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Theorem dchrvmasumiflem2 22726
Description: Lemma for dchrvmasum 22749. (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 9393 . 2  |-  ( ph  ->  1  e.  RR )
2 fzfid 11787 . . . . . 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 11445 . . . . . . . . 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 22559 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  d
) )  e.  CC )
12 elfznn 11470 . . . . . . . . . . . 12  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
1312adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
14 mucl 22454 . . . . . . . . . . 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 10739 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  RR )
1716, 13nndivred 10362 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  RR )
1817recnd 9404 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  CC )
1911, 18mulcld 9398 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  e.  CC )
202, 19fsumcl 13202 . . . . 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 12969 . . . . . . 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 9398 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  S )  e.  CC )
26 0cnd 9371 . . . . . 6  |-  ( (
ph  /\  S  = 
0 )  ->  0  e.  CC )
27 df-ne 2603 . . . . . . 7  |-  ( S  =/=  0  <->  -.  S  =  0 )
28 dchrvmasumif.t . . . . . . . . . 10  |-  ( ph  ->  seq 1 (  +  ,  K )  ~~>  T )
29 climcl 12969 . . . . . . . . . 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 10099 . . . . . . 7  |-  ( (
ph  /\  S  =/=  0 )  ->  ( T  /  S )  e.  CC )
3527, 34sylan2br 476 . . . . . 6  |-  ( (
ph  /\  -.  S  =  0 )  -> 
( T  /  S
)  e.  CC )
3626, 35ifclda 3816 . . . . 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 22718 . . . 4  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  S ) )  e.  O(1) )
45 rpssre 10993 . . . . 5  |-  RR+  C_  RR
46 o1const 13089 . . . . 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 13094 . . 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 11787 . . . . . . 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 11445 . . . . . . . . . 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 22559 . . . . . . . 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 11018 . . . . . . . . . . . . 13  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
56 rpdivcl 11005 . . . . . . . . . . . . 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 11470 . . . . . . . . . . . . 13  |-  ( k  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  k  e.  NN )
5958nnrpd 11018 . . . . . . . . . . . 12  |-  ( k  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  k  e.  RR+ )
60 ifcl 3826 . . . . . . . . . . . 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 22047 . . . . . . . . . 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 10362 . . . . . . . . 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 9404 . . . . . . . 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 9398 . . . . . . 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 13202 . . . . . 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 9398 . . . . 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 13202 . . . 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 9398 . . . 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 9370 . . . . . . . . . 10  |-  0  e.  CC
7230ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  T  e.  CC )
73 ifcl 3826 . . . . . . . . . 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 9792 . . . . . . . 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 13172 . . . . . . 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 9398 . . . . . . . 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 13247 . . . . . . 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 9401 . . . . . . . . 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 6094 . . . . . . . . . . . . 13  |-  ( if ( S  =  0 ,  0 ,  ( T  /  S ) )  =  0  -> 
( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  ( S  x.  0 ) )
81 oveq2 6094 . . . . . . . . . . . . 13  |-  ( if ( S  =  0 ,  0 ,  ( T  /  S ) )  =  ( T  /  S )  -> 
( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  ( S  x.  ( T  /  S ) ) )
8280, 81ifsb 3797 . . . . . . . . . . . 12  |-  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S
) ) )
8323mul01d 9560 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S  x.  0 )  =  0 )
8483ifeq1d 3802 . . . . . . . . . . . . 13  |-  ( ph  ->  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  ( S  x.  ( T  /  S ) ) ) )
8531, 32, 33divcan2d 10101 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  S  =/=  0 )  ->  ( S  x.  ( T  /  S ) )  =  T )
8627, 85sylan2br 476 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  S  =  0 )  -> 
( S  x.  ( T  /  S ) )  =  T )
8786ifeq2da 3815 . . . . . . . . . . . . 13  |-  ( ph  ->  if ( S  =  0 ,  0 ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T ) )
8884, 87eqtrd 2470 . . . . . . . . . . . 12  |-  ( ph  ->  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T ) )
8982, 88syl5eq 2482 . . . . . . . . . . 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 6102 . . . . . . . . 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 13244 . . . . . . . . 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 2475 . . . . . . . 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 6102 . . . . . . 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 2474 . . . . . 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 4373 . . . . 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 22725 . . . . 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 2513 . . . 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 13099 . . 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 11445 . . . . . . 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 22559 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  n
) )  e.  CC )
110 elfznn 11470 . . . . . . . 8  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
111110adantl 466 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
112 vmacl 22431 . . . . . . . 8  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
113 nndivre 10349 . . . . . . . 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 9404 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  /  n
)  e.  CC )
117109, 116mulcld 9398 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  n ) )  x.  ( (Λ `  n
)  /  n ) )  e.  CC )
1182, 117fsumcl 13202 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n
) )  e.  CC )
119 relogcl 22002 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
120119adantl 466 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
121120recnd 9404 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
122 ifcl 3826 . . . 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 9397 . 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 12914 . . . 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 22721 . . . 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 5690 . . 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 9469 . . 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 13122 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 1369    e. wcel 1756    =/= wne 2601   A.wral 2710    C_ wss 3323   ifcif 3786   class class class wbr 4287    e. cmpt 4345   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279   +oocpnf 9407    <_ cle 9411    - cmin 9587    / cdiv 9985   NNcn 10314   3c3 10364   ZZcz 10638   RR+crp 10983   [,)cico 11294   ...cfz 11429   |_cfl 11632    seqcseq 11798   abscabs 12715    ~~> cli 12954   O(1)co1 12956   sum_csu 13155   Basecbs 14166   0gc0g 14370   ZRHomczrh 17906  ℤ/nczn 17909   logclog 21981  Λcvma 22404   mmucmu 22407  DChrcdchr 22546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-omul 6917  df-er 7093  df-ec 7095  df-qs 7099  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-o1 12960  df-lo1 12961  df-sum 13156  df-ef 13345  df-e 13346  df-sin 13347  df-cos 13348  df-pi 13350  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-divs 14439  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-mhm 15456  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mulg 15539  df-subg 15669  df-nsg 15670  df-eqg 15671  df-ghm 15736  df-cntz 15826  df-od 16023  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-rng 16635  df-cring 16636  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-dvr 16763  df-rnghom 16794  df-drng 16812  df-subrg 16841  df-lmod 16928  df-lss 16991  df-lsp 17030  df-sra 17230  df-rgmod 17231  df-lidl 17232  df-rsp 17233  df-2idl 17291  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-zring 17859  df-zrh 17910  df-zn 17913  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-cmp 18965  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317  df-log 21983  df-cxp 21984  df-em 22361  df-vma 22410  df-mu 22413  df-dchr 22547
This theorem is referenced by:  dchrvmasumif  22727
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