ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  geo2lim Unicode version

Theorem geo2lim 11253
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 9329 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 9049 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
3 halfcn 8902 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
43a1i 9 . . . . . 6  |-  ( A  e.  CC  ->  (
1  /  2 )  e.  CC )
5 halfre 8901 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
6 halfge0 8904 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
7 absid 10811 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  0  <_  ( 1  / 
2 ) )  -> 
( abs `  (
1  /  2 ) )  =  ( 1  /  2 ) )
85, 6, 7mp2an 422 . . . . . . . 8  |-  ( abs `  ( 1  /  2
) )  =  ( 1  /  2 )
9 halflt1 8905 . . . . . . . 8  |-  ( 1  /  2 )  <  1
108, 9eqbrtri 3919 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
1110a1i 9 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( 1  / 
2 ) )  <  1 )
124, 11expcnv 11241 . . . . 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 8694 . . . . . . . 8  |-  NN  e.  _V
1615mptex 5614 . . . . . . 7  |-  ( k  e.  NN  |->  ( A  /  ( 2 ^ k ) ) )  e.  _V
1714, 16eqeltri 2190 . . . . . 6  |-  F  e. 
_V
1817a1i 9 . . . . 5  |-  ( A  e.  CC  ->  F  e.  _V )
19 nnnn0 8952 . . . . . . . 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 10392 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( 1  / 
2 ) ^ j
)  e.  CC )
23 oveq2 5750 . . . . . . . 8  |-  ( k  =  j  ->  (
( 1  /  2
) ^ k )  =  ( ( 1  /  2 ) ^
j ) )
24 eqid 2117 . . . . . . . 8  |-  ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) )  =  ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) )
2523, 24fvmptg 5465 . . . . . . 7  |-  ( ( j  e.  NN0  /\  ( ( 1  / 
2 ) ^ j
)  e.  CC )  ->  ( ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) ) `
 j )  =  ( ( 1  / 
2 ) ^ j
) )
2620, 22, 25syl2anc 408 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( ( 1  /  2
) ^ j ) )
2726, 22eqeltrd 2194 . . . . 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 8849 . . . . . . . . 9  |-  2  e.  NN
30 nnexpcl 10274 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  j  e.  NN0 )  -> 
( 2 ^ j
)  e.  NN )
3129, 20, 30sylancr 410 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  NN )
3231nncnd 8702 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  CC )
3331nnap0d 8734 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
) #  0 )
3428, 32, 33divrecapd 8521 . . . . . 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 8518 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  e.  CC )
37 oveq2 5750 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
3837oveq2d 5758 . . . . . . . 8  |-  ( k  =  j  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ j ) ) )
3938, 14fvmptg 5465 . . . . . . 7  |-  ( ( j  e.  NN  /\  ( A  /  (
2 ^ j ) )  e.  CC )  ->  ( F `  j )  =  ( A  /  ( 2 ^ j ) ) )
4035, 36, 39syl2anc 408 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  =  ( A  /  ( 2 ^ j ) ) )
41 2cn 8759 . . . . . . . . 9  |-  2  e.  CC
42 2ap0 8781 . . . . . . . . 9  |-  2 #  0
43 nnz 9041 . . . . . . . . . 10  |-  ( j  e.  NN  ->  j  e.  ZZ )
4443adantl 275 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ZZ )
45 exprecap 10302 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  2 #  0  /\  j  e.  ZZ )  ->  (
( 1  /  2
) ^ j )  =  ( 1  / 
( 2 ^ j
) ) )
4641, 42, 44, 45mp3an12i 1304 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( 1  / 
2 ) ^ j
)  =  ( 1  /  ( 2 ^ j ) ) )
4726, 46eqtrd 2150 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( 1  /  ( 2 ^ j ) ) )
4847oveq2d 5758 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  x.  (
( k  e.  NN0  |->  ( ( 1  / 
2 ) ^ k
) ) `  j
) )  =  ( A  x.  ( 1  /  ( 2 ^ j ) ) ) )
4934, 40, 483eqtr4d 2160 . . . . 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 11068 . . . 4  |-  ( A  e.  CC  ->  F  ~~>  ( A  x.  0
) )
51 mul01 8119 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
5250, 51breqtrd 3924 . . 3  |-  ( A  e.  CC  ->  F  ~~>  0 )
53 seqex 10188 . . . 4  |-  seq 1
(  +  ,  F
)  e.  _V
5453a1i 9 . . 3  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  e.  _V )
5540, 36eqeltrd 2194 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  e.  CC )
5640oveq2d 5758 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  -  ( F `  j )
)  =  ( A  -  ( A  / 
( 2 ^ j
) ) ) )
57 geo2sum 11251 . . . . 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 9330 . . . . . . . 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 503 . . . . . . 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 8998 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  n  e.  NN0 )
6563, 64expcld 10392 . . . . . . 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 9140 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  n  e.  ZZ )
6863, 66, 67expap0d 10398 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( 2 ^ n ) #  0 )
6962, 65, 68divclapd 8518 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( A  / 
( 2 ^ n
) )  e.  CC )
70 oveq2 5750 . . . . . . . 8  |-  ( k  =  n  ->  (
2 ^ k )  =  ( 2 ^ n ) )
7170oveq2d 5758 . . . . . . 7  |-  ( k  =  n  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ n ) ) )
7271, 14fvmptg 5465 . . . . . 6  |-  ( ( n  e.  NN  /\  ( A  /  (
2 ^ n ) )  e.  CC )  ->  ( F `  n )  =  ( A  /  ( 2 ^ n ) ) )
7361, 69, 72syl2anc 408 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( F `  n )  =  ( A  /  ( 2 ^ n ) ) )
7435, 1eleqtrdi 2210 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ( ZZ>= ` 
1 ) )
7573, 74, 69fsum3ser 11134 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  (  seq 1
(  +  ,  F
) `  j )
)
7656, 58, 753eqtr2rd 2157 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 j )  =  ( A  -  ( F `  j )
) )
771, 2, 52, 13, 54, 55, 76climsubc2 11070 . 2  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  ~~>  ( A  -  0 ) )
78 subid1 7950 . 2  |-  ( A  e.  CC  ->  ( A  -  0 )  =  A )
7977, 78breqtrd 3924 1  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   _Vcvv 2660   class class class wbr 3899    |-> cmpt 3959   ` cfv 5093  (class class class)co 5742   CCcc 7586   RRcr 7587   0cc0 7588   1c1 7589    + caddc 7591    x. cmul 7593    < clt 7768    <_ cle 7769    - cmin 7901   # cap 8311    / cdiv 8400   NNcn 8688   2c2 8739   NN0cn0 8945   ZZcz 9022   ZZ>=cuz 9294   ...cfz 9758    seqcseq 10186   ^cexp 10260   abscabs 10737    ~~> cli 11015   sum_csu 11090
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-isom 5102  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-frec 6256  df-1o 6281  df-oadd 6285  df-er 6397  df-en 6603  df-dom 6604  df-fin 6605  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-q 9380  df-rp 9410  df-fz 9759  df-fzo 9888  df-seqfrec 10187  df-exp 10261  df-ihash 10490  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739  df-clim 11016  df-sumdc 11091
This theorem is referenced by:  trilpolemeq1  13160
  Copyright terms: Public domain W3C validator