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Theorem dchrvmasumiflem2 22887
Description: Lemma for dchrvmasum 22910. (Contributed by Mario Carneiro, 5-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.  )
dchrvmasumif.f  |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )
dchrvmasumif.c  |-  ( ph  ->  C  e.  ( 0 [,) +oo ) )
dchrvmasumif.s  |-  ( ph  ->  seq 1 (  +  ,  F )  ~~>  S )
dchrvmasumif.1  |-  ( ph  ->  A. y  e.  ( 1 [,) +oo )
( abs `  (
(  seq 1 (  +  ,  F ) `  ( |_ `  y ) )  -  S ) )  <_  ( C  /  y ) )
dchrvmasumif.g  |-  K  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  x.  ( ( log `  a )  /  a ) ) )
dchrvmasumif.e  |-  ( ph  ->  E  e.  ( 0 [,) +oo ) )
dchrvmasumif.t  |-  ( ph  ->  seq 1 (  +  ,  K )  ~~>  T )
dchrvmasumif.2  |-  ( ph  ->  A. y  e.  ( 3 [,) +oo )
( abs `  (
(  seq 1 (  +  ,  K ) `  ( |_ `  y ) )  -  T ) )  <_  ( E  x.  ( ( log `  y
)  /  y ) ) )
Assertion
Ref Expression
dchrvmasumiflem2  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n
) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  e.  O(1) )
Distinct variable groups:    x, n, y,  .1.    C, n, x, y   
n, F, x, y   
x, a, y    x, E, y    y, K    n, N, x, y    ph, n, x    T, n, x, y    S, n, x, y    n, Z, x, y    D, n, x, y    n, a, L, x, y    X, a, n, x, y
Allowed substitution hints:    ph( y, a)    C( a)    D( a)    S( a)    T( a)    .1. ( a)    E( n, a)    F( a)    G( x, y, n, a)    K( x, n, a)    N( a)    Z( a)

Proof of Theorem dchrvmasumiflem2
Dummy variables  k 
d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1red 9515 . 2  |-  ( ph  ->  1  e.  RR )
2 fzfid 11915 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
3 rpvmasum.g . . . . . . . 8  |-  G  =  (DChr `  N )
4 rpvmasum.z . . . . . . . 8  |-  Z  =  (ℤ/n `  N )
5 rpvmasum.d . . . . . . . 8  |-  D  =  ( Base `  G
)
6 rpvmasum.l . . . . . . . 8  |-  L  =  ( ZRHom `  Z
)
7 dchrisum.b . . . . . . . . 9  |-  ( ph  ->  X  e.  D )
87ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  X  e.  D )
9 elfzelz 11573 . . . . . . . . 9  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  ZZ )
109adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  ZZ )
113, 4, 5, 6, 8, 10dchrzrhcl 22720 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  d
) )  e.  CC )
12 elfznn 11598 . . . . . . . . . . . 12  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
1312adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
14 mucl 22615 . . . . . . . . . . 11  |-  ( d  e.  NN  ->  (
mmu `  d )  e.  ZZ )
1513, 14syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  ZZ )
1615zred 10861 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  RR )
1716, 13nndivred 10484 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  RR )
