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Theorem dchrvmasumiflem2 23813
Description: Lemma for dchrvmasum 23836. (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 9628 . 2  |-  ( ph  ->  1  e.  RR )
2 fzfid 12086 . . . . . 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 11713 . . . . . . . . 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 23646 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  d
) )  e.  CC )
12 elfznn 11739 . . . . . . . . . . . 12  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
1312adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
14 mucl 23541 . . . . . . . . . . 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 10990 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  RR )
1716, 13nndivred 10605 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  RR )
1817recnd 9639 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  CC )
1911, 18mulcld 9633 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  e.  CC )
202, 19fsumcl 13567 . . . . 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 13334 . . . . . . 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 9633 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  S )  e.  CC )
26 0cnd 9606 . . . . . 6  |-  ( (
ph  /\  S  = 
0 )  ->  0  e.  CC )
27 df-ne 2654 . . . . . . 7  |-  ( S  =/=  0  <->  -.  S  =  0 )
28 dchrvmasumif.t . . . . . . . . . 10  |-  ( ph  ->  seq 1 (  +  ,  K )  ~~>  T )
29 climcl 13334 . . . . . . . . . 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 10341 . . . . . . 7  |-  ( (
ph  /\  S  =/=  0 )  ->  ( T  /  S )  e.  CC )
3527, 34sylan2br 476 . . . . . 6  |-  ( (
ph  /\  -.  S  =  0 )  -> 
( T  /  S
)  e.  CC )
3626, 35ifclda 3976 . . . . 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 23805 . . . 4  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  S ) )  e.  O(1) )
45 rpssre 11255 . . . . 5  |-  RR+  C_  RR
46 o1const 13454 . . . . 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 13459 . . 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 12086 . . . . . . 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 11713 . . . . . . . . . 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 23646 . . . . . . . 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 11280 . . . . . . . . . . . . 13  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
56 rpdivcl 11267 . . . . . . . . . . . . 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 11739 . . . . . . . . . . . . 13  |-  ( k  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  k  e.  NN )
5958nnrpd 11280 . . . . . . . . . . . 12  |-  ( k  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  k  e.  RR+ )
60 ifcl 3986 . . . . . . . . . . . 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 23134 . . . . . . . . . 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 10605 . . . . . . . . 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 9639 . . . . . . . 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 9633 . . . . . . 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 13567 . . . . . 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 9633 . . . . 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 13567 . . . 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 9633 . . . 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 9605 . . . . . . . . . 10  |-  0  e.  CC
7230ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  T  e.  CC )
73 ifcl 3986 . . . . . . . . . 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 10033 . . . . . . . 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 13537 . . . . . . 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 9633 . . . . . . . 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 13615 . . . . . . 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 9636 . . . . . . . . 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 ovif2 6379 . . . . . . . . . . . 12  |-  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S
) ) )
8123mul01d 9796 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S  x.  0 )  =  0 )
8281ifeq1d 3962 . . . . . . . . . . . . 13  |-  ( ph  ->  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  ( S  x.  ( T  /  S ) ) ) )
8331, 32, 33divcan2d 10343 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  S  =/=  0 )  ->  ( S  x.  ( T  /  S ) )  =  T )
8427, 83sylan2br 476 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  S  =  0 )  -> 
( S  x.  ( T  /  S ) )  =  T )
8584ifeq2da 3975 . . . . . . . . . . . . 13  |-  ( ph  ->  if ( S  =  0 ,  0 ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T ) )
8682, 85eqtrd 2498 . . . . . . . . . . . 12  |-  ( ph  ->  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T ) )
8780, 86syl5eq 2510 . . . . . . . . . . 11  |-  ( ph  ->  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T
) )
8887adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S
) ) )  =  if ( S  =  0 ,  0 ,  T ) )
8988oveq2d 6312 . . . . . . . . 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 ) ) )
9071, 30, 73sylancr 663 . . . . . . . . . . 11  |-  ( ph  ->  if ( S  =  0 ,  0 ,  T )  e.  CC )
9190adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  if ( S  =  0 , 
0 ,  T )  e.  CC )
922, 91, 19fsummulc1 13612 . . . . . . . . 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
) ) )
9379, 89, 923eqtrrd 2503 . . . . . . . 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
) ) ) )
