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Theorem aaliou3lem3 19724
Description: Lemma for aaliou3 19731. (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 2283 . . 3  |-  ( ZZ>= `  A )  =  (
ZZ>= `  A )
2 nnz 10045 . . . 4  |-  ( A  e.  NN  ->  A  e.  ZZ )
3 uzid 10242 . . . 4  |-  ( A  e.  ZZ  ->  A  e.  ( ZZ>= `  A )
)
42, 3syl 15 . . 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 19722 . . 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 19723 . . . . 5  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  e.  ( 0 (,] ( G `  b ) ) )
9 0xr 8878 . . . . . 6  |-  0  e.  RR*
10 elioc2 10713 . . . . . 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 644 . . . . 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 201 . . . 4  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( ( F `  b )  e.  RR  /\  0  <  ( F `
 b )  /\  ( F `  b )  <_  ( G `  b ) ) )
1312simp1d 967 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  e.  RR )
14 2cn 9816 . . . . . . 7  |-  2  e.  CC
15 2ne0 9829 . . . . . . 7  |-  2  =/=  0
1614, 15reccli 9490 . . . . . 6  |-  ( 1  /  2 )  e.  CC
1716a1i 10 . . . . 5  |-  ( A  e.  NN  ->  (
1  /  2 )  e.  CC )
18 2re 9815 . . . . . . . . . 10  |-  2  e.  RR
1918, 15rereccli 9525 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
20 halfgt0 9932 . . . . . . . . 9  |-  0  <  ( 1  /  2
)
2119, 20elrpii 10357 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR+
22 rprege0 10368 . . . . . . . 8  |-  ( ( 1  /  2 )  e.  RR+  ->  ( ( 1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
) ) )
23 absid 11781 . . . . . . . 8  |-  ( ( ( 1  /  2
)  e.  RR  /\  0  <_  ( 1  / 
2 ) )  -> 
( abs `  (
1  /  2 ) )  =  ( 1  /  2 ) )
2421, 22, 23mp2b 9 . . . . . . 7  |-  ( abs `  ( 1  /  2
) )  =  ( 1  /  2 )
25 halflt1 9933 . . . . . . 7  |-  ( 1  /  2 )  <  1
2624, 25eqbrtri 4042 . . . . . 6  |-  ( abs `  ( 1  /  2
) )  <  1
2726a1i 10 . . . . 5  |-  ( A  e.  NN  ->  ( abs `  ( 1  / 
2 ) )  <  1 )
28 2rp 10359 . . . . . . 7  |-  2  e.  RR+
29 nnnn0 9972 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  NN0 )
30 faccl 11298 . . . . . . . . . 10  |-  ( A  e.  NN0  ->  ( ! `
 A )  e.  NN )
3129, 30syl 15 . . . . . . . . 9  |-  ( A  e.  NN  ->  ( ! `  A )  e.  NN )
3231nnzd 10116 . . . . . . . 8  |-  ( A  e.  NN  ->  ( ! `  A )  e.  ZZ )
3332znegcld 10119 . . . . . . 7  |-  ( A  e.  NN  ->  -u ( ! `  A )  e.  ZZ )
34 rpexpcl 11122 . . . . . . 7  |-  ( ( 2  e.  RR+  /\  -u ( ! `  A )  e.  ZZ )  ->  (
2 ^ -u ( ! `  A )
)  e.  RR+ )
3528, 33, 34sylancr 644 . . . . . 6  |-  ( A  e.  NN  ->  (
2 ^ -u ( ! `  A )
)  e.  RR+ )
3635rpcnd 10392 . . . . 5  |-  ( A  e.  NN  ->  (
2 ^ -u ( ! `  A )
)  e.  CC )
372, 17, 27, 36, 5geolim3 19719 . . . 4  |-  ( A  e.  NN  ->  seq  A (  +  ,  G
)  ~~>  ( ( 2 ^ -u ( ! `
 A ) )  /  ( 1  -  ( 1  /  2
) ) ) )
38 seqex 11048 . . . . 5  |-  seq  A
(  +  ,  G
)  e.  _V
39 ovex 5883 . . . . 5  |-  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  e.  _V
4038, 39breldm 4883 . . . 4  |-  (  seq 
A (  +  ,  G )  ~~>  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  ->  seq  A (  +  ,  G )  e.  dom  ~~>  )
4137, 40syl 15 . . 3  |-  ( A  e.  NN  ->  seq  A (  +  ,  G
)  e.  dom  ~~>  )
4212simp2d 968 . . . . 5  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
0  <  ( F `  b ) )
4313, 42elrpd 10388 . . . 4  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  e.  RR+ )
4443rpge0d 10394 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
0  <_  ( F `  b ) )
4512simp3d 969 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  <_  ( G `  b ) )
461, 4, 6, 13, 41, 44, 45cvgcmp 12274 . 2  |-  ( A  e.  NN  ->  seq  A (  +  ,  F