1817recnd 9526 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  CC )
1911, 18mulcld 9520 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  e.  CC )
202, 19fsumcl 13331 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  e.  CC )
21 dchrvmasumif.s . . . . . . 7  |-  ( ph  ->  seq 1 (  +  ,  F )  ~~>  S )
22 climcl 13098 . . . . . . 7  |-  (  seq 1 (  +  ,  F )  ~~>  S  ->  S  e.  CC )
2321, 22syl 16 . . . . . 6  |-  ( ph  ->  S  e.  CC )
2423adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  S  e.  CC )
2520, 24mulcld 9520 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  S )  e.  CC )
26 0cnd 9493 . . . . . 6  |-  ( (
ph  /\  S  = 
0 )  ->  0  e.  CC )
27 df-ne 2650 . . . . . . 7  |-  ( S  =/=  0  <->  -.  S  =  0 )
28 dchrvmasumif.t . . . . . . . . . 10  |-  ( ph  ->  seq 1 (  +  ,  K )  ~~>  T )
29 climcl 13098 . . . . . . . . . 10  |-  (  seq 1 (  +  ,  K )  ~~>  T  ->  T  e.  CC )
3028, 29syl 16 . . . . . . . . 9  |-  ( ph  ->  T  e.  CC )
3130adantr 465 . . . . . . . 8  |-  ( (
ph  /\  S  =/=  0 )  ->  T  e.  CC )
3223adantr 465 . . . . . . . 8  |-  ( (
ph  /\  S  =/=  0 )  ->  S  e.  CC )
33 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  S  =/=  0 )  ->  S  =/=  0 )
3431, 32, 33divcld 10221 . . . . . . 7  |-  ( (
ph  /\  S  =/=  0 )  ->  ( T  /  S )  e.  CC )
3527, 34sylan2br 476 . . . . . 6  |-  ( (
ph  /\  -.  S  =  0 )  -> 
( T  /  S
)  e.  CC )
3626, 35ifclda 3932 . . . . 5  |-  ( ph  ->  if ( S  =  0 ,  0 ,  ( T  /  S
) )  e.  CC )
3736adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  if ( S  =  0 , 
0 ,  ( T  /  S ) )  e.  CC )
38 rpvmasum.a . . . . 5  |-  ( ph  ->  N  e.  NN )
39 rpvmasum.1 . . . . 5  |-  .1.  =  ( 0g `  G )
40 dchrisum.n1 . . . . 5  |-  ( ph  ->  X  =/=  .1.  )
41 dchrvmasumif.f . . . . 5  |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )
42 dchrvmasumif.c . . . . 5  |-  ( ph  ->  C  e.  ( 0 [,) +oo ) )
43 dchrvmasumif.1 . . . . 5  |-  ( ph  ->  A. y  e.  ( 1 [,) +oo )
( abs `  (
(  seq 1 (  +  ,  F ) `  ( |_ `  y ) )  -  S ) )  <_  ( C  /  y ) )
444, 6, 38, 3, 5, 39, 7, 40, 41, 42, 21, 43dchrmusum2 22879 . . . 4  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  S ) )  e.  O(1) )
45 rpssre 11115 . . . . 5  |-  RR+  C_  RR
46 o1const 13218 . . . . 5  |-  ( (
RR+  C_  RR  /\  if ( S  =  0 ,  0 ,  ( T  /  S ) )  e.  CC )  ->  ( x  e.  RR+  |->  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  e.  O(1) )
4745, 36, 46sylancr 663 . . . 4  |-  ( ph  ->  ( x  e.  RR+  |->  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  e.  O(1) )
4825, 37, 44, 47o1mul2 13223 . . 3  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  S )  x.  if ( S  =  0 ,  0 ,  ( T  /  S
) ) ) )  e.  O(1) )
49 fzfid 11915 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1 ... ( |_ `  ( x  /  d
) ) )  e. 
