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Theorem aaliou3lem3 24099
Description: Lemma for aaliou3 24106. (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 2622 . . 3 |- ( ZZ>= ` A ) = ( ZZ>= ` A )
2 nnz 11399 . . . 4 |- ( A e. NN -> A e. ZZ )
3 uzid 11702 . . . 4 |- ( A e. ZZ -> A e. ( ZZ>= ` A ) )
42, 3syl 17 . . 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 24097 . . 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 24098 . . . . 5 |- ( ( A e. NN /\ b e. ( ZZ>= ` A ) ) -> ( F ` b ) e. ( 0 (,] ( G ` b ) ) )
9 0xr 10086 . . . . . 6 |- 0 e. RR*
10 elioc2 12236 . . . . . 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 695 . . . . 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 222 . . . 4 |- ( ( A e. NN /\ b e. ( ZZ>= ` A ) ) -> ( ( F ` b ) e. RR /\ 0 < ( F ` b ) /\ ( F ` b ) <_ ( G ` b ) ) )
1312simp1d 1073 . . 3 |- ( ( A e. NN /\ b e. ( ZZ>= ` A ) ) -> ( F ` b ) e. RR )
14 halfcn 11247 . . . . . 6 |- ( 1 / 2 ) e. CC
1514a1i 11 . . . . 5 |- ( A e. NN -> ( 1 / 2 ) e. CC )
16 halfre 11246 . . . . . . . . 9 |- ( 1 / 2 ) e. RR
17 halfgt0 11248 . . . . . . . . 9 |- 0 < ( 1 / 2 )
1816, 17elrpii 11835 . . . . . . . 8 |- ( 1 / 2 ) e. RR+
19 rprege0 11847 . . . . . . . 8 |- ( ( 1 / 2 ) e. RR+ -> ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) ) )
20 absid 14036 . . . . . . . 8 |- ( ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) ) -> ( abs ` ( 1 / 2 ) ) = ( 1 / 2 ) )
2118, 19, 20mp2b 10 . . . . . . 7 |- ( abs ` ( 1 / 2 ) ) = ( 1 / 2 )
22 halflt1 11250 . . . . . . 7 |- ( 1 / 2 ) < 1
2321, 22eqbrtri 4674 . . . . . 6 |- ( abs ` ( 1 / 2 ) ) < 1
2423a1i 11 . . . . 5 |- ( A e. NN -> ( abs ` ( 1 / 2 ) ) < 1 )
25 2rp 11837 . . . . . . 7 |- 2 e. RR+
26 nnnn0 11299 . . . . . . . . . 10 |- ( A e. NN -> A e. NN0 )
27 faccl 13070 . . . . . . . . . 10 |- ( A e. NN0 -> ( ! ` A ) e. NN )
2826, 27syl 17 . . . . . . . . 9 |- ( A e. NN -> ( ! ` A ) e. NN )
2928nnzd 11481 . . . . . . . 8 |- ( A e. NN -> ( ! ` A ) e. ZZ )
3029znegcld 11484 . . . . . . 7 |- ( A e. NN -> -u ( ! ` A ) e. ZZ )
31 rpexpcl 12879 . . . . . . 7 |- ( ( 2 e. RR+ /\ -u ( ! ` A ) e. ZZ ) -> ( 2 ^ -u ( ! ` A ) ) e. RR+ )
3225, 30, 31sylancr 695 . . . . . 6 |- ( A e. NN -> ( 2 ^ -u ( ! ` A ) ) e. RR+ )
3332rpcnd 11874 . . . . 5 |- ( A e. NN -> ( 2 ^ -u ( ! ` A ) ) e. CC )
342, 15, 24, 33, 5geolim3 24094 . . . 4 |- ( A e. NN -> seq A ( + , G ) ~~> ( ( 2 ^ -u ( ! ` A ) ) / ( 1 - ( 1 / 2 ) ) ) )
35 seqex 12803 . . . . 5 |- seq A ( + , G ) e. _V
36 ovex 6678 . . . . 5 |- ( ( 2 ^ -u ( ! ` A ) ) / ( 1 - ( 1 / 2 ) ) ) e. _V
3735, 36breldm 5329 . . . 4 |- ( seq A ( + , G ) ~~> ( ( 2 ^ -u ( ! ` A ) ) / ( 1 - ( 1 / 2 ) ) ) -> seq A ( + , G ) e. dom ~~> )
3834, 37syl 17 . . 3 |- ( A e. NN -> seq A ( + , G ) e. dom ~~> )
3912simp2d 1074 . . . . 5 |- ( ( A e. NN /\ b e. ( ZZ>= ` A ) ) -> 0 < ( F ` b ) )
4013, 39elrpd 11869 . . . 4 |- ( ( A e. NN /\ b e. ( ZZ>= ` A ) ) -> ( F ` b ) e. RR+ )
4140rpge0d 11876 . . 3 |- ( ( A e. NN /\ b e. ( ZZ>= ` A ) ) -> 0 <_ ( F ` b ) )
4212simp3d 1075 . . 3 |- ( ( A e. NN /\ b e. ( ZZ>= ` A ) ) -> ( F ` b ) <_ ( G ` b ) )
