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Theorem geo2lim 12347
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 10279 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1z 10069 . . . 4  |-  1  e.  ZZ
32a1i 10 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
4 1re 8853 . . . . . . . . 9  |-  1  e.  RR
5 rehalfcl 9954 . . . . . . . . 9  |-  ( 1  e.  RR  ->  (
1  /  2 )  e.  RR )
64, 5ax-mp 8 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR
76recni 8865 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
87a1i 10 . . . . . 6  |-  ( A  e.  CC  ->  (
1  /  2 )  e.  CC )
9 0re 8854 . . . . . . . . . 10  |-  0  e.  RR
10 halfgt0 9948 . . . . . . . . . 10  |-  0  <  ( 1  /  2
)
119, 6, 10ltleii 8957 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
12 absid 11797 . . . . . . . . 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 9949 . . . . . . . 8  |-  ( 1  /  2 )  <  1
1513, 14eqbrtri 4058 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
1615a1i 10 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( 1  / 
2 ) )  <  1 )
178, 16expcnv 12338 . . . . 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 9768 . . . . . . . 8  |-  NN  e.  _V
2120mptex 5762 . . . . . . 7  |-  ( k  e.  NN  |->  ( A  /  ( 2 ^ k ) ) )  e.  _V
2219, 21eqeltri 2366 . . . . . 6  |-  F  e. 
_V
2322a1i 10 . . . . 5  |-  ( A  e.  CC  ->  F  e.  _V )
24 nnnn0 9988 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  NN0 )
2524adantl 452 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  NN0 )
26 oveq2 5882 . . . . . . . . 9  |-  ( k  =  j  ->  (
( 1  /  2
) ^ k )  =  ( ( 1  /  2 ) ^
j ) )
27 eqid 2296 . . . . . . . . 9  |-  ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) )  =  ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) )
28 ovex 5899 . . . . . . . . 9  |-  ( ( 1  /  2 ) ^ j )  e. 
_V
2926, 27, 28fvmpt 5618 . . . . . . . 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 10061 . . . . . . . . 9  |-  ( j  e.  NN  ->  j  e.  ZZ )
3231adantl 452 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ZZ )
33 2cn 9832 . . . . . . . . 9  |-  2  e.  CC
34 2ne0 9845 . . . . . . . . 9  |-  2  =/=  0
35 exprec 11159 . . . . . . . . 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 2328 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( 1  /  ( 2 ^ j ) ) )
39 2nn 9893 . . . . . . . . 9  |-  2  e.  NN
40 nnexpcl 11132 . . . . . . . . 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 9807 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 1  /  (
2 ^ j ) )  e.  RR )
4342recnd 8877 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 1  /  (
2 ^ j ) )  e.  CC )
4438, 43eqeltrd 2370 . . . . 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 9778 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  CC )
4741nnne0d 9806 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  =/=  0 )
4845, 46, 47divrecd 9555 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  =  ( A  x.  ( 1  / 
( 2 ^ j
) ) ) )
49 oveq2 5882 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
5049oveq2d 5890 . . . . . . . 8  |-  ( k  =  j  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ j ) ) )
51 ovex 5899 . . . . . . . 8  |-  ( A  /  ( 2 ^ j ) )  e. 
_V
5250, 19, 51fvmpt 5618 . . . . . . 7  |-  ( j  e.  NN  ->  ( F `  j )  =  ( A  / 
( 2 ^ j
) ) )
5352adantl 452 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  =  ( A  /  ( 2 ^ j ) ) )
5438oveq2d 5890 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  x.  (
( k  e.  NN0  |->  ( ( 1  / 
2 ) ^ k
) ) `  j
) )  =  ( A  x.  ( 1  /  ( 2 ^ j ) ) ) )
5548, 53, 543eqtr4d 2338 . . . . 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 12126 . . . 4  |-  ( A  e.  CC  ->  F  ~~>  ( A  x.  0
) )
57 mul01 9007 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
5856, 57breqtrd 4063 . . 3  |-  ( A  e.  CC  ->  F  ~~>  0 )
59 seqex 11064 . . . 4  |-  seq  1
(  +  ,  F
)  e.  _V
6059a1i 10 . . 3  |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  e.  _V )
6145, 46, 47divcld 9552 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  e.  CC )
6253, 61eqeltrd 2370 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  e.  CC )
6353oveq2d 5890 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  -  ( F `  j )
)  =  ( A  -  ( A  / 
( 2 ^ j
) ) ) )
64 geo2sum 12345 . . . . 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 10835 . . . . . . 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 5882 . . . . . . . 8  |-  ( k  =  n  ->  (
2 ^ k )  =  ( 2 ^ n ) )
6968oveq2d 5890 . . . . . . 7  |-  ( k  =  n  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ n ) ) )
70 ovex 5899 . . . . . . 7  |-  ( A  /  ( 2 ^ n ) )  e. 
_V
7169, 19, 70fvmpt 5618 . . . . . 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 2386 . . . . 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 9988 . . . . . . . . 9  |-  ( n  e.  NN  ->  n  e.  NN0 )
77 nnexpcl 11132 . . . . . . . . 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 9778 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( 2 ^ n )  e.  CC )
8179nnne0d 9806 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( 2 ^ n )  =/=  0 )
8275, 80, 81divcld 9552 . . . . 5  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  ( 1 ... j ) )  ->  ( A  /  ( 2 ^ n ) )  e.  CC )
8372, 74, 82fsumser 12219 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  (  seq  1
(  +  ,  F
) `  j )
)
8463, 65, 833eqtr2rd 2335 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  (  seq  1 (  +  ,  F ) `
 j )  =  ( A  -  ( F `  j )
) )
851, 3, 58, 18, 60, 62, 84climsubc2 12128 . 2  |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  ~~>  ( A  -  0 ) )
86 subid1 9084 . 2  |-  ( A  e.  CC  ->  ( A  -  0 )  =  A )
8785, 86breqtrd 4063 1  |-  ( A  e.  CC  ->  seq  1 (  +  ,  F )  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062   ^cexp 11120   abscabs 11735    ~~> cli 11974   sum_csu 12174
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175
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