MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dchrvmasumlem3 Structured version   Unicode version

Theorem dchrvmasumlem3 22746
Description: Lemma for dchrvmasum 22772. (Contributed by Mario Carneiro, 3-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.  )
dchrvmasum.f  |-  ( (
ph  /\  m  e.  RR+ )  ->  F  e.  CC )
dchrvmasum.g  |-  ( m  =  ( x  / 
d )  ->  F  =  K )
dchrvmasum.c  |-  ( ph  ->  C  e.  ( 0 [,) +oo ) )
dchrvmasum.t  |-  ( ph  ->  T  e.  CC )
dchrvmasum.1  |-  ( (
ph  /\  m  e.  ( 3 [,) +oo ) )  ->  ( abs `  ( F  -  T ) )  <_ 
( C  x.  (
( log `  m
)  /  m ) ) )
dchrvmasum.r  |-  ( ph  ->  R  e.  RR )
dchrvmasum.2  |-  ( ph  ->  A. m  e.  ( 1 [,) 3 ) ( abs `  ( F  -  T )
)  <_  R )
Assertion
Ref Expression
dchrvmasumlem3  |-  ( ph  ->  ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  ( K  -  T ) ) )  e.  O(1) )
Distinct variable groups:    x, m,  .1.    m, d, x, C    F, d, x    m, K   
m, N, x    ph, d, m, x    T, d, m, x    R, d, m, x   
m, Z, x    D, m, x    L, d, m, x    X, d, m, x
Allowed substitution hints:    D( d)    .1. ( d)    F( m)    G( x, m, d)    K( x, d)    N( d)    Z( d)

Proof of Theorem dchrvmasumlem3
StepHypRef Expression
1 1red 9399 . 2  |-  ( ph  ->  1  e.  RR )
2 rpvmasum.z . . 3  |-  Z  =  (ℤ/n `  N )
3 rpvmasum.l . . 3  |-  L  =  ( ZRHom `  Z
)
4 rpvmasum.a . . 3  |-  ( ph  ->  N  e.  NN )
5 rpvmasum.g . . 3  |-  G  =  (DChr `  N )
6 rpvmasum.d . . 3  |-  D  =  ( Base `  G
)
7 rpvmasum.1 . . 3  |-  .1.  =  ( 0g `  G )
8 dchrisum.b . . 3  |-  ( ph  ->  X  e.  D )
9 dchrisum.n1 . . 3  |-  ( ph  ->  X  =/=  .1.  )
10 dchrvmasum.f . . 3  |-  ( (
ph  /\  m  e.  RR+ )  ->  F  e.  CC )
11 dchrvmasum.g . . 3  |-  ( m  =  ( x  / 
d )  ->  F  =  K )
12 dchrvmasum.c . . 3  |-  ( ph  ->  C  e.  ( 0 [,) +oo ) )
13 dchrvmasum.t . . 3  |-  ( ph  ->  T  e.  CC )
14 dchrvmasum.1 . . 3  |-  ( (
ph  /\  m  e.  ( 3 [,) +oo ) )  ->  ( abs `  ( F  -  T ) )  <_ 
( C  x.  (
( log `  m
)  /  m ) ) )
15 dchrvmasum.r . . 3  |-  ( ph  ->  R  e.  RR )
16 dchrvmasum.2 . . 3  |-  ( ph  ->  A. m  e.  ( 1 [,) 3 ) ( abs `  ( F  -  T )
)  <_  R )
172, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16dchrvmasumlem2 22745 . 2  |-  ( ph  ->  ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) )  e.  O(1) )
18 fzfid 11793 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
19 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
20 elfznn 11476 . . . . . . . . 9  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
2120nnrpd 11024 . . . . . . . 8  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
22 rpdivcl 11011 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
2319, 21, 22syl2an 477 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
2410ralrimiva 2797 . . . . . . . 8  |-  ( ph  ->  A. m  e.  RR+  F  e.  CC )
2524ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  A. m  e.  RR+  F  e.  CC )
2611eleq1d 2507 . . . . . . . 8  |-  ( m  =  ( x  / 
d )  ->  ( F  e.  CC  <->  K  e.  CC ) )
2726rspcv 3067 . . . . . . 7  |-  ( ( x  /  d )  e.  RR+  ->  ( A. m  e.  RR+  F  e.  CC  ->  K  e.  CC ) )
2823, 25, 27sylc 60 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  K  e.  CC )
2913ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  T  e.  CC )
3028, 29subcld 9717 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( K  -  T )  e.  CC )
3130abscld 12920 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( K  -  T
) )  e.  RR )
3220adantl 466 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
3331, 32nndivred 10368 . . 3  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( K  -  T ) )  / 
d )  e.  RR )
3418, 33fsumrecl 13209 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d )  e.  RR )
358ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  X  e.  D )
36 elfzelz 11451 . . . . . . 7  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  ZZ )
3736adantl 466 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  ZZ )
385, 2, 6, 3, 35, 37dchrzrhcl 22582 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  d
) )  e.  CC )
39 mucl 22477 . . . . . . . . 9  |-  ( d  e.  NN  ->  (
mmu `  d )  e.  ZZ )
4032, 39syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  ZZ )
4140zred 10745 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  RR )
4241, 32nndivred 10368 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  RR )
4342recnd 9410 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  CC )
4438, 43mulcld 9404 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  e.  CC )
4544, 30mulcld 9404 . . 3  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  ( K  -  T ) )  e.  CC )
4618, 45fsumcl 13208 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
)  e.  CC )
