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Theorem dchrvmasumiflem2 20651
Description: Lemma for dchrvmasum 20674. (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 1re 8837 . . 3  |-  1  e.  RR
21a1i 10 . 2  |-  ( ph  ->  1  e.  RR )
3 fzfid 11035 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
4 rpvmasum.g . . . . . . . 8  |-  G  =  (DChr `  N )
5 rpvmasum.z . . . . . . . 8  |-  Z  =  (ℤ/n `  N )
6 rpvmasum.d . . . . . . . 8  |-  D  =  ( Base `  G
)
7 rpvmasum.l . . . . . . . 8  |-  L  =  ( ZRHom `  Z
)
8 dchrisum.b . . . . . . . . 9  |-  ( ph  ->  X  e.  D )
98ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  X  e.  D )
10 elfzelz 10798 . . . . . . . . 9  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  ZZ )
1110adantl 452 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  ZZ )
124, 5, 6, 7, 9, 11dchrzrhcl 20484 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  d
) )  e.  CC )
13 elfznn 10819 . . . . . . . . . . . 12  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
1413adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
15 mucl 20379 . . . . . . . . . . 11  |-  ( d  e.  NN  ->  (
mmu `  d )  e.  ZZ )
1614, 15syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  ZZ )
1716zred 10117 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  RR )
1817, 14nndivred 9794 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  RR )
1918recnd 8861 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  CC )
2012, 19mulcld 8855 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  e.  CC )
213, 20fsumcl 12206 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  e.  CC )
22 dchrvmasumif.s . . . . . . 7  |-  ( ph  ->  seq  1 (  +  ,  F )  ~~>  S )
23 climcl 11973 . . . . . . 7  |-  (  seq  1 (  +  ,  F )  ~~>  S  ->  S  e.  CC )
2422, 23syl 15 . . . . . 6  |-  ( ph  ->  S  e.  CC )
2524adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  S  e.  CC )
2621, 25mulcld 8855 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  S )  e.  CC )
27 0cn 8831 . . . . . . 7  |-  0  e.  CC
2827a1i 10 . . . . . 6  |-  ( (
ph  /\  S  = 
0 )  ->  0  e.  CC )
29 df-ne 2448 . . . . . . 7  |-  ( S  =/=  0  <->  -.  S  =  0 )
30 dchrvmasumif.t . . . . . . . . . 10  |-  ( ph  ->  seq  1 (  +  ,  K )  ~~>  T )
31 climcl 11973 . . . . . . . . . 10  |-  (  seq  1 (  +  ,  K )  ~~>  T  ->  T  e.  CC )
3230, 31syl 15 . . . . . . . . 9  |-  ( ph  ->  T  e.  CC )
3332adantr 451 . . . . . . . 8  |-  ( (
ph  /\  S  =/=  0 )  ->  T  e.  CC )
3424adantr 451 . . . . . . . 8  |-  ( (
ph  /\  S  =/=  0 )  ->  S  e.  CC )
35 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  S  =/=  0 )  ->  S  =/=  0 )
3633, 34, 35divcld 9536 . . . . . . 7  |-  ( (
ph  /\  S  =/=  0 )  ->  ( T  /  S )  e.  CC )
3729, 36sylan2br 462 . . . . . 6  |-  ( (
ph  /\  -.  S  =  0 )  -> 
( T  /  S
)  e.  CC )
3828, 37ifclda 3592 . . . . 5  |-  ( ph  ->  if ( S  =  0 ,  0 ,  ( T  /  S
) )  e.  CC )
3938adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  if ( S  =  0 , 
0 ,  ( T  /  S ) )  e.  CC )
40 rpvmasum.a . . . . 5  |-  ( ph  ->  N  e.  NN )
41 rpvmasum.1 . . . . 5  |-  .1.  =  ( 0g `  G )
42 dchrisum.n1 . . . . 5  |-  ( ph  ->  X  =/=  .1.  )
43 dchrvmasumif.f . . . . 5  |-  F  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  /  a ) )
44 dchrvmasumif.c . . . . 5  |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )
45 dchrvmasumif.1 . . . . 5  |-  ( ph  ->  A. y  e.  ( 1 [,)  +oo )
( abs `  (
(  seq  1 (  +  ,  F ) `
 ( |_ `  y ) )  -  S ) )  <_ 
( C  /  y
) )
465, 7, 40, 4, 6, 41, 8, 42, 43, 44, 22, 45dchrmusum2 20643 . . . 4  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  S ) )  e.  O ( 1 ) )
47 rpssre 10364 . . . . 5  |-  RR+  C_  RR
48 o1const 12093 . . . . 5  |-  ( (
RR+  C_  RR  /\  if ( S  =  0 ,  0 ,  ( T  /  S ) )  e.  CC )  ->  ( x  e.  RR+  |->  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  e.  O ( 1 ) )
4947, 38, 48sylancr 644 . . . 4  |-  ( ph  ->  ( x  e.  RR+  |->  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  e.  O
( 1 ) )
5026, 39, 46, 49o1mul2 12098 . . 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 ) )
51 fzfid 11035 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1 ... ( |_ `  ( x  /  d
) ) )  e. 
