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Theorem aaliou3lem3 20263
Description: Lemma for aaliou3 20270. (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 2438 . . 3  |-  ( ZZ>= `  A )  =  (
ZZ>= `  A )
2 nnz 10305 . . . 4  |-  ( A  e.  NN  ->  A  e.  ZZ )
3 uzid 10502 . . . 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 20261 . . 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 20262 . . . . 5  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  e.  ( 0 (,] ( G `  b ) ) )
9 0xr 9133 . . . . . 6  |-  0  e.  RR*
10 elioc2 10975 . . . . . 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 646 . . . . 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 203 . . . 4  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( ( F `  b )  e.  RR  /\  0  <  ( F `
 b )  /\  ( F `  b )  <_  ( G `  b ) ) )
1312simp1d 970 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  e.  RR )
14 2cn 10072 . . . . . . 7  |-  2  e.  CC
15 2ne0 10085 . . . . . . 7  |-  2  =/=  0
1614, 15reccli 9746 . . . . . 6  |-  ( 1  /  2 )  e.  CC
1716a1i 11 . . . . 5  |-  ( A  e.  NN  ->  (
1  /  2 )  e.  CC )
18 2re 10071 . . . . . . . . . 10  |-  2  e.  RR
1918, 15rereccli 9781 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
20 halfgt0 10190 . . . . . . . . 9  |-  0  <  ( 1  /  2
)
2119, 20elrpii 10617 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR+
22 rprege0 10628 . . . . . . . 8  |-  ( ( 1  /  2 )  e.  RR+  ->  ( ( 1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
) ) )
23 absid 12103 . . . . . . . 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 10191 . . . . . . 7  |-  ( 1  /  2 )  <  1
2624, 25eqbrtri 4233 . . . . . 6  |-  ( abs `  ( 1  /  2
) )  <  1
2726a1i 11 . . . . 5  |-  ( A  e.  NN  ->  ( abs `  ( 1  / 
2 ) )  <  1 )
28 2rp 10619 . . . . . . 7  |-  2  e.  RR+
29 nnnn0 10230 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  NN0 )
30 faccl 11578 . . . . . . . . . 10  |-  ( A  e.  NN0  ->  ( ! `
 A )  e.  NN )
3129, 30syl 16 . . . . . . . . 9  |-  ( A  e.  NN  ->  ( ! `  A )  e.  NN )
3231nnzd 10376 . . . . . . . 8  |-  ( A  e.  NN  ->  ( ! `  A )  e.  ZZ )
3332znegcld 10379 . . . . . . 7  |-  ( A  e.  NN  ->  -u ( ! `  A )  e.  ZZ )
34 rpexpcl 11402 . . . . . . 7  |-  ( ( 2  e.  RR+  /\  -u ( ! `  A )  e.  ZZ )  ->  (
2 ^ -u ( ! `  A )
)  e.  RR+ )
3528, 33, 34sylancr 646 . . . . . 6  |-  ( A  e.  NN  ->  (
2 ^ -u ( ! `  A )
)  e.  RR+ )
3635rpcnd 10652 . . . . 5  |-  ( A  e.  NN  ->  (
2 ^ -u ( ! `  A )
)  e.  CC )
372, 17, 27, 36, 5geolim3 20258 . . . 4  |-  ( A  e.  NN  ->  seq  A (  +  ,  G
)  ~~>  ( ( 2 ^ -u ( ! `
 A ) )  /  ( 1  -  ( 1  /  2
) ) ) )
38 seqex 11327 . . . . 5  |-  seq  A
(  +  ,  G
)  e.  _V
39 ovex 6108 . . . . 5  |-  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  e.  _V
4038, 39breldm 5076 . . . 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 971 . . . . 5  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
0  <  ( F `  b ) )
4313, 42elrpd 10648 . . . 4  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  e.  RR+ )
4443rpge0d 10654 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
0  <_  ( F `  b ) )
4512simp3d 972 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  <_  ( G `  b ) )
461, 4, 6, 13, 41, 44, 45cvgcmp 12597 . 2  |-  ( A  e.  NN  ->  seq  A (  +  ,  F
)  e.  dom  ~~>  )
47 eqidd 2439 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  =  ( F `
