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Theorem dchrvmasumlem3 21195
Description: Lemma for dchrvmasum 21221. (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 1re 9092 . . 3  |-  1  e.  RR
21a1i 11 . 2  |-  ( ph  ->  1  e.  RR )
3 rpvmasum.z . . 3  |-  Z  =  (ℤ/n `  N )
4 rpvmasum.l . . 3  |-  L  =  ( ZRHom `  Z
)
5 rpvmasum.a . . 3  |-  ( ph  ->  N  e.  NN )
6 rpvmasum.g . . 3  |-  G  =  (DChr `  N )
7 rpvmasum.d . . 3  |-  D  =  ( Base `  G
)
8 rpvmasum.1 . . 3  |-  .1.  =  ( 0g `  G )
9 dchrisum.b . . 3  |-  ( ph  ->  X  e.  D )
10 dchrisum.n1 . . 3  |-  ( ph  ->  X  =/=  .1.  )
11 dchrvmasum.f . . 3  |-  ( (
ph  /\  m  e.  RR+ )  ->  F  e.  CC )
12 dchrvmasum.g . . 3  |-  ( m  =  ( x  / 
d )  ->  F  =  K )
13 dchrvmasum.c . . 3  |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )
14 dchrvmasum.t . . 3  |-  ( ph  ->  T  e.  CC )
15 dchrvmasum.1 . . 3  |-  ( (
ph  /\  m  e.  ( 3 [,)  +oo ) )  ->  ( abs `  ( F  -  T ) )  <_ 
( C  x.  (
( log `  m
)  /  m ) ) )
16 dchrvmasum.r . . 3  |-  ( ph  ->  R  e.  RR )
17 dchrvmasum.2 . . 3  |-  ( ph  ->  A. m  e.  ( 1 [,) 3 ) ( abs `  ( F  -  T )
)  <_  R )
183, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17dchrvmasumlem2 21194 . 2  |-  ( ph  ->  ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) )  e.  O ( 1 ) )
19 fzfid 11314 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
20 simpr 449 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
21 elfznn 11082 . . . . . . . . 9  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
2221nnrpd 10649 . . . . . . . 8  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
23 rpdivcl 10636 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
2420, 22, 23syl2an 465 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
2511ralrimiva 2791 . . . . . . . 8  |-  ( ph  ->  A. m  e.  RR+  F  e.  CC )
2625ad2antrr 708 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  A. m  e.  RR+  F  e.  CC )
2712eleq1d 2504 . . . . . . . 8  |-  ( m  =  ( x  / 
d )  ->  ( F  e.  CC  <->  K  e.  CC ) )
2827rspcv 3050 . . . . . . 7  |-  ( ( x  /  d )  e.  RR+  ->  ( A. m  e.  RR+  F  e.  CC  ->  K  e.  CC ) )
2924, 26, 28sylc 59 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  K  e.  CC )
3014ad2antrr 708 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  T  e.  CC )
3129, 30subcld 9413 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( K  -  T )  e.  CC )
3231abscld 12240 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( K  -  T
) )  e.  RR )
3321adantl 454 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
3432, 33nndivred 10050 . . 3  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( K  -  T ) )  / 
d )  e.  RR )
3519, 34fsumrecl 12530 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d )  e.  RR )
369ad2antrr 708 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  X  e.  D )
37 elfzelz 11061 . . . . . . 7  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  ZZ )
3837adantl 454 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  ZZ )
396, 3, 7, 4, 36, 38dchrzrhcl 21031 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  d
) )  e.  CC )
40 mucl 20926 . . . . . . . . 9  |-  ( d  e.  NN  ->  (
mmu `  d )  e.  ZZ )
4133, 40syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  ZZ )
4241zred 10377 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  RR )
4342, 33nndivred 10050 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  RR )
4443recnd 9116 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  CC )
4539, 44mulcld 9110 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  e.  CC )
4645, 31mulcld 9110 . . 3  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  ( K  -  T ) )  e.  CC )
4719, 46fsumcl 12529 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
)  e.  CC )
4847abscld 12240 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  e.  RR )
4935recnd 9116 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d )  e.  CC )
5049abscld 12240 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs ` 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) )  e.  RR )
5146abscld 12240 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  e.  RR )
5219, 51fsumrecl 12530 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( abs `  (
( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  ( K  -  T ) ) )  e.  RR )
5319, 46fsumabs 12582 . . . . 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 ) ) ) )
5445abscld 12240 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) ) )  e.  RR )
5533nnrecred 10047 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  /  d )  e.  RR )
5631absge0d 12248 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( K  -  T ) ) )
5739, 44absmuld 12258 . . . . . . . . 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 ) ) ) )
5839abscld 12240 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( X `  ( L `  d )
) )  e.  RR )
591a1i 11 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  1  e.  RR )
6044abscld 12240 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  e.  RR )
6139absge0d 12248 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( X `
 ( L `  d ) ) ) )
6244absge0d 12248 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( abs `  ( ( mmu `  d )  /  d ) ) )
63 eqid 2438 . . . . . . . . . . . 12  |-  ( Base `  Z )  =  (
Base `  Z )
645nnnn0d 10276 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  NN0 )
653, 63, 4znzrhfo 16830 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  L : ZZ -onto-> ( Base `  Z
) )
6664, 65syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  L : ZZ -onto-> ( Base `  Z ) )
67 fof 5655 . . . . . . . . . . . . . . 15  |-  ( L : ZZ -onto-> ( Base `  Z )  ->  L : ZZ --> ( Base `  Z
) )
6866, 67syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  L : ZZ --> ( Base `  Z ) )