Fin )
529adantr 451 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  X  e.  D )
53 elfzelz 10798 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  k  e.  ZZ )
5453adantl 452 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  k  e.  ZZ )
554, 5, 6, 7, 52, 54dchrzrhcl 20484 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( X `  ( L `  k
) )  e.  CC )
56 simpr 447 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
5713nnrpd 10389 . . . . . . . . . . . . 13  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
58 rpdivcl 10376 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
5956, 57, 58syl2an 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
60 elfznn 10819 . . . . . . . . . . . . 13  |-  ( k  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  k  e.  NN )
6160nnrpd 10389 . . . . . . . . . . . 12  |-  ( k  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  k  e.  RR+ )
62 ifcl 3601 . . . . . . . . . . . 12  |-  ( ( ( x  /  d
)  e.  RR+  /\  k  e.  RR+ )  ->  if ( S  =  0 ,  ( x  / 
d ) ,  k )  e.  RR+ )
6359, 61, 62syl2an 463 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  if ( S  =  0 , 
( x  /  d
) ,  k )  e.  RR+ )
6463relogcld 19974 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( log `  if ( S  =  0 ,  ( x  /  d ) ,  k ) )  e.  RR )
6560adantl 452 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  k  e.  NN )
6664, 65nndivred 9794 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( ( log `  if ( S  =  0 ,  ( x  /  d ) ,  k ) )  /  k )  e.  RR )
6766recnd 8861 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( ( log `  if ( S  =  0 ,  ( x  /  d ) ,  k ) )  /  k )  e.  CC )
6855, 67mulcld 8855 . . . . . . 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 )
6951, 68fsumcl 12206 . . . . . 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 )
7020, 69mulcld 8855 . . . . 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 )
713, 70fsumcl 12206 . . . 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 )
7226, 39mulcld 8855 . . . 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 )
7332ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  T  e.  CC )
74 ifcl 3601 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  T  e.  CC )  ->  if ( S  =  0 ,  0 ,  T )  e.  CC )
7527, 73, 74sylancr 644 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  if ( S  =  0 , 
0 ,  T )  e.  CC )
7620, 69, 75subdid 9235 . . . . . . . 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 ) ) ) )
7776sumeq2dv 12176 . . . . . . 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 ) ) ) )
7820, 75mulcld 8855 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  if ( S  =  0 ,  0 ,  T ) )  e.  CC )
793, 70, 78fsumsub 12250 . . . . . . 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 ) ) ) )
8021, 25, 39mulassd 8858 . . . . . . . . 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 ) ) ) ) )
81 oveq2 5866 . . . . . . . . . . . . 13  |-  ( if ( S  =  0 ,  0 ,  ( T  /  S ) )  =  0  -> 
( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  ( S  x.  0 ) )
82 oveq2 5866 . . . . . . . . . . . . 13  |-  ( if ( S  =  0 ,  0 ,  ( T  /  S ) )  =  ( T  /  S )  -> 
( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  ( S  x.  ( T  /  S ) ) )
8381, 82ifsb 3574 . . . . . . . . . . . 12  |-  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S
) ) )
8424mul01d 9011 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S  x.  0 )  =  0 )
8584ifeq1d 3579 . . . . . . . . . . . . 13  |-  ( ph  ->  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  ( S  x.  ( T  /  S ) ) ) )
8633, 34, 35divcan2d 9538 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  S  =/=  0 )  ->  ( S  x.  ( T  /  S ) )  =  T )
8729, 86sylan2br 462 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  S  =  0 )  -> 
( S  x.  ( T  /  S ) )  =  T )
8887ifeq2da 3591 . . . . . . . . . . . . 13  |-  ( ph  ->  if ( S  =  0 ,  0 ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T ) )
8985, 88eqtrd 2315 . . . . . . . . . . . 12  |-  ( ph  ->  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T ) )
9083, 89syl5eq 2327 . . . . . . . . . . 11  |-  ( ph  ->  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T
) )
9190adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S
) ) )  =  if ( S  =  0 ,  0 ,  T ) )
9291oveq2d 5874 . . . . . . . . 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 ) ) )
9327, 32, 74sylancr 644 . . . . . . . . . . 11  |-  ( ph  ->  if ( S  =  0 ,  0 ,  T )  e.  CC )
9493adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  if ( S  =  0 , 
0 ,  T )  e.  CC )
953, 94, 20fsummulc1 12247 . . . . . . . . 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
) ) )
9680, 92, 953eqtrrd 2320 . . . . . . . 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
) ) ) )
