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Theorem aaliou3lem3 20214
Description: Lemma for aaliou3 20221. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Hypotheses
Ref Expression
aaliou3lem.a  |-  G  =  ( c  e.  (
ZZ>= `  A )  |->  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( c  -  A ) ) ) )
aaliou3lem.b  |-  F  =  ( a  e.  NN  |->  ( 2 ^ -u ( ! `  a )
) )
Assertion
Ref Expression
aaliou3lem3  |-  ( A  e.  NN  ->  (  seq  A (  +  ,  F )  e.  dom  ~~>  /\ 
sum_ b  e.  (
ZZ>= `  A ) ( F `  b )  e.  RR+  /\  sum_ b  e.  ( ZZ>= `  A )
( F `  b
)  <_  ( 2  x.  ( 2 ^
-u ( ! `  A ) ) ) ) )
Distinct variable groups:    F, b,
c    A, a, b, c    G, a, b
Allowed substitution hints:    F( a)    G( c)

Proof of Theorem aaliou3lem3
StepHypRef Expression
1 eqid 2404 . . 3  |-  ( ZZ>= `  A )  =  (
ZZ>= `  A )
2 nnz 10259 . . . 4  |-  ( A  e.  NN  ->  A  e.  ZZ )
3 uzid 10456 . . . 4  |-  ( A  e.  ZZ  ->  A  e.  ( ZZ>= `  A )
)
42, 3syl 16 . . 3  |-  ( A  e.  NN  ->  A  e.  ( ZZ>= `  A )
)
5 aaliou3lem.a . . . 4  |-  G  =  ( c  e.  (
ZZ>= `  A )  |->  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( c  -  A ) ) ) )
65aaliou3lem1 20212 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( G `  b
)  e.  RR )
7 aaliou3lem.b . . . . . 6  |-  F  =  ( a  e.  NN  |->  ( 2 ^ -u ( ! `  a )
) )
85, 7aaliou3lem2 20213 . . . . 5  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  e.  ( 0 (,] ( G `  b ) ) )
9 0xr 9087 . . . . . 6  |-  0  e.  RR*
10 elioc2 10929 . . . . . 6  |-  ( ( 0  e.  RR*  /\  ( G `  b )  e.  RR )  ->  (
( F `  b
)  e.  ( 0 (,] ( G `  b ) )  <->  ( ( F `  b )  e.  RR  /\  0  < 
( F `  b
)  /\  ( F `  b )  <_  ( G `  b )
) ) )
119, 6, 10sylancr 645 . . . . 5  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( ( F `  b )  e.  ( 0 (,] ( G `
 b ) )  <-> 
( ( F `  b )  e.  RR  /\  0  <  ( F `
 b )  /\  ( F `  b )  <_  ( G `  b ) ) ) )
128, 11mpbid 202 . . . 4  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( ( F `  b )  e.  RR  /\  0  <  ( F `
 b )  /\  ( F `  b )  <_  ( G `  b ) ) )
1312simp1d 969 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  e.  RR )
14 2cn 10026 . . . . . . 7  |-  2  e.  CC
15 2ne0 10039 . . . . . . 7  |-  2  =/=  0
1614, 15reccli 9700 . . . . . 6  |-  ( 1  /  2 )  e.  CC
1716a1i 11 . . . . 5  |-  ( A  e.  NN  ->  (
1  /  2 )  e.  CC )
18 2re 10025 . . . . . . . . . 10  |-  2  e.  RR
1918, 15rereccli 9735 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
20 halfgt0 10144 . . . . . . . . 9  |-  0  <  ( 1  /  2
)
2119, 20elrpii 10571 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR+
22 rprege0 10582 . . . . . . . 8  |-  ( ( 1  /  2 )  e.  RR+  ->  ( ( 1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
) ) )
23 absid 12056 . . . . . . . 8  |-  ( ( ( 1  /  2
)  e.  RR  /\  0  <_  ( 1  / 
2 ) )  -> 
( abs `  (
1  /  2 ) )  =  ( 1  /  2 ) )
