) )
48 oveq2 6089 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
4948oveq2d 6097 . . . . . . . 8  |-  ( k  =  j  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ j ) ) )
50 ovex 6106 . . . . . . . 8  |-  ( A  /  ( 2 ^ j ) )  e. 
_V
5149, 18, 50fvmpt 5806 . . . . . . 7  |-  ( j  e.  NN  ->  ( F `  j )  =  ( A  / 
( 2 ^ j
) ) )
5251adantl 453 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  =  ( A  /  ( 2 ^ j ) ) )
5337oveq2d 6097 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  x.  (
( k  e.  NN0  |->  ( ( 1  / 
2 ) ^ k
) ) `  j
) )  =  ( A  x.  ( 1  /  ( 2 ^ j ) ) ) )
5447, 52, 533eqtr4d 2478 . . . . 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 12430 . . . 4  |-  ( A  e.  CC  ->  F  ~~>  ( A  x.  0
) )
56 mul01 9245 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
5755, 56breqtrd 4236 . . 3  |-  ( A  e.  CC  ->  F  ~~>  0 )
58 seqex 11325 . . . 4  |-  seq  1
(  +  ,  F
)  e.  _V
5958a1i 11 . . 3  |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  e.  _V )
6044, 45, 46divcld 9790 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  e.  CC )
6152, 60eqeltrd 2510 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  e.  CC )
6252oveq2d 6097 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  -  ( F `  j )
)  =  ( A  -  ( A  / 
( 2 ^ j
) ) ) )
63 geo2sum 12650 . . . . 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 11080 . . . . . . 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 6089 . . . . . . . 8  |-  ( k  =  n  ->  (
2 ^ k )  =  ( 2 ^ n ) )
6867oveq2d 6097 . . . . . . 7  |-  ( k  =  n  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ n ) ) )
69 ovex 6106 . . . . . . 7  |-  ( A  /  ( 2 ^ n ) )  e. 
_V
7068, 18, 69fvmpt 5806 . . . . . 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 2526 . . . . 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 10228 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  e.  NN0 )
76 nnexpcl 11394 . . . . . . . . 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 10016 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( 2 ^ n )  e.  CC )
8078nnne0d 10044 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( 2 ^ n )  =/=  0 )
8174, 79, 80divcld 9790 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( A  /  ( 2 ^ n ) )  e.  CC )
8271, 73, 81fsumser 12524 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  (  seq  1
(  +  ,  F
) `  j )
)
8362, 64, 823eqtr2rd 2475 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  (  seq  1 (  +  ,  F ) `
 j )  =  ( A  -  ( F `  j )
) )
841, 3, 57, 17, 59, 61, 83climsubc2 12432 . 2  |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  ~~>  ( A  -  0 ) )
85 subid1 9322 . 2  |-  ( A  e.  CC  ->  ( A  -  0 )  =  A )
8684, 85breqtrd 4236 1  |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956   class class class wbr 4212    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    <_ cle 9121    - cmin 9291    / cdiv 9677   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043    seq cseq 11323   ^cexp 11382   abscabs 12039    ~~> cli 12278   sum_csu 12479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-rlim 12283  df-sum 12480
  Copyright terms: Public domain W3C validator