6968ad2antrr 708 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  L : ZZ
--> ( Base `  Z
) )
7069, 38ffvelrnd 5873 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( L `  d )  e.  (
Base `  Z )
)
716, 7, 3, 63, 36, 70dchrabs2 21048 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( X `  ( L `  d )
) )  <_  1
)
7242recnd 9116 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  CC )
7333nncnd 10018 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  CC )
7433nnne0d 10046 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  =/=  0 )
7572, 73, 74absdivd 12259 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  ( abs `  d ) ) )
7633nnrpd 10649 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  RR+ )
7776rprege0d 10657 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( d  e.  RR  /\  0  <_ 
d ) )
78 absid 12103 . . . . . . . . . . . . . . 15  |-  ( ( d  e.  RR  /\  0  <_  d )  -> 
( abs `  d
)  =  d )
7977, 78syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  d )  =  d )
8079oveq2d 6099 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( mmu `  d ) )  / 
( abs `  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  d ) )
8175, 80eqtrd 2470 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  =  ( ( abs `  (
mmu `  d )
)  /  d ) )
8272abscld 12240 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( mmu `  d
) )  e.  RR )
83 mule1 20933 . . . . . . . . . . . . . 14  |-  ( d  e.  NN  ->  ( abs `  ( mmu `  d ) )  <_ 
1 )
8433, 83syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( mmu `  d
) )  <_  1
)
8582, 59, 76, 84lediv1dd 10704 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( mmu `  d ) )  / 
d )  <_  (
1  /  d ) )
8681, 85eqbrtrd 4234 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( mmu `  d )  /  d
) )  <_  (
1  /  d ) )
8758, 59, 60, 55, 61, 62, 71, 86lemul12ad 9955 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( X `  ( L `  d ) ) )  x.  ( abs `  ( ( mmu `  d )  /  d
) ) )  <_ 
( 1  x.  (
1  /  d ) ) )
8855recnd 9116 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  /  d )  e.  CC )
8988mulid2d 9108 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1  x.  ( 1  / 
d ) )  =  ( 1  /  d
) )
9087, 89breqtrd 4238 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( X `  ( L `  d ) ) )  x.  ( abs `  ( ( mmu `  d )  /  d
) ) )  <_ 
( 1  /  d
) )
9157, 90eqbrtrd 4234 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) ) )  <_  ( 1  /  d ) )
9254, 55, 32, 56, 91lemul1ad 9952 . . . . . . 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
) ) ) )
9345, 31absmuld 12258 . . . . . . 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
) ) ) )
9432recnd 9116 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( K  -  T
) )  e.  CC )
9594, 73, 74divrec2d 9796 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( abs `  ( K  -  T ) )  / 
d )  =  ( ( 1  /  d
)  x.  ( abs `  ( K  -  T
) ) ) )
9692, 93, 953brtr4d 4244 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( abs `  ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( K  -  T )
) )  <_  (
( abs `  ( K  -  T )
)  /  d ) )
9719, 51, 34, 96fsumle 12580 . . . . 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 ) )
9848, 52, 35, 53, 97letrd 9229 . . . 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 ) )
9935leabsd 12219 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d )  <_  ( abs `  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( abs `  ( K  -  T )
)  /  d ) ) )
10048, 35, 50, 98, 99letrd 9229 . . 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 ) ) )
101100adantrr 699 . 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 ) ) )
1022, 18, 35, 47, 101o1le 12448 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   class class class wbr 4214    e. cmpt 4268   -->wf 5452   -onto->wfo 5454   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    x. cmul 8997    +oocpnf 9119    <_ cle 9123    - cmin 9293    / cdiv 9679   NNcn 10002   3c3 10052   NN0cn0 10223   ZZcz 10284   RR+crp 10614   [,)cico 10920   ...cfz 11045   |_cfl 11203   abscabs 12041   O ( 1 )co1 12282   sum_csu 12481   Basecbs 13471   0gc0g 13725   ZRHomczrh 16780  ℤ/nczn 16783   logclog 20454   mmucmu 20879  DChrcdchr 21018
This theorem is referenced by:  dchrvmasumiflem1  21197
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-disj 4185  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-omul 6731  df-er 6907  df-ec 6909  df-qs 6913  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-acn 7831  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ioc 10923  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-fac 11569  df-bc 11596  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-o1 12286  df-lo1 12287  df-sum 12482  df-ef 12672  df-e 12673  df-sin 12674  df-cos 12675  df-pi 12677  df-dvds 12855  df-prm 13082  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-divs 13737  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-mhm 14740  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-mulg 14817  df-subg 14943  df-nsg 14944  df-eqg 14945  df-ghm 15006  df-cntz 15118  df-od 15169  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-cring 15666  df-ur 15667  df-oppr 15730  df-dvdsr 15748  df-unit 15749  df-invr 15779  df-dvr 15790  df-rnghom 15821  df-drng 15839  df-subrg 15868  df-lmod 15954  df-lss 16011  df-lsp 16050  df-sra 16246  df-rgmod 16247  df-lidl 16248  df-rsp 16249  df-2idl 16305  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-zrh 16784  df-zn 16787  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cn 17293  df-cnp 17294  df-haus 17381  df-cmp 17452  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-limc 19755  df-dv 19756  df-log 20456  df-cxp 20457  df-em 20833  df-mu 20885  df-dchr 21019
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