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Theorem mulog2sumlem3 21222
Description: Lemma for mulog2sum 21223. (Contributed by Mario Carneiro, 13-May-2016.)
Hypotheses
Ref Expression
logdivsum.1  |-  F  =  ( y  e.  RR+  |->  ( sum_ i  e.  ( 1 ... ( |_
`  y ) ) ( ( log `  i
)  /  i )  -  ( ( ( log `  y ) ^ 2 )  / 
2 ) ) )
mulog2sumlem.1  |-  ( ph  ->  F  ~~> r  L )
Assertion
Ref Expression
mulog2sumlem3  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x
) ) ) )  e.  O ( 1 ) )
Distinct variable groups:    i, n, x, y    x, F    n, L, x    ph, n, x
Allowed substitution hints:    ph( y, i)    F( y, i, n)    L( y, i)

Proof of Theorem mulog2sumlem3
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 2cn 10062 . . . . . 6  |-  2  e.  CC
21a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  2  e.  CC )
3 fzfid 11304 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
4 elfznn 11072 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
54adantl 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
6 mucl 20916 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
mmu `  n )  e.  ZZ )
75, 6syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  ZZ )
87zred 10367 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  RR )
98, 5nndivred 10040 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  RR )
109recnd 9106 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  CC )
11 simpr 448 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
124nnrpd 10639 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
13 rpdivcl 10626 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  /  n )  e.  RR+ )
1411, 12, 13syl2an 464 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR+ )
1514relogcld 20510 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  RR )
1615recnd 9106 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  CC )
1716sqcld 11513 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  ( x  /  n ) ) ^
2 )  e.  CC )
1817halfcld 10204 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( log `  (
x  /  n ) ) ^ 2 )  /  2 )  e.  CC )
1910, 18mulcld 9100 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 ) )  e.  CC )
203, 19fsumcl 12519 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) )  e.  CC )
21 relogcl 20465 . . . . . . 7  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2221adantl 453 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
2322recnd 9106 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
242, 20, 23subdid 9481 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) )  -  ( log `  x
) ) )  =  ( ( 2  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) ) )  -  ( 2  x.  ( log `  x
) ) ) )
253, 2, 19fsummulc2 12559 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 2  x.  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 2  x.  (
( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) ) ) )
261a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  2  e.  CC )
2726, 10, 18mul12d 9267 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( ( mmu `  n )  /  n )  x.  ( ( ( log `  ( x  /  n
) ) ^ 2 )  /  2 ) ) )  =  ( ( ( mmu `  n )  /  n
)  x.  ( 2  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 ) ) ) )
28 2ne0 10075 . . . . . . . . . . 11  |-  2  =/=  0
2928a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  2  =/=  0 )
3017, 26, 29divcan2d 9784 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 ) )  =  ( ( log `  (
x  /  n ) ) ^ 2 ) )
3130oveq2d 6089 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( 2  x.  ( ( ( log `  ( x  /  n
) ) ^ 2 )  /  2 ) ) )  =  ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) ) )
3227, 31eqtrd 2467 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( ( mmu `  n )  /  n )  x.  ( ( ( log `  ( x  /  n
) ) ^ 2 )  /  2 ) ) )  =  ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) ) )
3332sumeq2dv 12489 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 2  x.  (
( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) ) )
3425, 33eqtrd 2467 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 2  x.  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) ) )
3534oveq1d 6088 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( (
2  x.  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) ) )  -  ( 2  x.  ( log `  x
) ) )  =  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x
) ) ) )
3624, 35eqtrd 2467 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) )  -  ( log `  x
) ) )  =  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x
) ) ) )
3736mpteq2dva 4287 . 2  |-  ( ph  ->  ( x  e.  RR+  |->  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) )  -  ( log `  x
) ) ) )  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x
) ) ) ) )
3820, 23subcld 9403 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 ) )  -  ( log `  x ) )  e.  CC )
39 rpssre 10614 . . . . 5  |-  RR+  C_  RR
40 o1const 12405 . . . . 5  |-  ( (
