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Theorem dchrvmasumif 20646
Description: An asymptotic approximation for the sum of  X ( n )Λ
( n )  /  n conditional on the value of the infinite sum  S. (We will later show that the case  S  =  0 is impossible, and hence establish dchrvmasum 20668.) (Contributed by Mario Carneiro, 5-May-2016.)
Hypotheses
Ref Expression
rpvmasum.z  |-  Z  =  (ℤ/n `  N )
rpvmasum.l  |-  L  =  ( ZRHom `  Z
)
rpvmasum.a  |-  ( ph  ->  N  e.  NN )
rpvmasum.g  |-  G  =  (DChr `  N )
rpvmasum.d  |-  D  =  ( Base `  G
)
rpvmasum.1  |-  .1.  =  ( 0g `  G )
dchrisum.b  |-  ( ph  ->  X  e.  D )
dchrisum.n1  |-  ( ph  ->  X  =/=  .1.  )
dchrvmasumif.f  |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )
dchrvmasumif.c  |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )
dchrvmasumif.s  |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  S )
dchrvmasumif.1  |-  ( ph  ->  A. y  e.  ( 1 [,)  +oo )
( abs `  (
(  seq  1 (  +  ,  F ) `
 ( |_ `  y ) )  -  S ) )  <_ 
( C  /  y
) )
Assertion
Ref Expression
dchrvmasumif  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n
) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  e.  O
( 1 ) )
Distinct variable groups:    x, n, y,  .1.    C, n, x, y   
n, F, x, y   
x, a, y    n, N, x, y    ph, n, x    S, n, x, y   
n, Z, x, y    D, n, x, y    n, a, L, x, y    X, a, n, x, y
Dummy variables  c 
t are mutually distinct and distinct from all other variables.
Allowed substitution hints:    ph( y, a)    C( a)    D( a)    S( a)    .1. ( a)    F( a)    G( x, y, n, a)    N( a)    Z( a)

Proof of Theorem dchrvmasumif
StepHypRef Expression
1 rpvmasum.z . . 3  |-  Z  =  (ℤ/n `  N )
2 rpvmasum.l . . 3  |-  L  =  ( ZRHom `  Z
)
3 rpvmasum.a . . 3  |-  ( ph  ->  N  e.  NN )
4 rpvmasum.g . . 3  |-  G  =  (DChr `  N )
5 rpvmasum.d . . 3  |-  D  =  ( Base `  G
)
6 rpvmasum.1 . . 3  |-  .1.  =  ( 0g `  G )
7 dchrisum.b . . 3  |-  ( ph  ->  X  e.  D )
8 dchrisum.n1 . . 3  |-  ( ph  ->  X  =/=  .1.  )
9 eqid 2284 . . 3  |-  ( a  e.  NN  |->  ( ( X `  ( L `
 a ) )  x.  ( ( log `  a )  /  a
) ) )  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  x.  ( ( log `  a )  /  a ) ) )
101, 2, 3, 4, 5, 6, 7, 8, 9dchrvmasumlema 20643 . 2  |-  ( ph  ->  E. t E. c  e.  ( 0 [,)  +oo ) (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) ) `  ( |_ `  y ) )  -  t ) )  <_  ( c  x.  ( ( log `  y
)  /  y ) ) ) )
113adantr 453 . . . . . 6  |-  ( (
ph  /\  ( c  e.  ( 0 [,)  +oo )  /\  (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) ) `  ( |_ `  y ) )  -  t ) )  <_  ( c  x.  ( ( log `  y
)  /  y ) ) ) ) )  ->  N  e.  NN )
127adantr 453 . . . . . 6  |-  ( (
ph  /\  ( c  e.  ( 0 [,)  +oo )  /\  (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) ) `  ( |_ `  y ) )  -  t ) )  <_  ( c  x.  ( ( log `  y
)  /  y ) ) ) ) )  ->  X  e.  D
)
138adantr 453 . . . . . 6  |-  ( (
ph  /\  ( c  e.  ( 0 [,)  +oo )  /\  (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) ) `  ( |_ `  y ) )  -  t ) )  <_  ( c  x.  ( ( log `  y
)  /  y ) ) ) ) )  ->  X  =/=  .1.  )
14 dchrvmasumif.f . . . . . 6  |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )
15 dchrvmasumif.c . . . . . . 7  |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )
1615adantr 453 . . . . . 6  |-  ( (
ph  /\  ( c  e.  ( 0 [,)  +oo )  /\  (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) ) `  ( |_ `  y ) )  -  t ) )  <_  ( c  x.  ( ( log `  y
