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Theorem dchrvmasumlem3 23440
Description: Lemma for dchrvmasum 23466. (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 9611 . 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 23439 . 2  |-  ( ph  ->  ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) )  e.  O(1) )
18 fzfid 12051 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
19 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
20 elfznn 11714 . . . . . . . . 9  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
2120nnrpd 11255 . . . . . . . 8  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
22 rpdivcl 11242 . . . . . . . 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 2878 . . . . . . . 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 2536 . . . . . . . 8  |-  ( m  =  ( x  / 
d )  ->  ( F  e.  CC  <->  K  e.  CC ) )
2726rspcv 3210 . . . . . . 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 9930 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( K  -  T )  e.  CC )
3130abscld 13230 . . . 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 10584 . . 3  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( K  -  T ) )  / 
d )  e.  RR )
3418, 33fsumrecl 13519 . 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 11688 . . . . . . 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 23276 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  d
) )  e.  CC )
39 mucl 23171 . . . . . . . . 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 10966 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  RR )
4241, 32nndivred 10584 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  RR )
4342recnd 9622 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  CC )
4438, 43mulcld 9616 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  e.  CC )
4544, 30mulcld 9616 . . 3  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  ( K  -  T ) )  e.  CC )
4618, 45fsumcl 13518 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
)  e.  CC )
4746abscld 13230 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  e.  RR )
4834recnd 9622 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d )  e.  CC )
4948abscld 13230 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) )  e.  RR )
5045abscld 13230 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  e.  RR )
5118, 50fsumrecl 13519 . . . . 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 13578 . . . . 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 13230 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) ) )  e.  RR )
5432nnrecred 10581 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  /  d )  e.  RR )
5530absge0d 13238 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( K  -  T ) ) )
5638, 43absmuld 13248 . . . . . . . . 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 13230 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( X `  ( L `  d )
) )  e.  RR )
58 1red 9611 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  1  e.  RR )
5943abscld 13230 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  e.  RR )
6038absge0d 13238 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( X `
 ( L `  d ) ) ) )
6143absge0d 13238 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( ( mmu `  d )  /  d ) ) )
62 eqid 2467 . . . . . . . . . . . 12  |-  ( Base `  Z )  =  (
Base `  Z )
634nnnn0d 10852 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  NN0 )
642, 62, 3znzrhfo 18381 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
6563, 64syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  L : ZZ -onto-> ( Base `  Z ) )
66 fof 5795 . . . . . . . . . . . . . . 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 6022 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( L `  d )  e.  (
Base `  Z )
)
705, 6, 2, 62, 35, 69dchrabs2 23293 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( X `  ( L `  d )
) )  <_  1
)
7141recnd 9622 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  CC )
7232nncnd 10552 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  CC )
7332nnne0d 10580 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  =/=  0 )
7471, 72, 73absdivd 13249 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  ( abs `  d ) ) )
7532nnrpd 11255 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  RR+ )
7675rprege0d 11263 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( d  e.  RR  /\  0  <_ 
d ) )
77 absid 13092 . . . . . . . . . . . . . . 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 6300 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( mmu `  d ) )  / 
( abs `  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  d ) )
8074, 79eqtrd 2508 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  d ) )
8171abscld 13230 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( mmu `  d
) )  e.  RR )
82 mule1 23178 . . . . . . . . . . . . . 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 11310 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( mmu `  d ) )  / 
d )  <_  (
1  /  d ) )
8580, 84eqbrtrd 4467 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  <_  (
1  /  d ) )
8657, 58, 59, 54, 60, 61, 70, 85lemul12ad 10488 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( X `  ( L `  d ) ) )  x.  ( abs `  ( ( mmu `  d )  /  d
) ) )  <_ 
( 1  x.  (
1  /  d ) ) )
8754recnd 9622 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  /  d )  e.  CC )
8887mulid2d 9614 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  x.  ( 1  / 
d ) )  =  ( 1  /  d
) )
8986, 88breqtrd 4471 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( X `  ( L `  d ) ) )  x.  ( abs `  ( ( mmu `  d )  /  d
) ) )  <_ 
( 1  /  d
) )
9056, 89eqbrtrd 4467 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) ) )  <_  ( 1  /  d ) )
9153, 54, 31, 55, 90lemul1ad 10485 . . . . . . 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 13248 . . . . . . 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 9622 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( K  -  T
) )  e.  CC )
9493, 72, 73divrec2d 10324 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( K  -  T ) )  / 
d )  =  ( ( 1  /  d
)  x.  ( abs `  ( K  -  T
) ) ) )
9591, 92, 943brtr4d 4477 . . . . . 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 13576 . . . . 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 9738 . . . 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 13209 . . . 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 9738 . . 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 13438 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   class class class wbr 4447    |-> cmpt 4505   -->wf 5584   -onto->wfo 5586   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    x. cmul 9497   +oocpnf 9625    <_ cle 9629    - cmin 9805    / cdiv 10206   NNcn 10536   3c3 10586   NN0cn0 10795   ZZcz 10864   RR+crp 11220   [,)cico 11531   ...cfz 11672   |_cfl 11895   abscabs 13030   O(1)co1 13272   sum_csu 13471   Basecbs 14490   0gc0g 14695   ZRHomczrh 18332  ℤ/nczn 18335   logclog 22698   mmucmu 23124  DChrcdchr 23263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-o1 13276  df-lo1 13277  df-sum 13472  df-ef 13665  df-e 13666  df-sin 13667  df-cos 13668  df-pi 13670  df-dvds 13848  df-prm 14077  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-divs 14764  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-nsg 16004  df-eqg 16005  df-ghm 16070  df-cntz 16160  df-od 16359  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-cring 17003  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-dvr 17133  df-rnghom 17165  df-drng 17198  df-subrg 17227  df-lmod 17314  df-lss 17379  df-lsp 17418  df-sra 17618  df-rgmod 17619  df-lidl 17620  df-rsp 17621  df-2idl 17679  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-zring 18285  df-zrh 18336  df-zn 18339  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-haus 19610  df-cmp 19681  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145  df-limc 22033  df-dv 22034  df-log 22700  df-cxp 22701  df-em 23078  df-mu 23130  df-dchr 23264
This theorem is referenced by:  dchrvmasumiflem1  23442
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