RR+  C_  RR  /\  2  e.  CC )  ->  (
x  e.  RR+  |->  2 )  e.  O ( 1 ) )
4139, 1, 40mp2an 654 . . . 4  |-  ( x  e.  RR+  |->  2 )  e.  O ( 1 )
4241a1i 11 . . 3  |-  ( ph  ->  ( x  e.  RR+  |->  2 )  e.  O
( 1 ) )
43 emre 20836 . . . . . . . . . . . . 13  |-  gamma  e.  RR
4443recni 9094 . . . . . . . . . . . 12  |-  gamma  e.  CC
45 mulcl 9066 . . . . . . . . . . . 12  |-  ( (
gamma  e.  CC  /\  ( log `  ( x  /  n ) )  e.  CC )  ->  ( gamma  x.  ( log `  (
x  /  n ) ) )  e.  CC )
4644, 16, 45sylancr 645 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( gamma  x.  ( log `  (
x  /  n ) ) )  e.  CC )
47 mulog2sumlem.1 . . . . . . . . . . . . 13  |-  ( ph  ->  F  ~~> r  L )
48 rlimcl 12289 . . . . . . . . . . . . 13  |-  ( F  ~~> r  L  ->  L  e.  CC )
4947, 48syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  CC )
5049ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  L  e.  CC )
5146, 50subcld 9403 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
)  e.  CC )
5218, 51addcld 9099 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) )  e.  CC )
5310, 52mulcld 9100 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  e.  CC )
543, 53fsumcl 12519 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  e.  CC )
5510, 51mulcld 9100 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) )  e.  CC )
563, 55fsumcl 12519 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( (
gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) )  e.  CC )
5754, 23, 56sub32d 9435 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  ( log `  x ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( (
gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )  =  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( (
gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )  -  ( log `  x ) ) )
583, 53, 55fsumsub 12563 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( ( mmu `  n )  /  n )  x.  ( ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  ( ( ( mmu `  n
)  /  n )  x.  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )  =  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( (
gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) ) )
5910, 52, 51subdid 9481 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) )  -  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )  =  ( ( ( ( mmu `  n )  /  n )  x.  ( ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  ( ( ( mmu `  n
)  /  n )  x.  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) ) )
6018, 51pncand 9404 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( ( ( log `  ( x  /  n
) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) )  -  (
( gamma  x.  ( log `  ( x  /  n
) ) )  -  L ) )  =  ( ( ( log `  ( x  /  n
) ) ^ 2 )  /  2 ) )
6160oveq2d 6089 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) )  -  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )  =  ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) ) )
6259, 61eqtr3d 2469 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( ( mmu `  n )  /  n
)  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  ( ( ( mmu `  n
)  /  n )  x.  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )  =  ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) ) )
6362sumeq2dv 12489 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( ( mmu `  n )  /  n )  x.  ( ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  ( ( ( mmu `  n
)  /  n )  x.  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )  = 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) ) )
6458, 63eqtr3d 2469 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( (
gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )  = 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) ) )
6564oveq1d 6088 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( (
gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )  -  ( log `  x ) )  =  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 ) )  -  ( log `  x ) ) )
6657, 65eqtrd 2467 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  ( log `  x ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( (
gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )  =  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) )  -  ( log `  x
) ) )
6766mpteq2dva 4287 . . . 4  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  ( log `  x ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( (
gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) ) )  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) )  -  ( log `  x
) ) ) )
6854, 23subcld 9403 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  ( log `  x ) )  e.  CC )
69 logdivsum.1 . . . . . 6  |-  F  =  ( y  e.  RR+  |->  ( sum_ i  e.  ( 1 ... ( |_
`  y ) ) ( ( log `  i
)  /  i )  -  ( ( ( log `  y ) ^ 2 )  / 
2 ) ) )
70 eqid 2435 . . . . . 6  |-  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) )  =  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) )
