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Theorem dchrvmasumiflem2 21196
Description: Lemma for dchrvmasum 21219. (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 9090 . . 3  |-  1  e.  RR
21a1i 11 . 2  |-  ( ph  ->  1  e.  RR )
3 fzfid 11312 . . . . . 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 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  X  e.  D )
10 elfzelz 11059 . . . . . . . . 9  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  ZZ )
1110adantl 453 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  ZZ )
124, 5, 6, 7, 9, 11dchrzrhcl 21029 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  d
) )  e.  CC )
13 elfznn 11080 . . . . . . . . . . . 12  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
1413adantl 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
15 mucl 20924 . . . . . . . . . . 11  |-  ( d  e.  NN  ->  (
mmu `  d )  e.  ZZ )
1614, 15syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  ZZ )
1716zred 10375 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  d )  e.  RR )
1817, 14nndivred 10048 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  RR )
1918recnd 9114 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  d )  /  d )  e.  CC )
2012, 19mulcld 9108 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d
) )  e.  CC )
213, 20fsumcl 12527 . . . . 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 12293 . . . . . . 7  |-  (  seq  1 (  +  ,  F )  ~~>  S  ->  S  e.  CC )
2422, 23syl 16 . . . . . 6  |-  ( ph  ->  S  e.  CC )
2524adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  S  e.  CC )
2621, 25mulcld 9108 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  d )
)  x.  ( ( mmu `  d )  /  d ) )  x.  S )  e.  CC )
27 0cn 9084 . . . . . . 7  |-  0  e.  CC
2827a1i 11 . . . . . 6  |-  ( (
ph  /\  S  = 
0 )  ->  0  e.  CC )
29 df-ne 2601 . . . . . . 7  |-  ( S  =/=  0  <->  -.  S  =  0 )
30 dchrvmasumif.t . . . . . . . . . 10  |-  ( ph  ->  seq  1 (  +  ,  K )  ~~>  T )
31 climcl 12293 . . . . . . . . . 10  |-  (  seq  1 (  +  ,  K )  ~~>  T  ->  T  e.  CC )
3230, 31syl 16 . . . . . . . . 9  |-  ( ph  ->  T  e.  CC )
3332adantr 452 . . . . . . . 8  |-  ( (
ph  /\  S  =/=  0 )  ->  T  e.  CC )
3424adantr 452 . . . . . . . 8  |-  ( (
ph  /\  S  =/=  0 )  ->  S  e.  CC )
35 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  S  =/=  0 )  ->  S  =/=  0 )
3633, 34, 35divcld 9790 . . . . . . 7  |-  ( (
ph  /\  S  =/=  0 )  ->  ( T  /  S )  e.  CC )
3729, 36sylan2br 463 . . . . . 6  |-  ( (
ph  /\  -.  S  =  0 )  -> 
( T  /  S
)  e.  CC )
3828, 37ifclda 3766 . . . . 5  |-  ( ph  ->  if ( S  =  0 ,  0 ,  ( T  /  S
) )  e.  CC )
3938adantr 452 . . . 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 21188 . . . 4  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( X `  ( L `  d ) )  x.  ( ( mmu `  d )  /  d ) )  x.  S ) )  e.  O ( 1 ) )
47 rpssre 10622 . . . . 5  |-  RR+  C_  RR
48 o1const 12413 . . . . 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 645 . . . 4  |-  ( ph  ->  ( x  e.  RR+  |->  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  e.  O
( 1 ) )
5026, 39, 46, 49o1mul2 12418 . . 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 11312 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 1 ... ( |_ `  ( x  /  d
) ) )  e. 
