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Theorem geo2lim 11479
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 9522 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 9239 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
3 halfcn 9092 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
43a1i 9 . . . . . 6  |-  ( A  e.  CC  ->  (
1  /  2 )  e.  CC )
5 halfre 9091 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
6 halfge0 9094 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
7 absid 11035 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  0  <_  ( 1  / 
2 ) )  -> 
( abs `  (
1  /  2 ) )  =  ( 1  /  2 ) )
85, 6, 7mp2an 424 . . . . . . . 8  |-  ( abs `  ( 1  /  2
) )  =  ( 1  /  2 )
9 halflt1 9095 . . . . . . . 8  |-  ( 1  /  2 )  <  1
108, 9eqbrtri 4010 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
1110a1i 9 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( 1  / 
2 ) )  <  1 )
124, 11expcnv 11467 . . . . 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 8884 . . . . . . . 8  |-  NN  e.  _V
1615mptex 5722 . . . . . . 7  |-  ( k  e.  NN  |->  ( A  /  ( 2 ^ k ) ) )  e.  _V
1714, 16eqeltri 2243 . . . . . 6  |-  F  e. 
_V
1817a1i 9 . . . . 5  |-  ( A  e.  CC  ->  F  e.  _V )
19 nnnn0 9142 . . . . . . . 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 10609 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( 1  / 
2 ) ^ j
)  e.  CC )
23 oveq2 5861 . . . . . . . 8  |-  ( k  =  j  ->  (
( 1  /  2
) ^ k )  =  ( ( 1  /  2 ) ^
j ) )
24 eqid 2170 . . . . . . . 8  |-  ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) )  =  ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) )
2523, 24fvmptg 5572 . . . . . . 7  |-  ( ( j  e.  NN0  /\  ( ( 1  / 
2 ) ^ j
)  e.  CC )  ->  ( ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) ) `
 j )  =  ( ( 1  / 
2 ) ^ j
) )
2620, 22, 25syl2anc 409 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( ( 1  /  2
) ^ j ) )
2726, 22eqeltrd 2247 . . . . 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 9039 . . . . . . . . 9  |-  2  e.  NN
30 nnexpcl 10489 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  j  e.  NN0 )  -> 
( 2 ^ j
)  e.  NN )
3129, 20, 30sylancr 412 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  NN )
3231nncnd 8892 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  CC )
3331nnap0d 8924 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
) #  0 )
3428, 32, 33divrecapd 8710 . . . . . 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 8707 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  e.  CC )
37 oveq2 5861 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
3837oveq2d 5869 . . . . . . . 8  |-  ( k  =  j  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ j ) ) )
3938, 14fvmptg 5572 . . . . . . 7  |-  ( ( j  e.  NN  /\  ( A  /  (
2 ^ j ) )  e.  CC )  ->  ( F `  j )  =  ( A  /  ( 2 ^ j ) ) )
4035, 36, 39syl2anc 409 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  =  ( A  /  ( 2 ^ j ) ) )
41 2cn 8949 . . . . . . . . 9  |-  2  e.  CC
42 2ap0 8971 . . . . . . . . 9  |-  2 #  0
43 nnz 9231 . . . . . . . . . 10  |-  ( j  e.  NN  ->  j  e.  ZZ )
4443adantl 275 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ZZ )
45 exprecap 10517 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  2 #  0  /\  j  e.  ZZ )  ->  (
( 1  /  2
) ^ j )  =  ( 1  / 
( 2 ^ j
) ) )
4641, 42, 44, 45mp3an12i 1336 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( 1  / 
2 ) ^ j
)  =  ( 1  /  ( 2 ^ j ) ) )
4726, 46eqtrd 2203 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( 1  /  ( 2 ^ j ) ) )
4847oveq2d 5869 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  x.  (
( k  e.  NN0  |->  ( ( 1  / 
2 ) ^ k
) ) `  j
) )  =  ( A  x.  ( 1  /  ( 2 ^ j ) ) ) )
4934, 40, 483eqtr4d 2213 . . . . 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 11294 . . . 4  |-  ( A  e.  CC  ->  F  ~~>  ( A  x.  0
) )
51 mul01 8308 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
5250, 51breqtrd 4015 . . 3  |-  ( A  e.  CC  ->  F  ~~>  0 )
53 seqex 10403 . . . 4  |-  seq 1
(  +  ,  F
)  e.  _V
5453a1i 9 . . 3  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  e.  _V )
5540, 36eqeltrd 2247 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  e.  CC )
5640oveq2d 5869 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  -  ( F `  j )
)  =  ( A  -  ( A  / 
( 2 ^ j
) ) ) )
57 geo2sum 11477 . . . . 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 9523 . . . . . . . 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 524 . . . . . . 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 9188 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  n  e.  NN0 )
6563, 64expcld 10609 . . . . . . 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 9333 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  n  e.  ZZ )
6863, 66, 67expap0d 10615 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( 2 ^ n ) #  0 )
6962, 65, 68divclapd 8707 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( A  / 
( 2 ^ n
) )  e.  CC )
70 oveq2 5861 . . . . . . . 8  |-  ( k  =  n  ->  (
2 ^ k )  =  ( 2 ^ n ) )
7170oveq2d 5869 . . . . . . 7  |-  ( k  =  n  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ n ) ) )
7271, 14fvmptg 5572 . . . . . 6  |-  ( ( n  e.  NN  /\  ( A  /  (
2 ^ n ) )  e.  CC )  ->  ( F `  n )  =  ( A  /  ( 2 ^ n ) ) )
7361, 69, 72syl2anc 409 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( F `  n )  =  ( A  /  ( 2 ^ n ) ) )
7435, 1eleqtrdi 2263 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ( ZZ>= ` 
1 ) )
7573, 74, 69fsum3ser 11360 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  (  seq 1
(  +  ,  F
) `  j )
)
7656, 58, 753eqtr2rd 2210 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 j )  =  ( A  -  ( F `  j )
) )
771, 2, 52, 13, 54, 55, 76climsubc2 11296 . 2  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  ~~>  ( A  -  0 ) )
78 subid1 8139 . 2  |-  ( A  e.  CC  ->  ( A  -  0 )  =  A )
7977, 78breqtrd 4015 1  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   _Vcvv 2730   class class class wbr 3989    |-> cmpt 4050   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775    + caddc 7777    x. cmul 7779    < clt 7954    <_ cle 7955    - cmin 8090   # cap 8500    / cdiv 8589   NNcn 8878   2c2 8929   NN0cn0 9135   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965    seqcseq 10401   ^cexp 10475   abscabs 10961    ~~> cli 11241   sum_csu 11316
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317
This theorem is referenced by:  trilpolemeq1  14072
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