9493oveq2d 6312 . . . . . . 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
) ) ) ) )
9576, 78, 943eqtrd 2502 . . . . . 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
) ) ) ) )
9695mpteq2dva 4543 . . . . 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
) ) ) ) ) )
97 dchrvmasumif.g . . . . . 6  |-  K  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  x.  ( ( log `  a )  /  a ) ) )
98 dchrvmasumif.e . . . . . 6  |-  ( ph  ->  E  e.  ( 0 [,) +oo ) )
99 dchrvmasumif.2 . . . . . 6  |-  ( ph  ->  A. y  e.  ( 3 [,) +oo )
( abs `  (
(  seq 1 (  +  ,  K ) `  ( |_ `  y ) )  -  T ) )  <_  ( E  x.  ( ( log `  y
)  /  y ) ) )
1004, 6, 38, 3, 5, 39, 7, 40, 41, 42, 21, 43, 97, 98, 28, 99dchrvmasumiflem1 23812 . . . . 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) )
10196, 100eqeltrrd 2546 . . . 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) )
10269, 70, 101o1dif 13464 . . 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) ) )
10348, 102mpbird 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) )
1047ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  X  e.  D )
105 elfzelz 11713 . . . . . . 7  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  ZZ )
106105adantl 466 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  ZZ )
1073, 4, 5, 6, 104, 106dchrzrhcl 23646 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  n
) )  e.  CC )
108 elfznn 11739 . . . . . . . 8  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
109108adantl 466 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
110 vmacl 23518 . . . . . . . 8  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
111 nndivre 10592 . . . . . . . 8  |-  ( ( (Λ `  n )  e.  RR  /\  n  e.  NN )  ->  (
(Λ `  n )  /  n )  e.  RR )
112110, 111mpancom 669 . . . . . . 7  |-  ( n  e.  NN  ->  (
(Λ `  n )  /  n )  e.  RR )
113109, 112syl 16 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  /  n
)  e.  RR )
114113recnd 9639 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  /  n
)  e.  CC )
115107, 114mulcld 9633 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  n ) )  x.  ( (Λ `  n
)  /  n ) )  e.  CC )
1162, 115fsumcl 13567 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n
) )  e.  CC )
117 relogcl 23089 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
118117adantl 466 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
119118recnd 9639 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
120 ifcl 3986 . . . 4  |-  ( ( ( log `  x
)  e.  CC  /\  0  e.  CC )  ->  if ( S  =  0 ,  ( log `  x ) ,  0 )  e.  CC )
121119, 71, 120sylancl 662 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  if ( S  =  0 , 
( log `  x
) ,  0 )  e.  CC )
122116, 121addcld 9632 . 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 )
123122abscld 13279 . . . 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 )
124123adantrr 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 )
12538adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  N  e.  NN )
1267adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  X  e.  D )
12740adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  X  =/=  .1.  )
128 simprl 756 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  e.  RR+ )
129 simprr 757 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
1  <_  x )
1304, 6, 125, 3, 5, 39, 126, 127, 128, 129dchrvmasum2if 23808 . . . 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 ) ) ) )
131130fveq2d 5876 . . 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 ) ) ) ) )
132 eqle 9704 . . 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 ) ) ) ) )
133124, 131, 132syl2anc 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 ) ) ) ) )
1341, 103, 69, 122, 133o1le 13487 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807    C_ wss 3471   ifcif 3944   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   +oocpnf 9642    <_ cle 9646    - cmin 9824    / cdiv 10227   NNcn 10556   3c3 10607   ZZcz 10885   RR+crp 11245   [,)cico 11556   ...cfz 11697   |_cfl 11930    seqcseq 12110   abscabs 13079    ~~> cli 13319   O(1)co1 13321   sum_csu 13520   Basecbs 14644   0gc0g 14857   ZRHomczrh 18664  ℤ/nczn 18667   logclog 23068  Λcvma 23491   mmucmu 23494  DChrcdchr 23633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-ec 7331  df-qs 7335  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-o1 13325  df-lo1 13326  df-sum 13521  df-ef 13815  df-e 13816  df-sin 13817  df-cos 13818  df-pi 13820  df-dvds 13999  df-gcd 14157  df-prm 14230  df-pc 14373  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-qus 14926  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mulg 16187  df-subg 16325  df-nsg 16326  df-eqg 16327  df-ghm 16392  df-cntz 16482  df-od 16680  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-ring 17327  df-cring 17328  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-dvr 17459  df-rnghom 17491  df-drng 17525  df-subrg 17554  df-lmod 17641  df-lss 17706  df-lsp 17745  df-sra 17945  df-rgmod 17946  df-lidl 17947  df-rsp 17948  df-2idl 18007  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-zring 18616  df-zrh 18668  df-zn 18671  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-cmp 20014  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-limc 22396  df-dv 22397  df-log 23070  df-cxp 23071  df-em 23448  df-vma 23497  df-mu 23500  df-dchr 23634
This theorem is referenced by:  dchrvmasumif  23814
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