431, 4, 6, 13, 38, 41, 42cvgcmp 14548 . 2 |- ( A e. NN -> seq A ( + , F ) e. dom ~~> )
44 eqidd 2623 . . 3 |- ( ( A e. NN /\ b e. ( ZZ>= ` A ) ) -> ( F ` b ) = ( F ` b ) )
451, 1, 4, 44, 40, 43isumrpcl 14575 . 2 |- ( A e. NN -> sum_ b e. ( ZZ>= ` A ) ( F ` b ) e. RR+ )
46 eqidd 2623 . . . 4 |- ( ( A e. NN /\ b e. ( ZZ>= ` A ) ) -> ( G ` b ) = ( G ` b ) )
471, 2, 44, 13, 46, 6, 42, 43, 38isumle 14576 . . 3 |- ( A e. NN -> sum_ b e. ( ZZ>= ` A ) ( F ` b ) <_ sum_ b e. ( ZZ>= ` A ) ( G ` b ) )
486recnd 10068 . . . . 5 |- ( ( A e. NN /\ b e. ( ZZ>= ` A ) ) -> ( G ` b ) e. CC )
491, 2, 46, 48, 34isumclim 14488 . . . 4 |- ( A e. NN -> sum_ b e. ( ZZ>= ` A ) ( G ` b ) = ( ( 2 ^ -u ( ! ` A ) ) / ( 1 - ( 1 / 2 ) ) ) )
50 1mhlfehlf 11251 . . . . . 6 |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
5150oveq2i 6661 . . . . 5 |- ( ( 2 ^ -u ( ! ` A ) ) / ( 1 - ( 1 / 2 ) ) ) = ( ( 2 ^ -u ( ! ` A ) ) / ( 1 / 2 ) )
52 2cn 11091 . . . . . . . 8 |- 2 e. CC
53 mulcl 10020 . . . . . . . 8 |- ( ( ( 2 ^ -u ( ! ` A ) ) e. CC /\ 2 e. CC ) -> ( ( 2 ^ -u ( ! ` A ) ) x. 2 ) e. CC )
5433, 52, 53sylancl 694 . . . . . . 7 |- ( A e. NN -> ( ( 2 ^ -u ( ! ` A ) ) x. 2 ) e. CC )
5554div1d 10793 . . . . . 6 |- ( A e. NN -> ( ( ( 2 ^ -u ( ! ` A ) ) x. 2 ) / 1 ) = ( ( 2 ^ -u ( ! ` A ) ) x. 2 ) )
56 1rp 11836 . . . . . . . . 9 |- 1 e. RR+
57 rpcnne0 11850 . . . . . . . . 9 |- ( 1 e. RR+ -> ( 1 e. CC /\ 1 =/= 0 ) )
5856, 57ax-mp 5 . . . . . . . 8 |- ( 1 e. CC /\ 1 =/= 0 )
59 2cnne0 11242 . . . . . . . 8 |- ( 2 e. CC /\ 2 =/= 0 )
60 divdiv2 10737 . . . . . . . 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 ) )
6158, 59, 60mp3an23 1416 . . . . . . 7 |- ( ( 2 ^ -u ( ! ` A ) ) e. CC -> ( ( 2 ^ -u ( ! ` A ) ) / ( 1 / 2 ) ) = ( ( ( 2 ^ -u ( ! ` A ) ) x. 2 ) / 1 ) )
6233, 61syl 17 . . . . . 6 |- ( A e. NN -> ( ( 2 ^ -u ( ! ` A ) ) / ( 1 / 2 ) ) = ( ( ( 2 ^ -u ( ! ` A ) ) x. 2 ) / 1 ) )
63 mulcom 10022 . . . . . . 7 |- ( ( 2 e. CC /\ ( 2 ^ -u ( ! ` A ) ) e. CC ) -> ( 2 x. ( 2 ^ -u ( ! ` A ) ) ) = ( ( 2 ^ -u ( ! ` A ) ) x. 2 ) )
6452, 33, 63sylancr 695 . . . . . 6 |- ( A e. NN -> ( 2 x. ( 2 ^ -u ( ! ` A ) ) ) = ( ( 2 ^ -u ( ! ` A ) ) x. 2 ) )
6555, 62, 643eqtr4d 2666 . . . . 5 |- ( A e. NN -> ( ( 2 ^ -u ( ! ` A ) ) / ( 1 / 2 ) ) = ( 2 x. ( 2 ^ -u ( ! ` A ) ) ) )
6651, 65syl5eq 2668 . . . 4 |- ( A e. NN -> ( ( 2 ^ -u ( ! ` A ) ) / ( 1 - ( 1 / 2 ) ) ) = ( 2 x. ( 2 ^ -u ( ! ` A ) ) ) )
6749, 66eqtrd 2656 . . 3 |- ( A e. NN -> sum_ b e. ( ZZ>= ` A ) ( G ` b ) = ( 2 x. ( 2 ^ -u ( ! ` A ) ) ) )
6847, 67breqtrd 4679 . 2 |- ( A e. NN -> sum_ b e. ( ZZ>= ` A ) ( F ` b ) <_ ( 2 x. ( 2 ^ -u ( ! ` A ) ) ) )
6943, 45, 683jca 1242 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 setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   (,]cioc 12176   seqcseq 12801   ^cexp 12860   !cfa 13060   abscabs 13974   ~~> cli 14215   sum_csu 14416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ioc 12180  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417
This theorem is referenced by:  aaliou3lem4  24101  aaliou3lem7  24104
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