)  e.  dom  ~~>  )
47 eqidd 2284 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  =  ( F `
 b ) )
481, 1, 4, 47, 43, 46isumrpcl 12302 . 2  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( F `  b
)  e.  RR+ )
49 eqidd 2284 . . . 4  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( G `  b
)  =  ( G `
 b ) )
501, 2, 47, 13, 49, 6, 45, 46, 41isumle 12303 . . 3  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( F `  b
)  <_  sum_ b  e.  ( ZZ>= `  A )
( G `  b
) )
516recnd 8861 . . . . 5  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( G `  b
)  e.  CC )
521, 2, 49, 51, 37isumclim 12220 . . . 4  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( G `  b
)  =  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) ) )
53 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
54 2halves 9940 . . . . . . . 8  |-  ( 1  e.  CC  ->  (
( 1  /  2
)  +  ( 1  /  2 ) )  =  1 )
5553, 54ax-mp 8 . . . . . . 7  |-  ( ( 1  /  2 )  +  ( 1  / 
2 ) )  =  1
5653, 16, 16, 55subaddrii 9135 . . . . . 6  |-  ( 1  -  ( 1  / 
2 ) )  =  ( 1  /  2
)
5756oveq2i 5869 . . . . 5  |-  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  =  ( ( 2 ^ -u ( ! `
 A ) )  /  ( 1  / 
2 ) )
58 mulcl 8821 . . . . . . . 8  |-  ( ( ( 2 ^ -u ( ! `  A )
)  e.  CC  /\  2  e.  CC )  ->  ( ( 2 ^
-u ( ! `  A ) )  x.  2 )  e.  CC )
5936, 14, 58sylancl 643 . . . . . . 7  |-  ( A  e.  NN  ->  (
( 2 ^ -u ( ! `  A )
)  x.  2 )  e.  CC )
6059div1d 9528 . . . . . 6  |-  ( A  e.  NN  ->  (
( ( 2 ^
-u ( ! `  A ) )  x.  2 )  /  1
)  =  ( ( 2 ^ -u ( ! `  A )
)  x.  2 ) )
61 1rp 10358 . . . . . . . . 9  |-  1  e.  RR+
62 rpcnne0 10371 . . . . . . . . 9  |-  ( 1  e.  RR+  ->  ( 1  e.  CC  /\  1  =/=  0 ) )
6361, 62ax-mp 8 . . . . . . . 8  |-  ( 1  e.  CC  /\  1  =/=  0 )
64 rpcnne0 10371 . . . . . . . . 9  |-  ( 2  e.  RR+  ->  ( 2  e.  CC  /\  2  =/=  0 ) )
6528, 64ax-mp 8 . . . . . . . 8  |-  ( 2  e.  CC  /\  2  =/=  0 )
66 divdiv2 9472 . . . . . . . 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
) )
6763, 65, 66mp3an23 1269 . . . . . . 7  |-  ( ( 2 ^ -u ( ! `  A )
)  e.  CC  ->  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  /  2 ) )  =  ( ( ( 2 ^ -u ( ! `  A )
)  x.  2 )  /  1 ) )
6836, 67syl 15 . . . . . 6  |-  ( A  e.  NN  ->  (
( 2 ^ -u ( ! `  A )
)  /  ( 1  /  2 ) )  =  ( ( ( 2 ^ -u ( ! `  A )
)  x.  2 )  /  1 ) )
69 mulcom 8823 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( 2 ^ -u ( ! `  A )
)  e.  CC )  ->  ( 2  x.  ( 2 ^ -u ( ! `  A )
) )  =  ( ( 2 ^ -u ( ! `  A )
)  x.  2 ) )
7014, 36, 69sylancr 644 . . . . . 6  |-  ( A  e.  NN  ->  (
2  x.  ( 2 ^ -u ( ! `
 A ) ) )  =  ( ( 2 ^ -u ( ! `  A )
)  x.  2 ) )
7160, 68, 703eqtr4d 2325 . . . . 5  |-  ( A  e.  NN  ->  (
( 2 ^ -u ( ! `  A )
)  /  ( 1  /  2 ) )  =  ( 2  x.  ( 2 ^ -u ( ! `  A )
) ) )
7257, 71syl5eq 2327 . . . 4  |-  ( A  e.  NN  ->  (
( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  =  ( 2  x.  ( 2 ^ -u ( ! `  A )
) ) )
7352, 72eqtrd 2315 . . 3  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( G `  b
)  =  ( 2  x.  ( 2 ^
-u ( ! `  A ) ) ) )
7450, 73breqtrd 4047 . 2  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( F `  b
)  <_  ( 2  x.  ( 2 ^
-u ( ! `  A ) ) ) )
7546, 48, 743jca 1132 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   (,]cioc 10657    seq cseq 11046   ^cexp 11104   !cfa 11288   abscabs 11719    ~~> cli 11958   sum_csu 12158
This theorem is referenced by:  aaliou3lem4  19726  aaliou3lem7  19729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ioc 10661  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159
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