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Theorem dchrvmasumlem3 24416
Description: Lemma for dchrvmasum 24442. (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 9676 . 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 24415 . 2  |-  ( ph  ->  ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) )  e.  O(1) )
18 fzfid 12224 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
19 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
20 elfznn 11854 . . . . . . . . 9  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
2120nnrpd 11362 . . . . . . . 8  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
22 rpdivcl 11348 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
2319, 21, 22syl2an 485 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
2410ralrimiva 2809 . . . . . . . 8  |-  ( ph  ->  A. m  e.  RR+  F  e.  CC )
2524ad2antrr 740 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  A. m  e.  RR+  F  e.  CC )
2611eleq1d 2533 . . . . . . . 8  |-  ( m  =  ( x  / 
d )  ->  ( F  e.  CC  <->  K  e.  CC ) )
2726rspcv 3132 . . . . . . 7  |-  ( ( x  /  d )  e.  RR+  ->  ( A. m  e.  RR+  F  e.  CC  ->  K  e.  CC ) )
2823, 25, 27sylc 61 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  K  e.  CC )
2913ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  T  e.  CC )
3028, 29subcld 10005 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( K  -  T )  e.  CC )
3130abscld 13575 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( K  -  T
) )  e.  RR )
3220adantl 473 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
3331, 32nndivred 10680 . . 3  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( K  -  T ) )  / 
d )  e.  RR )
3418, 33fsumrecl 13877 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d )  e.  RR )
358ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  X  e.  D )
36 elfzelz 11826 . . . . . . 7  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  ZZ )
3736adantl 473 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  ZZ )
385, 2, 6, 3, 35, 37dchrzrhcl 24252 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  d
) )  e.  CC )
39 mucl 24147 . . . . . . . . 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 11063 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  RR )
4241, 32nndivred 10680 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  RR )
4342recnd 9687 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  CC )
4438, 43mulcld 9681 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  e.  CC )
4544, 30mulcld 9681 . . 3  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  ( K  -  T ) )  e.  CC )
4618, 45fsumcl 13876 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
)  e.  CC )
4746abscld 13575 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  e.  RR )
4834recnd 9687 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d )  e.  CC )
4948abscld 13575 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) )  e.  RR )
5045abscld 13575 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  e.  RR )
5118, 50fsumrecl 13877 . . . . 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 13938 . . . . 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 13575 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) ) )  e.  RR )
5432nnrecred 10677 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  /  d )  e.  RR )
5530absge0d 13583 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( K  -  T ) ) )
5638, 43absmuld 13593 . . . . . . . . 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 13575 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( X `  ( L `  d )
) )  e.  RR )
58 1red 9676 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  1  e.  RR )
5943abscld 13575 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  e.  RR )
6038absge0d 13583 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( X `
 ( L `  d ) ) ) )
6143absge0d 13583 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( ( mmu `  d )  /  d ) ) )
62 eqid 2471 . . . . . . . . . . . 12  |-  ( Base `  Z )  =  (
Base `  Z )
634nnnn0d 10949 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  NN0 )
642, 62, 3znzrhfo 19195 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
6563, 64syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  L : ZZ -onto-> ( Base `  Z ) )
66 fof 5806 . . . . . . . . . . . . . . 15  |-  ( L : ZZ -onto-> ( Base `  Z )  ->  L : ZZ --> ( Base `  Z
) )
6765, 66syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  L : ZZ --> ( Base `  Z ) )
6867ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  L : ZZ
--> ( Base `  Z
) )
6968, 37ffvelrnd 6038 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( L `  d )  e.  (
Base `  Z )
)
705, 6, 2, 62, 35, 69dchrabs2 24269 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( X `  ( L `  d )
) )  <_  1
)
7141recnd 9687 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  CC )
7232nncnd 10647 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  CC )
7332nnne0d 10676 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  =/=  0 )
7471, 72, 73absdivd 13594 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  ( abs `  d ) ) )
7532nnrpd 11362 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  RR+ )
7675rprege0d 11371 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( d  e.  RR  /\  0  <_ 
d ) )
77 absid 13436 . . . . . . . . . . . . . . 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 6324 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( mmu `  d ) )  / 
( abs `  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  d ) )
8074, 79eqtrd 2505 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  d ) )
8171abscld 13575 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( mmu `  d
) )  e.  RR )
82 mule1 24154 . . . . . . . . . . . . . 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 11419 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( mmu `  d ) )  / 
d )  <_  (
1  /  d ) )
8580, 84eqbrtrd 4416 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  <_  (
1  /  d ) )
8657, 58, 59, 54, 60, 61, 70, 85lemul12ad 10571 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( X `  ( L `  d ) ) )  x.  ( abs `  ( ( mmu `  d )  /  d
) ) )  <_ 
( 1  x.  (
1  /  d ) ) )
8754recnd 9687 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  /  d )  e.  CC )
8887mulid2d 9679 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  x.  ( 1  / 
d ) )  =  ( 1  /  d
) )
8986, 88breqtrd 4420 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( X `  ( L `  d ) ) )  x.  ( abs `  ( ( mmu `  d )  /  d
) ) )  <_ 
( 1  /  d
) )
9056, 89eqbrtrd 4416 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) ) )  <_  ( 1  /  d ) )
9153, 54, 31, 55, 90lemul1ad 10568 . . . . . . 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 13593 . . . . . . 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 9687 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( K  -  T
) )  e.  CC )
9493, 72, 73divrec2d 10409 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( K  -  T ) )  / 
d )  =  ( ( 1  /  d
)  x.  ( abs `  ( K  -  T
) ) ) )
9591, 92, 943brtr4d 4426 . . . . . 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 13936 . . . . 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 9809 . . . 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 13553 . . . 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 9809 . . 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 731 . 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 13793 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 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   class class class wbr 4395    |-> cmpt 4454   -->wf 5585   -onto->wfo 5587   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562   +oocpnf 9690    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   3c3 10682   NN0cn0 10893   ZZcz 10961   RR+crp 11325   [,)cico 11662   ...cfz 11810   |_cfl 12059   abscabs 13374   O(1)co1 13627   sum_csu 13829   Basecbs 15199   0gc0g 15416   ZRHomczrh 19148  ℤ/nczn 19151   logclog 23583   mmucmu 24100  DChrcdchr 24239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-ec 7383  df-qs 7387  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-o1 13631  df-lo1 13632  df-sum 13830  df-ef 14198  df-e 14199  df-sin 14200  df-cos 14201  df-pi 14203  df-dvds 14383  df-prm 14702  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-qus 15487  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-nsg 16893  df-eqg 16894  df-ghm 16959  df-cntz 17049  df-od 17250  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-rnghom 18021  df-drng 18055  df-subrg 18084  df-lmod 18171  df-lss 18234  df-lsp 18273  df-sra 18473  df-rgmod 18474  df-lidl 18475  df-rsp 18476  df-2idl 18533  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-zring 19117  df-zrh 19152  df-zn 19155  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586  df-em 23997  df-mu 24106  df-dchr 24240
This theorem is referenced by:  dchrvmasumiflem1  24418
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