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Theorem geo2lim 11413
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 9474 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 9194 . . 3  |-  ( A  e.  CC  ->  1  e.  ZZ )
3 halfcn 9047 . . . . . . 7  |-  ( 1  /  2 )  e.  CC
43a1i 9 . . . . . 6  |-  ( A  e.  CC  ->  (
1  /  2 )  e.  CC )
5 halfre 9046 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
6 halfge0 9049 . . . . . . . . 9  |-  0  <_  ( 1  /  2
)
7 absid 10971 . . . . . . . . 9  |-  ( ( ( 1  /  2
)  e.  RR  /\  0  <_  ( 1  / 
2 ) )  -> 
( abs `  (
1  /  2 ) )  =  ( 1  /  2 ) )
85, 6, 7mp2an 423 . . . . . . . 8  |-  ( abs `  ( 1  /  2
) )  =  ( 1  /  2 )
9 halflt1 9050 . . . . . . . 8  |-  ( 1  /  2 )  <  1
108, 9eqbrtri 3985 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  <  1
1110a1i 9 . . . . . 6  |-  ( A  e.  CC  ->  ( abs `  ( 1  / 
2 ) )  <  1 )
124, 11expcnv 11401 . . . . 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 8839 . . . . . . . 8  |-  NN  e.  _V
1615mptex 5693 . . . . . . 7  |-  ( k  e.  NN  |->  ( A  /  ( 2 ^ k ) ) )  e.  _V
1714, 16eqeltri 2230 . . . . . 6  |-  F  e. 
_V
1817a1i 9 . . . . 5  |-  ( A  e.  CC  ->  F  e.  _V )
19 nnnn0 9097 . . . . . . . 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 10551 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( 1  / 
2 ) ^ j
)  e.  CC )
23 oveq2 5832 . . . . . . . 8  |-  ( k  =  j  ->  (
( 1  /  2
) ^ k )  =  ( ( 1  /  2 ) ^
j ) )
24 eqid 2157 . . . . . . . 8  |-  ( k  e.  NN0  |->  ( ( 1  /  2 ) ^ k ) )  =  ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) )
2523, 24fvmptg 5544 . . . . . . 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 2234 . . . . 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 8994 . . . . . . . . 9  |-  2  e.  NN
30 nnexpcl 10432 . . . . . . . . 9  |-  ( ( 2  e.  NN  /\  j  e.  NN0 )  -> 
( 2 ^ j
)  e.  NN )
3129, 20, 30sylancr 411 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  NN )
3231nncnd 8847 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
)  e.  CC )
3331nnap0d 8879 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( 2 ^ j
) #  0 )
3428, 32, 33divrecapd 8666 . . . . . 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 8663 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  /  (
2 ^ j ) )  e.  CC )
37 oveq2 5832 . . . . . . . . 9  |-  ( k  =  j  ->  (
2 ^ k )  =  ( 2 ^ j ) )
3837oveq2d 5840 . . . . . . . 8  |-  ( k  =  j  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ j ) ) )
3938, 14fvmptg 5544 . . . . . . 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 8904 . . . . . . . . 9  |-  2  e.  CC
42 2ap0 8926 . . . . . . . . 9  |-  2 #  0
43 nnz 9186 . . . . . . . . . 10  |-  ( j  e.  NN  ->  j  e.  ZZ )
4443adantl 275 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ZZ )
45 exprecap 10460 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  2 #  0  /\  j  e.  ZZ )  ->  (
( 1  /  2
) ^ j )  =  ( 1  / 
( 2 ^ j
) ) )
4641, 42, 44, 45mp3an12i 1323 . . . . . . . 8  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( 1  / 
2 ) ^ j
)  =  ( 1  /  ( 2 ^ j ) ) )
4726, 46eqtrd 2190 . . . . . . 7  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( ( k  e. 