)  /  y ) ) ) ) )  ->  C  e.  ( 0 [,)  +oo )
)
17 dchrvmasumif.s . . . . . . 7  |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  S )
1817adantr 453 . . . . . 6  |-  ( (
ph  /\  ( c  e.  ( 0 [,)  +oo )  /\  (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) ) `  ( |_ `  y ) )  -  t ) )  <_  ( c  x.  ( ( log `  y
)  /  y ) ) ) ) )  ->  seq  1 (  +  ,  F )  ~~>  S )
19 dchrvmasumif.1 . . . . . . 7  |-  ( ph  ->  A. y  e.  ( 1 [,)  +oo )
( abs `  (
(  seq  1 (  +  ,  F ) `
 ( |_ `  y ) )  -  S ) )  <_ 
( C  /  y
) )
2019adantr 453 . . . . . 6  |-  ( (
ph  /\  ( c  e.  ( 0 [,)  +oo )  /\  (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) ) `  ( |_ `  y ) )  -  t ) )  <_  ( c  x.  ( ( log `  y
)  /  y ) ) ) ) )  ->  A. y  e.  ( 1 [,)  +oo )
( abs `  (
(  seq  1 (  +  ,  F ) `
 ( |_ `  y ) )  -  S ) )  <_ 
( C  /  y
) )
21 simprl 734 . . . . . 6  |-  ( (
ph  /\  ( c  e.  ( 0 [,)  +oo )  /\  (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) ) `  ( |_ `  y ) )  -  t ) )  <_  ( c  x.  ( ( log `  y
)  /  y ) ) ) ) )  ->  c  e.  ( 0 [,)  +oo )
)
22 simprrl 742 . . . . . 6  |-  ( (
ph  /\  ( c  e.  ( 0 [,)  +oo )  /\  (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) ) `  ( |_ `  y ) )  -  t ) )  <_  ( c  x.  ( ( log `  y
)  /  y ) ) ) ) )  ->  seq  1 (  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `
 a ) )  x.  ( ( log `  a )  /  a
) ) ) )  ~~>  t )
23 simprrr 743 . . . . . 6  |-  ( (
ph  /\  ( c  e.  ( 0 [,)  +oo )  /\  (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) ) `  ( |_ `  y ) )  -  t ) )  <_  ( c  x.  ( ( log `  y
)  /  y ) ) ) ) )  ->  A. y  e.  ( 3 [,)  +oo )
( abs `  (
(  seq  1 (  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `
 a ) )  x.  ( ( log `  a )  /  a
) ) ) ) `
 ( |_ `  y ) )  -  t ) )  <_ 
( c  x.  (
( log `  y
)  /  y ) ) )
241, 2, 11, 4, 5, 6, 12, 13, 14, 16, 18, 20, 9, 21, 22, 23dchrvmasumiflem2 20645 . . . . 5  |-  ( (
ph  /\  ( c  e.  ( 0 [,)  +oo )  /\  (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) ) `  ( |_ `  y ) )  -  t ) )  <_  ( c  x.  ( ( log `  y
)  /  y ) ) ) ) )  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n
) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  e.  O
( 1 ) )
2524expr 600 . . . 4  |-  ( (
ph  /\  c  e.  ( 0 [,)  +oo ) )  ->  (
(  seq  1 (  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `
 a ) )  x.  ( ( log `  a )  /  a
) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  (
(  seq  1 (  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `
 a ) )  x.  ( ( log `  a )  /  a
) ) ) ) `
 ( |_ `  y ) )  -  t ) )  <_ 
( c  x.  (
( log `  y
)  /  y ) ) )  ->  (
x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( X `  ( L `
 n ) )  x.  ( (Λ `  n
)  /  n ) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  e.  O
( 1 ) ) )
2625rexlimdva 2668 . . 3  |-  ( ph  ->  ( E. c  e.  ( 0 [,)  +oo ) (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) ) `  ( |_ `  y ) )  -  t ) )  <_  ( c  x.  ( ( log `  y
)  /  y ) ) )  ->  (
x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( X `  ( L `
 n ) )  x.  ( (Λ `  n
)  /  n ) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  e.  O
( 1 ) ) )
2726exlimdv 1665 . 2  |-  ( ph  ->  ( E. t E. c  e.  ( 0 [,)  +oo ) (  seq  1 (  +  , 
( a  e.  NN  |->  ( ( X `  ( L `  a ) )  x.  ( ( log `  a )  /  a ) ) ) )  ~~>  t  /\  A. y  e.  ( 3 [,)  +oo ) ( abs `  ( (  seq  1
(  +  ,  ( a  e.  NN  |->  ( ( X `  ( L `  a )