Fin )
508adantr 465 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  X  e.  D )
51 elfzelz 11573 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  k  e.  ZZ )
5251adantl 466 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  k  e.  ZZ )
533, 4, 5, 6, 50, 52dchrzrhcl 22720 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( X `  ( L `  k
) )  e.  CC )
54 simpr 461 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
5512nnrpd 11140 . . . . . . . . . . . . 13  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
56 rpdivcl 11127 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
5754, 55, 56syl2an 477 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
58 elfznn 11598 . . . . . . . . . . . . 13  |-  ( k  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  k  e.  NN )
5958nnrpd 11140 . . . . . . . . . . . 12  |-  ( k  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  k  e.  RR+ )
60 ifcl 3942 . . . . . . . . . . . 12  |-  ( ( ( x  /  d
)  e.  RR+  /\  k  e.  RR+ )  ->  if ( S  =  0 ,  ( x  / 
d ) ,  k )  e.  RR+ )
6157, 59, 60syl2an 477 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  if ( S  =  0 , 
( x  /  d
) ,  k )  e.  RR+ )
6261relogcld 22208 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( log `  if ( S  =  0 ,  ( x  /  d ) ,  k ) )  e.  RR )
6358adantl 466 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  k  e.  NN )
6462, 63nndivred 10484 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( ( log `  if ( S  =  0 ,  ( x  /  d ) ,  k ) )  /  k )  e.  RR )
6564recnd 9526 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( ( log `  if ( S  =  0 ,  ( x  /  d ) ,  k ) )  /  k )  e.  CC )
6653, 65mulcld 9520 . . . . . . 7  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( ( X `  ( L `  k ) )  x.  ( ( log `  if ( S  =  0 ,  ( x  / 
d ) ,  k ) )  /  k
) )  e.  CC )
6749, 66fsumcl 13331 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  sum_ k  e.  ( 1 ... ( |_ `  ( x  / 
d ) ) ) ( ( X `  ( L `  k ) )  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) )  e.  CC )
6819, 67mulcld 9520 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  sum_ k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) ( ( X `  ( L `  k ) )  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) )  e.  CC )
692, 68fsumcl 13331 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) )  e.  CC )
7025, 37mulcld 9520 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  S )  x.  if ( S  =  0 ,  0 ,  ( T  /  S
) ) )  e.  CC )
71 0cn 9492 . . . . . . . . . 10  |-  0  e.  CC
7230ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  T  e.  CC )
73 ifcl 3942 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  T  e.  CC )  ->  if ( S  =  0 ,  0 ,  T )  e.  CC )
7471, 72, 73sylancr 663 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  if ( S  =  0 , 
0 ,  T )  e.  CC )
7519, 67, 74subdid 9914 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  ( sum_ k  e.  ( 1 ... ( |_ `  ( x  / 
d ) ) ) ( ( X `  ( L `  k ) )  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) )  -  if ( S  =  0 ,  0 ,  T ) ) )  =  ( ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) )  -  (
( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  if ( S  =  0 ,  0 ,  T ) ) ) )
7675sumeq2dv 13301 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) )  -  if ( S  =  0 ,  0 ,  T ) ) )  =  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) )  -  (
( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  if ( S  =  0 ,  0 ,  T ) ) ) )
7719, 74mulcld 9520 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  if ( S  =  0 ,  0 ,  T ) )  e.  CC )
782, 68, 77fsumsub 13376 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( ( X `  ( L `
 d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) )  -  (
( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  if ( S  =  0 ,  0 ,  T ) ) )  =  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  sum_ k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) ( ( X `  ( L `  k ) )  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) )  -  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  if ( S  =  0 ,  0 ,  T ) ) ) )
7920, 24, 37mulassd 9523 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  S )  x.  if ( S  =  0 ,  0 ,  ( T  /  S