Fin )
529adantr 452 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  X  e.  D )
53 elfzelz 11059 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  k  e.  ZZ )
5453adantl 453 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  k  e.  ZZ )
554, 5, 6, 7, 52, 54dchrzrhcl 21029 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( X `  ( L `  k
) )  e.  CC )
56 simpr 448 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
5713nnrpd 10647 . . . . . . . . . . . . 13  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
58 rpdivcl 10634 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
5956, 57, 58syl2an 464 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
60 elfznn 11080 . . . . . . . . . . . . 13  |-  ( k  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  k  e.  NN )
6160nnrpd 10647 . . . . . . . . . . . 12  |-  ( k  e.  ( 1 ... ( |_ `  (
x  /  d ) ) )  ->  k  e.  RR+ )
62 ifcl 3775 . . . . . . . . . . . 12  |-  ( ( ( x  /  d
)  e.  RR+  /\  k  e.  RR+ )  ->  if ( S  =  0 ,  ( x  / 
d ) ,  k )  e.  RR+ )
6359, 61, 62syl2an 464 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  if ( S  =  0 , 
( x  /  d
) ,  k )  e.  RR+ )
6463relogcld 20518 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  ( log `  if ( S  =  0 ,  ( x  /  d ) ,  k ) )  e.  RR )
6560adantl 453 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( x  / 
d ) ) ) )  ->  k  e.  NN )
6664, 65nndivred 10048 . . . . . . . . 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 9114 . . . . . . . 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 9108 . . . . . . 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 12527 . . . . . 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 9108 . . . . 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 12527 . . . 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 9108 . . . 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 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  T  e.  CC )
74 ifcl 3775 . . . . . . . . . 10  |-  ( ( 0  e.  CC  /\  T  e.  CC )  ->  if ( S  =  0 ,  0 ,  T )  e.  CC )
7527, 73, 74sylancr 645 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  if ( S  =  0 , 
0 ,  T )  e.  CC )
7620, 69, 75subdid 9489 . . . . . . . 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 12497 . . . . . . 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 9108 . . . . . . . 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 12571 . . . . . . 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 9111 . . . . . . . . 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 6089 . . . . . . . . . . . . 13  |-  ( if ( S  =  0 ,  0 ,  ( T  /  S ) )  =  0  -> 
( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  ( S  x.  0 ) )
82 oveq2 6089 . . . . . . . . . . . . 13  |-  ( if ( S  =  0 ,  0 ,  ( T  /  S ) )  =  ( T  /  S )  -> 
( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  ( S  x.  ( T  /  S ) ) )
8381, 82ifsb 3748 . . . . . . . . . . . 12  |-  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S
) ) )
8424mul01d 9265 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( S  x.  0 )  =  0 )
8584ifeq1d 3753 . . . . . . . . . . . . 13  |-  ( ph  ->  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  ( S  x.  ( T  /  S ) ) ) )
8633, 34, 35divcan2d 9792 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  S  =/=  0 )  ->  ( S  x.  ( T  /  S ) )  =  T )
8729, 86sylan2br 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  -.  S  =  0 )  -> 
( S  x.  ( T  /  S ) )  =  T )
8887ifeq2da 3765 . . . . . . . . . . . . 13  |-  ( ph  ->  if ( S  =  0 ,  0 ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T ) )
8985, 88eqtrd 2468 . . . . . . . . . . . 12  |-  ( ph  ->  if ( S  =  0 ,  ( S  x.  0 ) ,  ( S  x.  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T ) )
9083, 89syl5eq 2480 . . . . . . . . . . 11  |-  ( ph  ->  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S ) ) )  =  if ( S  =  0 ,  0 ,  T
) )
9190adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( S  x.  if ( S  =  0 ,  0 ,  ( T  /  S
) ) )  =  if ( S  =  0 ,  0 ,  T ) )
9291oveq2d 6097 . . . . . . . . 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 645 . . . . . . . . . . 11  |-  ( ph  ->  if ( S  =  0 ,  0 ,  T )  e.  CC )
9493adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  if ( S  =  0 , 
0 ,  T )  e.  CC )
953, 94, 20fsummulc1 12568 . . . . . . . . 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 2473 . . . . . . . 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 6097 . . . . . . 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 2472 . . . . . 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 4295 . . . . 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 21195 . . . . 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 2511 . . . 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 12423 . . 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 224 . 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 707 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  X  e.  D )