9796oveq2d 5874 . . . . . . 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
) ) ) ) )
9877, 79, 973eqtrd 2319 . . . . . 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
) ) ) ) )
9998mpteq2dva 4106 . . . . 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
) ) ) ) ) )
100 dchrvmasumif.g . . . . . 6  |-  K  =  ( a  e.  NN  |->  ( ( X `  ( L `  a ) )  x.  ( ( log `  a )  /  a ) ) )
101 dchrvmasumif.e . . . . . 6  |-  ( ph  ->  E  e.  ( 0 [,)  +oo ) )
102 dchrvmasumif.2 . . . . . 6  |-  ( ph  ->  A. y  e.  ( 3 [,)  +oo )
( abs `  (
(  seq  1 (  +  ,  K ) `
 ( |_ `  y ) )  -  T ) )  <_ 
( E  x.  (
( log `  y
)  /  y ) ) )
1035, 7, 40, 4, 6, 41, 8, 42, 43, 44, 22, 45, 100, 101, 30, 102dchrvmasumiflem1 20650 . . . . 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 ) )
10499, 103eqeltrrd 2358 . . . 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 ) )
10571, 72, 104o1dif 12103 . . 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 ) ) )
10650, 105mpbird 223 . 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 ) )
1078ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  X  e.  D )
108 elfzelz 10798 . . . . . . 7  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  ZZ )
109108adantl 452 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  ZZ )
1104, 5, 6, 7, 107, 109dchrzrhcl 20484 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  n
) )  e.  CC )
111 elfznn 10819 . . . . . . . 8  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
112111adantl 452 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
113 vmacl 20356 . . . . . . . 8  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
114 nndivre 9781 . . . . . . . 8  |-  ( ( (Λ `  n )  e.  RR  /\  n  e.  NN )  ->  (
(Λ `  n )  /  n )  e.  RR )
115113, 114mpancom 650 . . . . . . 7  |-  ( n  e.  NN  ->  (
(Λ `  n )  /  n )  e.  RR )
116112, 115syl 15 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  /  n
)  e.  RR )
117116recnd 8861 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  /  n
)  e.  CC )
118110, 117mulcld 8855 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  n ) )  x.  ( (Λ `  n
)  /  n ) )  e.  CC )
1193, 118fsumcl 12206 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n
) )  e.  CC )
120 relogcl 19932 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
121120adantl 452 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
122121recnd 8861 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
123 ifcl 3601 . . . 4  |-  ( ( ( log `  x
)  e.  CC  /\  0  e.  CC )  ->  if ( S  =  0 ,  ( log `  x ) ,  0 )  e.  CC )
124122, 27, 123sylancl 643 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  if ( S  =  0 , 
( log `  x
) ,  0 )  e.  CC )
125119, 124addcld 8854 . 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 )
126125abscld 11918 . . . 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 )
127126adantrr 697 . . 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 )
12840adantr 451 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  N  e.  NN )
1298adantr 451 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  X  e.  D )
13042adantr 451 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  X  =/=  .1.  )
131 simprl 732 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  e.  RR+ )
132 simprr 733 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
1  <_  x )
1335, 7, 128, 4, 6, 41, 129, 130, 131, 132dchrvmasum2if 20646 . . . 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 ) ) ) )
134133fveq2d 5529 . . 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 ) ) ) ) )
135 eqle 8923 . . 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 ) ) ) ) )
136127, 134, 135syl2anc 642 . 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 ) ) ) ) )
1372, 106, 71, 125, 136o1le 12126 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    +oocpnf 8864    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   3c3 9796   ZZcz 10024   RR+crp 10354   [,)cico 10658   ...cfz 10782   |_cfl 10924    seq cseq 11046   abscabs 11719    ~~> cli 11958   O (
1 )co1 11960   sum_csu 12158   Basecbs 13148   0gc0g 13400   ZRHomczrh 16451  ℤ/nczn 16454   logclog 19912  Λcvma 20329   mmucmu 20332  DChrcdchr 20471
This theorem is referenced by:  dchrvmasumif  20652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-o1 11964  df-lo1 11965  df-sum 12159  df-ef 12349  df-e 12350  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-divs 13412  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-cntz 14793  df-od 14844  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-rnghom 15496  df-drng 15514  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-zrh 16455  df-zn 16458  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915  df-em 20287  df-vma 20335  df-mu 20338  df-dchr 20472
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