 b ) )
481, 1, 4, 47, 43, 46isumrpcl 12625 . 2  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( F `  b
)  e.  RR+ )
49 eqidd 2439 . . . 4  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( G `  b
)  =  ( G `
 b ) )
501, 2, 47, 13, 49, 6, 45, 46, 41isumle 12626 . . 3  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( F `  b
)  <_  sum_ b  e.  ( ZZ>= `  A )
( G `  b
) )
516recnd 9116 . . . . 5  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( G `  b
)  e.  CC )
521, 2, 49, 51, 37isumclim 12543 . . . 4  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( G `  b
)  =  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) ) )
53 1mhlfehlf 10192 . . . . . 6  |-  ( 1  -  ( 1  / 
2 ) )  =  ( 1  /  2
)
5453oveq2i 6094 . . . . 5  |-  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  =  ( ( 2 ^ -u ( ! `
 A ) )  /  ( 1  / 
2 ) )
55 mulcl 9076 . . . . . . . 8  |-  ( ( ( 2 ^ -u ( ! `  A )
)  e.  CC  /\  2  e.  CC )  ->  ( ( 2 ^
-u ( ! `  A ) )  x.  2 )  e.  CC )
5636, 14, 55sylancl 645 . . . . . . 7  |-  ( A  e.  NN  ->  (
( 2 ^ -u ( ! `  A )
)  x.  2 )  e.  CC )
5756div1d 9784 . . . . . 6  |-  ( A  e.  NN  ->  (
( ( 2 ^
-u ( ! `  A ) )  x.  2 )  /  1
)  =  ( ( 2 ^ -u ( ! `  A )
)  x.  2 ) )
58 1rp 10618 . . . . . . . . 9  |-  1  e.  RR+
59 rpcnne0 10631 . . . . . . . . 9  |-  ( 1  e.  RR+  ->  ( 1  e.  CC  /\  1  =/=  0 ) )
6058, 59ax-mp 8 . . . . . . . 8  |-  ( 1  e.  CC  /\  1  =/=  0 )
61 rpcnne0 10631 . . . . . . . . 9  |-  ( 2  e.  RR+  ->  ( 2  e.  CC  /\  2  =/=  0 ) )
6228, 61ax-mp 8 . . . . . . . 8  |-  ( 2  e.  CC  /\  2  =/=  0 )
63 divdiv2 9728 . . . . . . . 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 1272 . . . . . . 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 9078 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( 2 ^ -u ( ! `  A )
)  e.  CC )  ->  ( 2  x.  ( 2 ^ -u ( ! `  A )
) )  =  ( ( 2 ^ -u ( ! `  A )
)  x.  2 ) )
6714, 36, 66sylancr 646 . . . . . 6  |-  ( A  e.  NN  ->  (
2  x.  ( 2 ^ -u ( ! `
 A ) ) )  =  ( ( 2 ^ -u ( ! `  A )
)  x.  2 ) )
6857, 65, 673eqtr4d 2480 . . . . 5  |-  ( A  e.  NN  ->  (
( 2 ^ -u ( ! `  A )
)  /  ( 1  /  2 ) )  =  ( 2  x.  ( 2 ^ -u ( ! `  A )
) ) )
6954, 68syl5eq 2482 . . . 4  |-  ( A  e.  NN  ->  (
( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  =  ( 2  x.  ( 2 ^ -u ( ! `  A )
) ) )
7052, 69eqtrd 2470 . . 3  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( G `  b
)  =  ( 2  x.  ( 2 ^
-u ( ! `  A ) ) ) )
7150, 70breqtrd 4238 . 2  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( F `  b
)  <_  ( 2  x.  ( 2 ^
-u ( ! `  A ) ) ) )
7246, 48, 713jca 1135 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214    e. cmpt 4268   dom cdm 4880   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997   RR*cxr 9121    < clt 9122    <_ cle 9123    - cmin 9293   -ucneg 9294    / cdiv 9679   NNcn 10002   2c2 10051   NN0cn0 10223   ZZcz 10284   ZZ>=cuz 10490   RR+crp 10614   (,]cioc 10919    seq cseq 11325   ^cexp 11384   !cfa 11568   abscabs 12041    ~~> cli 12280   sum_csu 12481
This theorem is referenced by:  aaliou3lem4  20265  aaliou3lem7  20268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-ioc 10923  df-ico 10924  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-fac 11569  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482
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