) ) )  =  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) ) ) )
80 oveq2 6211 . . . . . . . . . . . . 13  |-  ( if ( S  =  0 ,  0 ,  ( T  /  S ) )  =  0  -> 
( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  ( S  x.  0 ) )
81 oveq2 6211 . . . . . . . . . . . . 13  |-  ( if ( S  =  0 ,  0 ,  ( T  /  S ) )  =  ( T  /  S )  -> 
( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  ( S  x.  ( T  /  S ) ) )
8280, 81ifsb 3913 . . . . . . . . . . . 12  |-  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S
) ) )
8323mul01d 9682 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S  x.  0 )  =  0 )
8483ifeq1d 3918 . . . . . . . . . . . . 13  |-  ( ph  ->  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  ( S  x.  ( T  /  S ) ) ) )
8531, 32, 33divcan2d 10223 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  S  =/=  0 )  ->  ( S  x.  ( T  /  S ) )  =  T )
8627, 85sylan2br 476 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  S  =  0 )  -> 
( S  x.  ( T  /  S ) )  =  T )
8786ifeq2da 3931 . . . . . . . . . . . . 13  |-  ( ph  ->  if ( S  =  0 ,  0 ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T ) )
8884, 87eqtrd 2495 . . . . . . . . . . . 12  |-  ( ph  ->  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T ) )
8982, 88syl5eq 2507 . . . . . . . . . . 11  |-  ( ph  ->  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T
) )
9089adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S
) ) )  =  if ( S  =  0 ,  0 ,  T ) )
9190oveq2d 6219 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) ) )  =  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  if ( S  =  0 ,  0 ,  T ) ) )
9271, 30, 73sylancr 663 . . . . . . . . . . 11  |-  ( ph  ->  if ( S  =  0 ,  0 ,  T )  e.  CC )
9392adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  if ( S  =  0 , 
0 ,  T )  e.  CC )
942, 93, 19fsummulc1 13373 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  if ( S  =  0 ,  0 ,  T ) )  =  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  if ( S  =  0 ,  0 ,  T
) ) )
9579, 91, 943eqtrrd 2500 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  if ( S  =  0 ,  0 ,  T
) )  =  ( ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  S )  x.  if ( S  =  0 ,  0 ,  ( T  /  S
) ) ) )
9695oveq2d 6219 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  sum_ k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) ( ( X `  ( L `  k ) )  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) )  -  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  if ( S  =  0 ,  0 ,  T ) ) )  =  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  sum_ k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) ( ( X `  ( L `  k ) )  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) )  -  (
( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  S )  x.  if ( S  =  0 ,  0 ,  ( T  /  S
) ) ) ) )
9776, 78, 963eqtrd 2499 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  ( sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) )  -  if ( S  =  0 ,  0 ,  T ) ) )  =  (
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) )  -  (
( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  S )  x.  if ( S  =  0 ,  0 ,  ( T  /  S
) ) ) ) )
9897mpteq2dva 4489 . . . . 5  |-  ( ph  ->  ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  ( sum_ k  e.  ( 1 ... ( |_ `  ( x  / 
d ) ) ) ( ( X `  ( L `  k ) )  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) )  -  if ( S  =  0 ,  0 ,  T ) ) ) )  =  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) )  -  (
( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  S )  x.  if ( S  =  0 ,  0 ,  ( T  /  S
) ) ) ) ) )
99 dchrvmasumif.g . . . . . 6  |-  K  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  x.  ( ( log `  a )  /  a ) ) )
100 dchrvmasumif.e . . . . . 6  |-  ( ph  ->  E  e.  ( 0 [,) +oo ) )
101 dchrvmasumif.2 . . . . . 6  |-  ( ph  ->  A. y  e.  ( 3 [,) +oo )
( abs `  (
(  seq 1 (  +  ,  K ) `  ( |_ `  y ) )  -  T ) )  <_  ( E  x.  ( ( log `  y
)  /  y ) ) )
1024, 6, 38, 3, 5, 39, 7, 40, 41, 42, 21, 43, 99, 100, 28, 101dchrvmasumiflem1 22886 . . . . 5  |-  ( ph  ->  ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  ( sum_ k  e.  ( 1 ... ( |_ `  ( x  / 