NN0  |->  ( ( 1  /  2 ) ^
k ) ) `  j )  =  ( 1  /  ( 2 ^ j ) ) )
4847oveq2d 5840 . . . . . 6  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  x.  (
( k  e.  NN0  |->  ( ( 1  / 
2 ) ^ k
) ) `  j
) )  =  ( A  x.  ( 1  /  ( 2 ^ j ) ) ) )
4934, 40, 483eqtr4d 2200 . . . . 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 11228 . . . 4  |-  ( A  e.  CC  ->  F  ~~>  ( A  x.  0
) )
51 mul01 8264 . . . 4  |-  ( A  e.  CC  ->  ( A  x.  0 )  =  0 )
5250, 51breqtrd 3990 . . 3  |-  ( A  e.  CC  ->  F  ~~>  0 )
53 seqex 10346 . . . 4  |-  seq 1
(  +  ,  F
)  e.  _V
5453a1i 9 . . 3  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  e.  _V )
5540, 36eqeltrd 2234 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( F `  j
)  e.  CC )
5640oveq2d 5840 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  ( A  -  ( F `  j )
)  =  ( A  -  ( A  / 
( 2 ^ j
) ) ) )
57 geo2sum 11411 . . . . 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 9475 . . . . . . . 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 519 . . . . . . 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 9143 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  n  e.  NN0 )
6563, 64expcld 10551 . . . . . . 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 9285 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  n  e.  ZZ )
6863, 66, 67expap0d 10557 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( 2 ^ n ) #  0 )
6962, 65, 68divclapd 8663 . . . . . 6  |-  ( ( ( A  e.  CC  /\  j  e.  NN )  /\  n  e.  (
ZZ>= `  1 ) )  ->  ( A  / 
( 2 ^ n
) )  e.  CC )
70 oveq2 5832 . . . . . . . 8  |-  ( k  =  n  ->  (
2 ^ k )  =  ( 2 ^ n ) )
7170oveq2d 5840 . . . . . . 7  |-  ( k  =  n  ->  ( A  /  ( 2 ^ k ) )  =  ( A  /  (
2 ^ n ) ) )
7271, 14fvmptg 5544 . . . . . 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 2250 . . . . 5  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  j  e.  ( ZZ>= ` 
1 ) )
7573, 74, 69fsum3ser 11294 . . . 4  |-  ( ( A  e.  CC  /\  j  e.  NN )  -> 
sum_ n  e.  (
1 ... j ) ( A  /  ( 2 ^ n ) )  =  (  seq 1
(  +  ,  F
) `  j )
)
7656, 58, 753eqtr2rd 2197 . . 3  |-  ( ( A  e.  CC  /\  j  e.  NN )  ->  (  seq 1 (  +  ,  F ) `
 j )  =  ( A  -  ( F `  j )
) )
771, 2, 52, 13, 54, 55, 76climsubc2 11230 . 2  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  ~~>  ( A  -  0 ) )
78 subid1 8095 . 2  |-  ( A  e.  CC  ->  ( A  -  0 )  =  A )
7977, 78breqtrd 3990 1  |-  ( A  e.  CC  ->  seq 1 (  +  ,  F )  ~~>  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   _Vcvv 2712   class class class wbr 3965    |-> cmpt 4025   ` cfv 5170  (class class class)co 5824   CCcc 7730   RRcr 7731   0cc0 7732   1c1 7733    + caddc 7735    x. cmul 7737    < clt 7912    <_ cle 7913    - cmin 8046   # cap 8456    / cdiv 8545   NNcn 8833   2c2 8884   NN0cn0 9090   ZZcz 9167   ZZ>=cuz 9439   ...cfz 9912    seqcseq 10344   ^cexp 10418   abscabs 10897    ~~> cli 11175   sum_csu 11250
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-isom 5179  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-frec 6338  df-1o 6363  df-oadd 6367  df-er 6480  df-en 6686  df-dom 6687  df-fin 6688  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-fz 9913  df-fzo 10042  df-seqfrec 10345  df-exp 10419  df-ihash 10650  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-clim 11176  df-sumdc 11251
This theorem is referenced by:  trilpolemeq1  13622
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