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Theorem aaliou3lem3 19740
Description: Lemma for aaliou3 19747. (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 2296 . . 3  |-  ( ZZ>= `  A )  =  (
ZZ>= `  A )
2 nnz 10061 . . . 4  |-  ( A  e.  NN  ->  A  e.  ZZ )
3 uzid 10258 . . . 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 19738 . . 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 19739 . . . . 5  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  e.  ( 0 (,] ( G `  b ) ) )
9 0xr 8894 . . . . . 6  |-  0  e.  RR*
10 elioc2 10729 . . . . . 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 9832 . . . . . . 7  |-  2  e.  CC
15 2ne0 9845 . . . . . . 7  |-  2  =/=  0
1614, 15reccli 9506 . . . . . 6  |-  ( 1  /  2 )  e.  CC
1716a1i 10 . . . . 5  |-  ( A  e.  NN  ->  (
1  /  2 )  e.  CC )
18 2re 9831 . . . . . . . . . 10  |-  2  e.  RR
1918, 15rereccli 9541 . . . . . . . . 9  |-  ( 1  /  2 )  e.  RR
20 halfgt0 9948 . . . . . . . . 9  |-  0  <  ( 1  /  2
)
2119, 20elrpii 10373 . . . . . . . 8  |-  ( 1  /  2 )  e.  RR+
22 rprege0 10384 . . . . . . . 8  |-  ( ( 1  /  2 )  e.  RR+  ->  ( ( 1  /  2 )  e.  RR  /\  0  <_  ( 1  /  2
) ) )
23 absid 11797 . . . . . . . 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 9949 . . . . . . 7  |-  ( 1  /  2 )  <  1
2624, 25eqbrtri 4058 . . . . . 6  |-  ( abs `  ( 1  /  2
) )  <  1
2726a1i 10 . . . . 5  |-  ( A  e.  NN  ->  ( abs `  ( 1  / 
2 ) )  <  1 )
28 2rp 10375 . . . . . . 7  |-  2  e.  RR+
29 nnnn0 9988 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  e.  NN0 )
30 faccl 11314 . . . . . . . . . 10  |-  ( A  e.  NN0  ->  ( ! `
 A )  e.  NN )
3129, 30syl 15 . . . . . . . . 9  |-  ( A  e.  NN  ->  ( ! `  A )  e.  NN )
3231nnzd 10132 . . . . . . . 8  |-  ( A  e.  NN  ->  ( ! `  A )  e.  ZZ )
3332znegcld 10135 . . . . . . 7  |-  ( A  e.  NN  ->  -u ( ! `  A )  e.  ZZ )
34 rpexpcl 11138 . . . . . . 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 10408 . . . . 5  |-  ( A  e.  NN  ->  (
2 ^ -u ( ! `  A )
)  e.  CC )
372, 17, 27, 36, 5geolim3 19735 . . . 4  |-  ( A  e.  NN  ->  seq  A (  +  ,  G
)  ~~>  ( ( 2 ^ -u ( ! `
 A ) )  /  ( 1  -  ( 1  /  2
) ) ) )
38 seqex 11064 . . . . 5  |-  seq  A
(  +  ,  G
)  e.  _V
39 ovex 5899 . . . . 5  |-  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  e.  _V
4038, 39breldm 4899 . . . 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 10404 . . . 4  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  e.  RR+ )
4443rpge0d 10410 . . 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 12290 . 2  |-  ( A  e.  NN  ->  seq  A (  +  ,  F
)  e.  dom  ~~>  )
47 eqidd 2297 . . 3  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( F `  b
)  =  ( F `
 b ) )
481, 1, 4, 47, 43, 46isumrpcl 12318 . 2  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( F `  b
)  e.  RR+ )
49 eqidd 2297 . . . 4  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( G `  b
)  =  ( G `
 b ) )
501, 2, 47, 13, 49, 6, 45, 46, 41isumle 12319 . . 3  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( F `  b
)  <_  sum_ b  e.  ( ZZ>= `  A )
( G `  b
) )
516recnd 8877 . . . . 5  |-  ( ( A  e.  NN  /\  b  e.  ( ZZ>= `  A ) )  -> 
( G `  b
)  e.  CC )
521, 2, 49, 51, 37isumclim 12236 . . . 4  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( G `  b
)  =  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) ) )
53 ax-1cn 8811 . . . . . . 7  |-  1  e.  CC
54 2halves 9956 . . . . . . . 8  |-  ( 1  e.  CC  ->  (
( 1  /  2
)  +  ( 1  /  2 ) )  =  1 )
5553, 54ax-mp 8 . . . . . . 7  |-  ( ( 1  /  2 )  +  ( 1  / 
2 ) )  =  1
5653, 16, 16, 55subaddrii 9151 . . . . . 6  |-  ( 1  -  ( 1  / 
2 ) )  =  ( 1  /  2
)
5756oveq2i 5885 . . . . 5  |-  ( ( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  =  ( ( 2 ^ -u ( ! `
 A ) )  /  ( 1  / 
2 ) )
58 mulcl 8837 . . . . . . . 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 9544 . . . . . 6  |-  ( A  e.  NN  ->  (
( ( 2 ^
-u ( ! `  A ) )  x.  2 )  /  1
)  =  ( ( 2 ^ -u ( ! `  A )
)  x.  2 ) )
61 1rp 10374 . . . . . . . . 9  |-  1  e.  RR+
62 rpcnne0 10387 . . . . . . . . 9  |-  ( 1  e.  RR+  ->  ( 1  e.  CC  /\  1  =/=  0 ) )
6361, 62ax-mp 8 . . . . . . . 8  |-  ( 1  e.  CC  /\  1  =/=  0 )
64 rpcnne0 10387 . . . . . . . . 9  |-  ( 2  e.  RR+  ->  ( 2  e.  CC  /\  2  =/=  0 ) )
6528, 64ax-mp 8 . . . . . . . 8  |-  ( 2  e.  CC  /\  2  =/=  0 )
66 divdiv2 9488 . . . . . . . 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 8839 . . . . . . 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 2338 . . . . 5  |-  ( A  e.  NN  ->  (
( 2 ^ -u ( ! `  A )
)  /  ( 1  /  2 ) )  =  ( 2  x.  ( 2 ^ -u ( ! `  A )
) ) )
7257, 71syl5eq 2340 . . . 4  |-  ( A  e.  NN  ->  (
( 2 ^ -u ( ! `  A )
)  /  ( 1  -  ( 1  / 
2 ) ) )  =  ( 2  x.  ( 2 ^ -u ( ! `  A )
) ) )
7352, 72eqtrd 2328 . . 3  |-  ( A  e.  NN  ->  sum_ b  e.  ( ZZ>= `  A )
( G `  b
)  =  ( 2  x.  ( 2 ^
-u ( ! `  A ) ) ) )
7450, 73breqtrd 4063 . 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   (,]cioc 10673    seq cseq 11062   ^cexp 11120   !cfa 11304   abscabs 11735    ~~> cli 11974   sum_csu 12174
This theorem is referenced by:  aaliou3lem4  19742  aaliou3lem7  19745
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-ioc 10677  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175
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