d ) ) ) ( ( X `  ( L `  k ) )  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) )  -  if ( S  =  0 ,  0 ,  T ) ) ) )  e.  O(1) )
10398, 102eqeltrrd 2543 . . . 4  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) )  -  (
( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  S )  x.  if ( S  =  0 ,  0 ,  ( T  /  S
) ) ) ) )  e.  O(1) )
10469, 70, 103o1dif 13228 . . 3  |-  ( ph  ->  ( ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) ) )  e.  O(1)  <-> 
( x  e.  RR+  |->  ( ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  S )  x.  if ( S  =  0 ,  0 ,  ( T  /  S
) ) ) )  e.  O(1) ) )
10548, 104mpbird 232 . 2  |-  ( ph  ->  ( x  e.  RR+  |->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  sum_ k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) ( ( X `  ( L `  k ) )  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) ) )  e.  O(1) )
1067ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  X  e.  D )
107 elfzelz 11573 . . . . . . 7  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  ZZ )
108107adantl 466 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  ZZ )
1093, 4, 5, 6, 106, 108dchrzrhcl 22720 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  n
) )  e.  CC )
110 elfznn 11598 . . . . . . . 8  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
111110adantl 466 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
112 vmacl 22592 . . . . . . . 8  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
113 nndivre 10471 . . . . . . . 8  |-  ( ( (Λ `  n )  e.  RR  /\  n  e.  NN )  ->  (
(Λ `  n )  /  n )  e.  RR )
114112, 113mpancom 669 . . . . . . 7  |-  ( n  e.  NN  ->  (
(Λ `  n )  /  n )  e.  RR )
115111, 114syl 16 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  /  n
)  e.  RR )
116115recnd 9526 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  /  n
)  e.  CC )
117109, 116mulcld 9520 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  n ) )  x.  ( (Λ `  n
)  /  n ) )  e.  CC )
1182, 117fsumcl 13331 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n
) )  e.  CC )
119 relogcl 22163 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
120119adantl 466 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
121120recnd 9526 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
122 ifcl 3942 . . . 4  |-  ( ( ( log `  x
)  e.  CC  /\  0  e.  CC )  ->  if ( S  =  0 ,  ( log `  x ) ,  0 )  e.  CC )
123121, 71, 122sylancl 662 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  if ( S  =  0 , 
( log `  x
) ,  0 )  e.  CC )
124118, 123addcld 9519 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( X `  ( L `
 n ) )  x.  ( (Λ `  n
)  /  n ) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) )  e.  CC )
125124abscld 13043 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( abs `  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n
) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  e.  RR )
126125adantrr 716 . . 3  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n )
)  x.  ( (Λ `  n )  /  n
) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  e.  RR )
12738adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  N  e.  NN )
1287adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  X  e.  D )
12940adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  X  =/=  .1.  )
130 simprl 755 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  e.  RR+ )
131 simprr 756 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
1  <_  x )
1324, 6, 127, 3, 5, 39, 128, 129, 130, 131dchrvmasum2if 22882 . . . 4  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n
) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) )  =  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) ) )
133132fveq2d 5806 . . 3  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n )
)  x.  ( (Λ `  n )  /  n
) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  =  ( abs `  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) ) ) )
134 eqle 9591 . . 3  |-  ( ( ( abs `  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n )
)  x.  ( (Λ `  n )  /  n
) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  e.  RR  /\  ( abs `  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n )
)  x.  ( (Λ `  n )  /  n
) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  =  ( abs `  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) ) ) )  ->  ( abs `  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n )
)  x.  ( (Λ `  n )  /  n
) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  <_  ( abs `  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) ) ) )
135126, 133, 134syl2anc 661 . 2  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n )
)  x.  ( (Λ `  n )  /  n
) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  <_  ( abs `  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( ( X `
 ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  x.  sum_ k  e.  ( 1 ... ( |_ `  ( x  /  d
) ) ) ( ( X `  ( L `  k )
)  x.  ( ( log `  if ( S  =  0 ,  ( x  /  d
) ,  k ) )  /  k ) ) ) ) )
1361, 105, 69, 124, 135o1le 13251 1  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n
) )  +  if ( S  =  0 ,  ( log `  x
) ,  0 ) ) )  e.  O(1) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799    C_ wss 3439   ifcif 3902   class class class wbr 4403    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203   CCcc 9394   RRcr 9395   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401   +oocpnf 9529    <_ cle 9533    - cmin 9709    / cdiv 10107   NNcn 10436   3c3 10486   ZZcz 10760   RR+crp 11105   [,)cico 11416   ...cfz 11557   |_cfl 11760    seqcseq 11926   abscabs 12844    ~~> cli 13083   O(1)co1 13085   sum_csu 13284   Basecbs 14295   0gc0g 14500   ZRHomczrh 18059  ℤ/nczn 18062   logclog 22142  Λcvma 22565   mmucmu 22568  DChrcdchr 22707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474  ax-addf 9475  ax-mulf 9476
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-disj 4374  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-tpos 6858  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-omul 7038  df-er 7214  df-ec 7216  df-qs 7220  df-map 7329  df-pm 7330  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-fi 7775  df-sup 7805  df-oi 7838  df-card 8223  df-acn 8226  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-q 11068  df-rp 11106  df-xneg 11203  df-xadd 11204  df-xmul 11205  df-ioo 11418  df-ioc 11419  df-ico 11420  df-icc 11421  df-fz 11558  df-fzo 11669  df-fl 11762  df-mod 11829  df-seq 11927  df-exp 11986  df-fac 12172  df-bc 12199  df-hash 12224  df-shft 12677  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-limsup 13070  df-clim 13087  df-rlim 13088  df-o1 13089  df-lo1 13090  df-sum 13285  df-ef 13474  df-e 13475  df-sin 13476  df-cos 13477  df-pi 13479  df-dvds 13657  df-gcd 13812  df-prm 13885  df-pc 14025  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-ress 14302  df-plusg 14373  df-mulr 14374  df-starv 14375  df-sca 14376  df-vsca 14377  df-ip 14378  df-tset 14379  df-ple 14380  df-ds 14382  df-unif 14383  df-hom 14384  df-cco 14385  df-rest 14483  df-topn 14484  df-0g 14502  df-gsum 14503  df-topgen 14504  df-pt 14505  df-prds 14508  df-xrs 14562  df-qtop 14567  df-imas 14568  df-divs 14569  df-xps 14570  df-mre 14646  df-mrc 14647  df-acs 14649  df-mnd 15537  df-mhm 15586  df-submnd 15587  df-grp 15667  df-minusg 15668  df-sbg 15669  df-mulg 15670  df-subg 15800  df-nsg 15801  df-eqg 15802  df-ghm 15867  df-cntz 15957  df-od 16156  df-cmn 16403  df-abl 16404  df-mgp 16717  df-ur 16729  df-rng 16773  df-cring 16774  df-oppr 16841  df-dvdsr 16859  df-unit 16860  df-invr 16890  df-dvr 16901  df-rnghom 16932  df-drng 16960  df-subrg 16989  df-lmod 17076  df-lss 17140  df-lsp 17179  df-sra 17379  df-rgmod 17380  df-lidl 17381  df-rsp 17382  df-2idl 17440  df-psmet 17937  df-xmet 17938  df-met 17939  df-bl 17940  df-mopn 17941  df-fbas 17942  df-fg 17943  df-cnfld 17947  df-zring 18012  df-zrh 18063  df-zn 18066  df-top 18638  df-bases 18640  df-topon 18641  df-topsp 18642  df-cld 18758  df-ntr 18759  df-cls 18760  df-nei 18837  df-lp 18875  df-perf 18876  df-cn 18966  df-cnp 18967  df-haus 19054  df-cmp 19125  df-tx 19270  df-hmeo 19463  df-fil 19554  df-fm 19646  df-flim 19647  df-flf 19648  df-xms 20030  df-ms 20031  df-tms 20032  df-cncf 20589  df-limc 21477  df-dv 21478  df-log 22144  df-cxp 22145  df-em 22522  df-vma 22571  df-mu 22574  df-dchr 22708
This theorem is referenced by:  dchrvmasumif  22888
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