108 elfzelz 11059 . . . . . . 7  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  ZZ )
109108adantl 453 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  ZZ )
1104, 5, 6, 7, 107, 109dchrzrhcl 21029 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( X `  ( L `  n
) )  e.  CC )
111 elfznn 11080 . . . . . . . 8  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
112111adantl 453 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
113 vmacl 20901 . . . . . . . 8  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
114 nndivre 10035 . . . . . . . 8  |-  ( ( (Λ `  n )  e.  RR  /\  n  e.  NN )  ->  (
(Λ `  n )  /  n )  e.  RR )
115113, 114mpancom 651 . . . . . . 7  |-  ( n  e.  NN  ->  (
(Λ `  n )  /  n )  e.  RR )
116112, 115syl 16 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  /  n
)  e.  RR )
117116recnd 9114 . . . . 5  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  /  n
)  e.  CC )
118110, 117mulcld 9108 . . . 4  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( X `  ( L `  n ) )  x.  ( (Λ `  n
)  /  n ) )  e.  CC )
1193, 118fsumcl 12527 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( X `  ( L `  n ) )  x.  ( (Λ `  n )  /  n
) )  e.  CC )
120 relogcl 20473 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
121120adantl 453 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
122121recnd 9114 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
123 ifcl 3775 . . . 4  |-  ( ( ( log `  x
)  e.  CC  /\  0  e.  CC )  ->  if ( S  =  0 ,  ( log `  x ) ,  0 )  e.  CC )
124122, 27, 123sylancl 644 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  if ( S  =  0 , 
( log `  x
) ,  0 )  e.  CC )
125119, 124addcld 9107 . 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 12238 . . . 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 698 . . 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 452 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  N  e.  NN )
1298adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  X  e.  D )
13042adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  X  =/=  .1.  )
131 simprl 733 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  ->  x  e.  RR+ )
132 simprr 734 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
1  <_  x )
1335, 7, 128, 4, 6, 41, 129, 130, 131, 132dchrvmasum2if 21191 . . . 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 5732 . . 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 9176 . . 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 643 . 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 12446 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 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705    C_ wss 3320   ifcif 3739   class class class wbr 4212    e. cmpt 4266   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    +oocpnf 9117    <_ cle 9121    - cmin 9291    / cdiv 9677   NNcn 10000   3c3 10050   ZZcz 10282   RR+crp 10612   [,)cico 10918   ...cfz 11043   |_cfl 11201    seq cseq 11323   abscabs 12039    ~~> cli 12278   O (
1 )co1 12280   sum_csu 12479   Basecbs 13469   0gc0g 13723   ZRHomczrh 16778  ℤ/nczn 16781   logclog 20452  Λcvma 20874   mmucmu 20877  DChrcdchr 21016
This theorem is referenced by:  dchrvmasumif  21197
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-disj 4183  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-omul 6729  df-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-acn 7829  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ioc 10921  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-o1 12284  df-lo1 12285  df-sum 12480  df-ef 12670  df-e 12671  df-sin 12672  df-cos 12673  df-pi 12675  df-dvds 12853  df-gcd 13007  df-prm 13080  df-pc 13211  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-divs 13735  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-nsg 14942  df-eqg 14943  df-ghm 15004  df-cntz 15116  df-od 15167  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-dvr 15788  df-rnghom 15819  df-drng 15837  df-subrg 15866  df-lmod 15952  df-lss 16009  df-lsp 16048  df-sra 16244  df-rgmod 16245  df-lidl 16246  df-rsp 16247  df-2idl 16303  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-zrh 16782  df-zn 16785  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-cmp 17450  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-limc 19753  df-dv 19754  df-log 20454  df-cxp 20455  df-em 20831  df-vma 20880  df-mu 20883  df-dchr 21017
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