2421, 22, 23mp2b 10 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  =  ( 1  /  2 )
25 halflt1 10145 . . . . . . 7  |-  ( 1  /  2 )  <  1
2624, 25eqbrtri 4191 . . . . . 6  |-  ( abs `  ( 1  /  2
) )  <  1
2726a1i 11 . . . . 5  |-  ( A  e.  NN  ->  ( abs `  ( 1  / 
2 ) )  <  1 )
28 2rp 10573 . . . . . . 7  |-  2  e.  RR+
29 nnnn0 10184 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  NN0 )
30 faccl 11531 . . . . . . . . . 10  |-  ( A  e.  NN0  ->  ( ! `
 A )  e.  NN )
3129, 30syl 16 . . . . . . . . 9  |-  ( A  e.  NN  ->  ( ! `  A )  e.  NN )
3231nnzd 10330 . . . . . . . 8  |-  ( A  e.  NN  ->  ( ! `  A )  e.  ZZ )
3332znegcld 10333 . . . . . . 7  |-  ( A  e.  NN  ->  -u ( ! `  A )  e.  ZZ )
34 rpexpcl 11355 . . . . . . 7  |-  ( ( 2  e.  RR+  /\  -u ( ! `  A )  e.  ZZ )  ->  (
2 ^ -u ( ! `  A )
)  e.  RR+ )
3528, 33, 34sylancr 645 . . . . . 6  |-  ( A  e.  NN  ->  (
2 ^ -u ( ! `  A )
)  e.  RR+ )
3635rpcnd 10606 . . . . 5  |-  ( A  e.  NN  ->  (
2 ^ -u ( ! `  A )
)  e.  CC )
372, 17, 27, 36, 5geolim3 20209 . . . 4  |-  ( A  e.  NN  ->  seq  A (  +  ,  G
)  ~~>  ( ( 2 ^ -u ( ! `
 A ) )  /  ( 1  -  ( 1  /  2
) ) ) )
38 seqex 11280 . . . . 5  |-  seq  A
(  +  ,  G
)  e.  _V
39 ovex 6065 . . . . 5  |-  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  e.  _V
4038, 39breldm 5033 . . . 4  |-  (  seq 
A (  +  ,  G )  ~~>  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  ->  seq  A (  +  ,  G )  e.  dom  ~~>  )
4137, 40syl 16 . . 3  |-  ( A  e.  NN  ->  seq  A (  +  ,  G
)  e.  dom  ~~>  )
4212simp2d 970 . . . . 5  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
0  <  ( F `  b ) )
4313, 42elrpd 10602 . . . 4  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  e.  RR+ )
4443rpge0d 10608 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
0  <_  ( F `  b ) )
4512simp3d 971 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  <_  ( G `  b ) )
461, 4, 6, 13, 41, 44, 45cvgcmp 12550 . 2  |-  ( A  e.  NN  ->  seq  A (  +  ,  F
)  e.  dom  ~~>  )
47 eqidd 2405 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  =  ( F `
 b ) )
481, 1, 4, 47, 43, 46isumrpcl 12578 . 2  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( F `  b
)  e.  RR+ )
49 eqidd 2405 . . . 4  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( G `  b
)  =  ( G `
 b ) )
501, 2, 47, 13, 49, 6, 45, 46, 41isumle 12579 . . 3  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( F `  b
)  <_  sum_ b  e.  ( ZZ>= `  A )
( G `  b
) )
516recnd 9070 . . . . 5  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( G `  b
)  e.  CC )
521, 2, 49, 51, 37isumclim 12496 . . . 4  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( G `  b
)  =  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) ) )
53 1mhlfehlf 10146 . . . . . 6  |-  ( 1  -  ( 1  / 
2 ) )  =  ( 1  /  2
)
5453oveq2i 6051 . . . . 5  |-  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  =  ( ( 2 ^ -u ( ! `
