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Theorem dchrvmasumlem3 24337
Description: Lemma for dchrvmasum 24363. (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 9658 . 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 24336 . 2  |-  ( ph  ->  ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) )  e.  O(1) )
18 fzfid 12186 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
19 simpr 463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
20 elfznn 11828 . . . . . . . . 9  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
2120nnrpd 11339 . . . . . . . 8  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
22 rpdivcl 11325 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
2319, 21, 22syl2an 480 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
2410ralrimiva 2802 . . . . . . . 8  |-  ( ph  ->  A. m  e.  RR+  F  e.  CC )
2524ad2antrr 732 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  A. m  e.  RR+  F  e.  CC )
2611eleq1d 2513 . . . . . . . 8  |-  ( m  =  ( x  / 
d )  ->  ( F  e.  CC  <->  K  e.  CC ) )
2726rspcv 3146 . . . . . . 7  |-  ( ( x  /  d )  e.  RR+  ->  ( A. m  e.  RR+  F  e.  CC  ->  K  e.  CC ) )
2823, 25, 27sylc 62 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  K  e.  CC )
2913ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  T  e.  CC )
3028, 29subcld 9986 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( K  -  T )  e.  CC )
3130abscld 13498 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( K  -  T
) )  e.  RR )
3220adantl 468 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
3331, 32nndivred 10658 . . 3  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( K  -  T ) )  / 
d )  e.  RR )
3418, 33fsumrecl 13800 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d )  e.  RR )
358ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  X  e.  D )
36 elfzelz 11800 . . . . . . 7  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  ZZ )
3736adantl 468 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  ZZ )
385, 2, 6, 3, 35, 37dchrzrhcl 24173 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  d
) )  e.  CC )
39 mucl 24068 . . . . . . . . 9  |-  ( d  e.  NN  ->  (
mmu `  d )  e.  ZZ )
4032, 39syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  ZZ )
4140zred 11040 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  RR )
4241, 32nndivred 10658 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  RR )
4342recnd 9669 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  CC )
4438, 43mulcld 9663 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  e.  CC )
4544, 30mulcld 9663 . . 3  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  ( K  -  T ) )  e.  CC )
4618, 45fsumcl 13799 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
)  e.  CC )
4746abscld 13498 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  e.  RR )
4834recnd 9669 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d )  e.  CC )
4948abscld 13498 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) )  e.  RR )
5045abscld 13498 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  e.  RR )
5118, 50fsumrecl 13800 . . . . 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 13861 . . . . 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 13498 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) ) )  e.  RR )
5432nnrecred 10655 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  /  d )  e.  RR )
5530absge0d 13506 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( K  -  T ) ) )
5638, 43absmuld 13516 . . . . . . . . 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 13498 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( X `  ( L `  d )
) )  e.  RR )
58 1red 9658 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  1  e.  RR )
5943abscld 13498 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  e.  RR )
6038absge0d 13506 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( X `
 ( L `  d ) ) ) )
6143absge0d 13506 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( ( mmu `  d )  /  d ) ) )
62 eqid 2451 . . . . . . . . . . . 12  |-  ( Base `  Z )  =  (
Base `  Z )
634nnnn0d 10925 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  NN0 )
642, 62, 3znzrhfo 19118 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
6563, 64syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  L : ZZ -onto-> ( Base `  Z ) )
66 fof 5793 . . . . . . . . . . . . . . 15  |-  ( L : ZZ -onto-> ( Base `  Z )  ->  L : ZZ --> ( Base `  Z
) )
6765, 66syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  L : ZZ --> ( Base `  Z ) )
6867ad2antrr 732 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  L : ZZ
--> ( Base `  Z
) )
6968, 37ffvelrnd 6023 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( L `  d )  e.  (