)  x.  ( ( log `  a )  /  a ) ) ) ) `  ( |_ `  y ) )  -  t ) )  <_  ( c  x.  ( ( log `  y
)  /  y ) ) )  ->  (
x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( X `  ( L `
 n ) )  x.  ( (Λ `  n
)  /  n ) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  e.  O
( 1 ) ) )
2810, 27mpd 16 1  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n
) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  e.  O
( 1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1529    = wceq 1624    e. wcel 1685    =/= wne 2447   A.wral 2544   E.wrex 2545   ifcif 3566   class class class wbr 4024    e. cmpt 4078   ` cfv 5221  (class class class)co 5819   0cc0 8732   1c1 8733    + caddc 8735    x. cmul 8737    +oocpnf 8859    <_ cle 8863    - cmin 9032    / cdiv 9418   NNcn 9741   3c3 9791   RR+crp 10349   [,)cico 10652   ...cfz 10776   |_cfl 10918    seq cseq 11040   abscabs 11713    ~~> cli 11952   O (
1 )co1 11954   sum_csu 12152   Basecbs 13142   0gc0g 13394   ZRHomczrh 16445  ℤ/nczn 16448   logclog 19906  Λcvma 20323  DChrcdchr 20465
This theorem is referenced by:  rpvmasum2  20655  dchrvmasumlem  20666
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-disj 3995  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-tpos 6195  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6655  df-ec 6657  df-qs 6661  df-map 6769  df-pm 6770  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-acn 7570  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10654  df-ioc 10655  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-fl 10919  df-mod 10968  df-seq 11041  df-exp 11099  df-fac 11283  df-bc 11310  df-hash 11332  df-shft 11556  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-limsup 11939  df-clim 11956  df-rlim 11957  df-o1 11958  df-lo1 11959  df-sum 12153  df-ef 12343  df-e 12344  df-sin 12345  df-cos 12346  df-pi 12348  df-dvds 12526  df-gcd 12680  df-prm 12753  df-phi 12828  df-pc 12884  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-starv 13217  df-sca 13218  df-vsca 13219  df-tset 13221  df-ple 13222  df-ds 13224  df-hom 13226  df-cco 13227  df-rest 13321  df-topn 13322  df-topgen 13338  df-pt 13339  df-prds 13342  df-xrs 13397  df-0g 13398  df-gsum 13399  df-qtop 13404  df-imas 13405  df-divs 13406  df-xps 13407  df-mre 13482  df-mrc 13483  df-acs 13485  df-mnd 14361  df-mhm 14409  df-submnd 14410  df-grp 14483  df-minusg 14484  df-sbg 14485  df-mulg 14486  df-subg 14612  df-nsg 14613  df-eqg 14614  df-ghm 14675  df-cntz 14787  df-od 14838  df-cmn 15085  df-abl 15086  df-mgp 15320  df-rng 15334  df-cring 15335  df-ur 15336  df-oppr 15399  df-dvdsr 15417  df-unit 15418  df-invr 15448  df-dvr 15459  df-rnghom 15490  df-drng 15508  df-subrg 15537  df-lmod 15623  df-lss 15684  df-lsp 15723  df-sra 15919  df-rgmod 15920  df-lidl 15921  df-rsp 15922  df-2idl 15978  df-xmet 16367  df-met 16368  df-bl 16369  df-mopn 16370  df-cnfld 16372  df-zrh 16449  df-zn 16452  df-top 16630  df-bases 16632  df-topon 16633  df-topsp 16634  df-cld 16750  df-ntr 16751  df-cls 16752  df-nei 16829  df-lp 16862  df-perf 16863  df-cn 16951  df-cnp 16952  df-haus 17037  df-cmp 17108  df-tx 17251  df-hmeo 17440  df-fbas 17514  df-fg 17515  df-fil 17535  df-fm 17627  df-flim 17628  df-flf 17629  df-xms 17879  df-ms 17880  df-tms 17881  df-cncf 18376  df-limc 19210  df-dv 19211  df-log 19908  df-cxp 19909  df-em 20281  df-vma 20329  df-mu 20332  df-dchr 20466
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