 A ) )  /  ( 1  / 
2 ) )
55 mulcl 9030 . . . . . . . 8  |-  ( ( ( 2 ^ -u ( ! `  A )
)  e.  CC  /\  2  e.  CC )  ->  ( ( 2 ^
-u ( ! `  A ) )  x.  2 )  e.  CC )
5636, 14, 55sylancl 644 . . . . . . 7  |-  ( A  e.  NN  ->  (
( 2 ^ -u ( ! `  A )
)  x.  2 )  e.  CC )
5756div1d 9738 . . . . . 6  |-  ( A  e.  NN  ->  (
( ( 2 ^
-u ( ! `  A ) )  x.  2 )  /  1
)  =  ( ( 2 ^ -u ( ! `  A )
)  x.  2 ) )
58 1rp 10572 . . . . . . . . 9  |-  1  e.  RR+
59 rpcnne0 10585 . . . . . . . . 9  |-  ( 1  e.  RR+  ->  ( 1  e.  CC  /\  1  =/=  0 ) )
6058, 59ax-mp 8 . . . . . . . 8  |-  ( 1  e.  CC  /\  1  =/=  0 )
61 rpcnne0 10585 . . . . . . . . 9  |-  ( 2  e.  RR+  ->  ( 2  e.  CC  /\  2  =/=  0 ) )
6228, 61ax-mp 8 . . . . . . . 8  |-  ( 2  e.  CC  /\  2  =/=  0 )
63 divdiv2 9682 . . . . . . . 8  |-  ( ( ( 2 ^ -u ( ! `  A )
)  e.  CC  /\  ( 1  e.  CC  /\  1  =/=  0 )  /\  ( 2  e.  CC  /\  2  =/=  0 ) )  -> 
( ( 2 ^
-u ( ! `  A ) )  / 
( 1  /  2
) )  =  ( ( ( 2 ^
-u ( ! `  A ) )  x.  2 )  /  1
) )
6460, 62, 63mp3an23 1271 . . . . . . 7  |-  ( ( 2 ^ -u ( ! `  A )
)  e.  CC  ->  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  /  2 ) )  =  ( ( ( 2 ^ -u ( ! `  A )
)  x.  2 )  /  1 ) )
6536, 64syl 16 . . . . . 6  |-  ( A  e.  NN  ->  (
( 2 ^ -u ( ! `  A )
)  /  ( 1  /  2 ) )  =  ( ( ( 2 ^ -u ( ! `  A )
)  x.  2 )  /  1 ) )
66 mulcom 9032 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( 2 ^ -u ( ! `  A )
)  e.  CC )  ->  ( 2  x.  ( 2 ^ -u ( ! `  A )
) )  =  ( ( 2 ^ -u ( ! `  A )
)  x.  2 ) )
6714, 36, 66sylancr 645 . . . . . 6  |-  ( A  e.  NN  ->  (
2  x.  ( 2 ^ -u ( ! `
 A ) ) )  =  ( ( 2 ^ -u ( ! `  A )
)  x.  2 ) )
6857, 65, 673eqtr4d 2446 . . . . 5  |-  ( A  e.  NN  ->  (
( 2 ^ -u ( ! `  A )
)  /  ( 1  /  2 ) )  =  ( 2  x.  ( 2 ^ -u ( ! `  A )
) ) )
6954, 68syl5eq 2448 . . . 4  |-  ( A  e.  NN  ->  (
( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  =  ( 2  x.  ( 2 ^ -u ( ! `  A )
) ) )
7052, 69eqtrd 2436 . . 3  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( G `  b
)  =  ( 2  x.  ( 2 ^
-u ( ! `  A ) ) ) )
7150, 70breqtrd 4196 . 2  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( F `  b
)  <_  ( 2  x.  ( 2 ^
-u ( ! `  A ) ) ) )
7246, 48, 713jca 1134 1  |-  ( A  e.  NN  ->  (  seq  A (  +  ,  F )  e.  dom  ~~>  /\ 
sum_ b  e.  (
ZZ>= `  A ) ( F `  b )  e.  RR+  /\  sum_ b  e.  ( ZZ>= `  A )
( F `  b
)  <_  ( 2  x.  ( 2 ^
-u ( ! `  A ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568   (,]cioc 10873    seq cseq 11278   ^cexp 11337   !cfa 11521   abscabs 11994    ~~> cli 12233   sum_csu 12434
This theorem is referenced by:  aaliou3lem4  20216  aaliou3lem7  20219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ioc 10877  df-ico 10878  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-fac 11522  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435
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