Base `  Z )
)
705, 6, 2, 62, 35, 69dchrabs2 24190 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( X `  ( L `  d )
) )  <_  1
)
7141recnd 9669 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  CC )
7232nncnd 10625 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  CC )
7332nnne0d 10654 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  =/=  0 )
7471, 72, 73absdivd 13517 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  ( abs `  d ) ) )
7532nnrpd 11339 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  RR+ )
7675rprege0d 11348 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( d  e.  RR  /\  0  <_ 
d ) )
77 absid 13359 . . . . . . . . . . . . . . 15  |-  ( ( d  e.  RR  /\  0  <_  d )  -> 
( abs `  d
)  =  d )
7876, 77syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  d )  =  d )
7978oveq2d 6306 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( mmu `  d ) )  / 
( abs `  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  d ) )
8074, 79eqtrd 2485 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  d ) )
8171abscld 13498 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( mmu `  d
) )  e.  RR )
82 mule1 24075 . . . . . . . . . . . . . 14  |-  ( d  e.  NN  ->  ( abs `  ( mmu `  d ) )  <_ 
1 )
8332, 82syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( mmu `  d
) )  <_  1
)
8481, 58, 75, 83lediv1dd 11396 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( mmu `  d ) )  / 
d )  <_  (
1  /  d ) )
8580, 84eqbrtrd 4423 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  <_  (
1  /  d ) )
8657, 58, 59, 54, 60, 61, 70, 85lemul12ad 10549 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( X `  ( L `  d ) ) )  x.  ( abs `  ( ( mmu `  d )  /  d
) ) )  <_ 
( 1  x.  (
1  /  d ) ) )
8754recnd 9669 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  /  d )  e.  CC )
8887mulid2d 9661 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  x.  ( 1  / 
d ) )  =  ( 1  /  d
) )
8986, 88breqtrd 4427 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( X `  ( L `  d ) ) )  x.  ( abs `  ( ( mmu `  d )  /  d
) ) )  <_ 
( 1  /  d
) )
9056, 89eqbrtrd 4423 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) ) )  <_  ( 1  /  d ) )
9153, 54, 31, 55, 90lemul1ad 10546 . . . . . . 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 13516 . . . . . . 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 9669 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( K  -  T
) )  e.  CC )
9493, 72, 73divrec2d 10387 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( K  -  T ) )  / 
d )  =  ( ( 1  /  d
)  x.  ( abs `  ( K  -  T
) ) ) )
9591, 92, 943brtr4d 4433 . . . . . 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 13859 . . . . 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 9792 . . . 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 13476 . . . 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 9792 . . 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 723 . 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 13716 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 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   class class class wbr 4402    |-> cmpt 4461   -->wf 5578   -onto->wfo 5580   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    x. cmul 9544   +oocpnf 9672    <_ cle 9676    - cmin 9860    / cdiv 10269   NNcn 10609   3c3 10660   NN0cn0 10869   ZZcz 10937   RR+crp 11302   [,)cico 11637   ...cfz 11784   |_cfl 12026   abscabs 13297   O(1)co1 13550   sum_csu 13752   Basecbs 15121   0gc0g 15338   ZRHomczrh 19071  ℤ/nczn 19074   logclog 23504   mmucmu 24021  DChrcdchr 24160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-er 7363  df-ec 7365  df-qs 7369  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-acn 8376  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-o1 13554  df-lo1 13555  df-sum 13753  df-ef 14121  df-e 14122  df-sin 14123  df-cos 14124  df-pi 14126  df-dvds 14306  df-prm 14623  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-qus 15409  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-mulg 16676  df-subg 16814  df-nsg 16815  df-eqg 16816  df-ghm 16881  df-cntz 16971  df-od 17172  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-cring 17783  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-dvr 17911  df-rnghom 17943  df-drng 17977  df-subrg 18006  df-lmod 18093  df-lss 18156  df-lsp 18195  df-sra 18395  df-rgmod 18396  df-lidl 18397  df-rsp 18398  df-2idl 18456  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-zring 19040  df-zrh 19075  df-zn 19078  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-log 23506  df-cxp 23507  df-em 23918  df-mu 24027  df-dchr 24161
This theorem is referenced by:  dchrvmasumiflem1  24339
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