4746abscld 12920 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  e.  RR )
4834recnd 9410 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d )  e.  CC )
4948abscld 12920 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) )  e.  RR )
5045abscld 12920 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  e.  RR )
5118, 50fsumrecl 13209 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( abs `  (
( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  ( K  -  T ) ) )  e.  RR )
5218, 45fsumabs 13262 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  <_  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( abs `  (
( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  ( K  -  T ) ) ) )
5344abscld 12920 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) ) )  e.  RR )
5432nnrecred 10365 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  /  d )  e.  RR )
5530absge0d 12928 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( K  -  T ) ) )
5638, 43absmuld 12938 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) ) )  =  ( ( abs `  ( X `
 ( L `  d ) ) )  x.  ( abs `  (
( mmu `  d
)  /  d ) ) ) )
5738abscld 12920 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( X `  ( L `  d )
) )  e.  RR )
58 1red 9399 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  1  e.  RR )
5943abscld 12920 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  e.  RR )
6038absge0d 12928 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( X `
 ( L `  d ) ) ) )
6143absge0d 12928 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( ( mmu `  d )  /  d ) ) )
62 eqid 2441 . . . . . . . . . . . 12  |-  ( Base `  Z )  =  (
Base `  Z )
634nnnn0d 10634 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  NN0 )
642, 62, 3znzrhfo 17978 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
6563, 64syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  L : ZZ -onto-> ( Base `  Z ) )
66 fof 5618 . . . . . . . . . . . . . . 15  |-  ( L : ZZ -onto-> ( Base `  Z )  ->  L : ZZ --> ( Base `  Z
) )
6765, 66syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  L : ZZ --> ( Base `  Z ) )
6867ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  L : ZZ
--> ( Base `  Z
) )
6968, 37ffvelrnd 5842 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( L `  d )  e.  (
Base `  Z )
)
705, 6, 2, 62, 35, 69dchrabs2 22599 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( X `  ( L `  d )
) )  <_  1
)
7141recnd 9410 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  CC )
7232nncnd 10336 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  CC )
7332nnne0d 10364 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  =/=  0 )
7471, 72, 73absdivd 12939 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  ( abs `  d ) ) )
7532nnrpd 11024 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  RR+ )
7675rprege0d 11032 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( d  e.  RR  /\  0  <_ 
d ) )
77 absid 12783 . . . . . . . . . . . . . . 15  |-  ( ( d  e.  RR  /\  0  <_  d )  -> 
( abs `  d
)  =  d )
7876, 77syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  d )  =  d )
7978oveq2d 6105 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( mmu `  d ) )  / 
( abs `  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  d ) )
8074, 79eqtrd 2473 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  d ) )
8171abscld 12920 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( mmu `  d
) )  e.  RR )
82 mule1 22484 . . . . . . . . . . . . . 14  |-  ( d  e.  NN  ->  ( abs `  ( mmu `  d ) )  <_ 
1 )
8332, 82syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( mmu `  d
) )  <_  1
)
8481, 58, 75, 83lediv1dd 11079 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( mmu `  d ) )  / 
d )  <_  (
1  /  d ) )
8580, 84eqbrtrd 4310 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  <_  (
1  /  d ) )
8657, 58, 59, 54, 60, 61, 70, 85lemul12ad 10273 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( X `  ( L `  d ) ) )  x.  ( abs `  ( ( mmu `  d )  /  d
) ) )  <_ 
( 1  x.  (
1  /  d ) ) )
8754recnd 9410 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  /  d )  e.  CC )
8887mulid2d 9402 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  x.  ( 1  / 
d ) )  =  ( 1  /  d
) )
8986, 88breqtrd 4314 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( X `  ( L `  d ) ) )  x.  ( abs `  ( ( mmu `  d )  /  d
) ) )  <_ 
( 1  /  d
) )
9056, 89eqbrtrd 4310 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) ) )  <_  ( 1  /  d ) )
9153, 54, 31, 55, 90lemul1ad 10270 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) ) )  x.  ( abs `  ( K  -  T )
) )  <_  (
( 1  /  d
)  x.  ( abs `  ( K  -  T
) ) ) )
9244, 30absmuld 12938 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  =  ( ( abs `  (
( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) ) )  x.  ( abs `  ( K  -  T
) ) ) )
9331recnd 9410 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( K  -  T
) )  e.  CC )
9493, 72, 73divrec2d 10109 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( K  -  T ) )  / 
d )  =  ( ( 1  /  d
)  x.  ( abs `  ( K  -  T
) ) ) )
9591, 92, 943brtr4d 4320 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  <_  (
( abs `  ( K  -  T )
)  /  d ) )
9618, 50, 33, 95fsumle 13260 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( abs `  (
( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  ( K  -  T ) ) )  <_  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) )
9747, 51, 34, 52, 96letrd 9526 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  <_  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) )
9834leabsd 12899 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d )  <_  ( abs `  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) ) )
9947, 34, 49, 97, 98letrd 9526 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  <_  ( abs `  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) ) )
10099adantrr 716 . 2  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  ( K  -  T ) ) )  <_  ( abs `  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) ) )
1011, 17, 34, 46, 100o1le 13128 1  |-  ( ph  ->  ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  ( K  -  T ) ) )  e.  O(1) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   A.wral 2713   class class class wbr 4290    e. cmpt 4348   -->wf 5412   -onto->wfo 5414   ` cfv 5416  (class class class)co 6089   CCcc 9278   RRcr 9279   0cc0 9280   1c1 9281    x. cmul 9285   +oocpnf 9413    <_ cle 9417    - cmin 9593    / cdiv 9991   NNcn 10320   3c3 10370   NN0cn0 10577   ZZcz 10644   RR+crp 10989   [,)cico 11300   ...cfz 11435   |_cfl 11638   abscabs 12721   O(1)co1 12962   sum_csu 13161   Basecbs 14172   0gc0g 14376   ZRHomczrh 17929  ℤ/nczn 17932   logclog 22004   mmucmu 22430  DChrcdchr 22569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358  ax-addf 9359  ax-mulf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-disj 4261  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-of 6318  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-tpos 6743  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-omul 6923  df-er 7099  df-ec 7101  df-qs 7105  df-map 7214  df-pm 7215  df-ixp 7262  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fsupp 7619  df-fi 7659  df-sup 7689  df-oi 7722  df-card 8107  df-acn 8110  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-q 10952  df-rp 10990  df-xneg 11087  df-xadd 11088  df-xmul 11089  df-ioo 11302  df-ioc 11303  df-ico 11304  df-icc 11305  df-fz 11436  df-fzo 11547  df-fl 11640  df-mod 11707  df-seq 11805  df-exp 11864  df-fac 12050  df-bc 12077  df-hash 12102  df-shft 12554  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-limsup 12947  df-clim 12964  df-rlim 12965  df-o1 12966  df-lo1 12967  df-sum 13162  df-ef 13351  df-e 13352  df-sin 13353  df-cos 13354  df-pi 13356  df-dvds 13534  df-prm 13762  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-starv 14251  df-sca 14252  df-vsca 14253  df-ip 14254  df-tset 14255  df-ple 14256  df-ds 14258  df-unif 14259  df-hom 14260  df-cco 14261  df-rest 14359  df-topn 14360  df-0g 14378  df-gsum 14379  df-topgen 14380  df-pt 14381  df-prds 14384  df-xrs 14438  df-qtop 14443  df-imas 14444  df-divs 14445  df-xps 14446  df-mre 14522  df-mrc 14523  df-acs 14525  df-mnd 15413  df-mhm 15462  df-submnd 15463  df-grp 15543  df-minusg 15544  df-sbg 15545  df-mulg 15546  df-subg 15676  df-nsg 15677  df-eqg 15678  df-ghm 15743  df-cntz 15833  df-od 16030  df-cmn 16277  df-abl 16278  df-mgp 16590  df-ur 16602  df-rng 16645  df-cring 16646  df-oppr 16713  df-dvdsr 16731  df-unit 16732  df-invr 16762  df-dvr 16773  df-rnghom 16804  df-drng 16832  df-subrg 16861  df-lmod 16948  df-lss 17012  df-lsp 17051  df-sra 17251  df-rgmod 17252  df-lidl 17253  df-rsp 17254  df-2idl 17312  df-psmet 17807  df-xmet 17808  df-met 17809  df-bl 17810  df-mopn 17811  df-fbas 17812  df-fg 17813  df-cnfld 17817  df-zring 17882  df-zrh 17933  df-zn 17936  df-top 18501  df-bases 18503  df-topon 18504  df-topsp 18505  df-cld 18621  df-ntr 18622  df-cls 18623  df-nei 18700  df-lp 18738  df-perf 18739  df-cn 18829  df-cnp 18830  df-haus 18917  df-cmp 18988  df-tx 19133  df-hmeo 19326  df-fil 19417  df-fm 19509  df-flim 19510  df-flf 19511  df-xms 19893  df-ms 19894  df-tms 19895  df-cncf 20452  df-limc 21339  df-dv 21340  df-log 22006  df-cxp 22007  df-em 22384  df-mu 22436  df-dchr 22570
This theorem is referenced by:  dchrvmasumiflem1  22748
  Copyright terms: Public domain W3C validator