71 eqid 2435 . . . . . 6  |-  ( ( ( 1  /  2
)  +  ( gamma  +  ( abs `  L
) ) )  + 
sum_ m  e.  (
1 ... 2 ) ( ( log `  (
_e  /  m )
)  /  m ) )  =  ( ( ( 1  /  2
)  +  ( gamma  +  ( abs `  L
) ) )  + 
sum_ m  e.  (
1 ... 2 ) ( ( log `  (
_e  /  m )
)  /  m ) )
7269, 47, 70, 71mulog2sumlem2 21221 . . . . 5  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  ( log `  x ) ) )  e.  O ( 1 ) )
7344a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  gamma  e.  CC )
7410, 16mulcld 9100 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) )  e.  CC )
753, 73, 74fsummulc2 12559 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( gamma  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  =  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( gamma  x.  (
( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )
7649adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  L  e.  CC )
773, 76, 10fsummulc1 12560 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( mmu `  n )  /  n )  x.  L )  =  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  L ) )
7875, 77oveq12d 6091 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( gamma  x.  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  x.  L ) )  =  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( gamma  x.  ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  -  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  L ) ) )
79 mulcl 9066 . . . . . . . . . 10  |-  ( (
gamma  e.  CC  /\  (
( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) )  e.  CC )  ->  ( gamma  x.  ( ( ( mmu `  n )  /  n )  x.  ( log `  (
x  /  n ) ) ) )  e.  CC )
8044, 74, 79sylancr 645 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( gamma  x.  ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  CC )
8110, 50mulcld 9100 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  L )  e.  CC )
823, 80, 81fsumsub 12563 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( gamma  x.  (
( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  -  ( ( ( mmu `  n )  /  n )  x.  L ) )  =  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( gamma  x.  (
( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  -  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  L ) ) )
8344a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  gamma  e.  CC )
8483, 10, 16mul12d 9267 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( gamma  x.  ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  =  ( ( ( mmu `  n )  /  n )  x.  ( gamma  x.  ( log `  ( x  /  n ) ) ) ) )
8584oveq1d 6088 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( gamma  x.  ( ( ( mmu `  n )  /  n )  x.  ( log `  (
x  /  n ) ) ) )  -  ( ( ( mmu `  n )  /  n
)  x.  L ) )  =  ( ( ( ( mmu `  n )  /  n
)  x.  ( gamma  x.  ( log `  (
x  /  n ) ) ) )  -  ( ( ( mmu `  n )  /  n
)  x.  L ) ) )
8610, 46, 50subdid 9481 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) )  =  ( ( ( ( mmu `  n )  /  n
)  x.  ( gamma  x.  ( log `  (
x  /  n ) ) ) )  -  ( ( ( mmu `  n )  /  n
)  x.  L ) ) )
8785, 86eqtr4d 2470 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( gamma  x.  ( ( ( mmu `  n )  /  n )  x.  ( log `  (
x  /  n ) ) ) )  -  ( ( ( mmu `  n )  /  n
)  x.  L ) )  =  ( ( ( mmu `  n
)  /  n )  x.  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )
8887sumeq2dv 12489 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( gamma  x.  (
( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  -  ( ( ( mmu `  n )  /  n )  x.  L ) )  = 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( (
gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )
8978, 82, 883eqtr2d 2473 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( ( gamma  x.  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  x.  L ) )  =  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( (
gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )
9089mpteq2dva 4287 . . . . . 6  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( gamma  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( mmu `  n )  /  n
)  x.  L ) ) )  =  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( (
gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) ) )
913, 74fsumcl 12519 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) )  e.  CC )
92 mulcl 9066 . . . . . . . 8  |-  ( (
gamma  e.  CC  /\  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) )  e.  CC )  ->  ( gamma  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) ) )  e.  CC )
9344, 91, 92sylancr 645 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( gamma  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  CC )
943, 10fsumcl 12519 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  e.  CC )
9594, 76mulcld 9100 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( mmu `  n )  /  n )  x.  L )  e.  CC )
9644a1i 11 . . . . . . . . 9  |-  ( ph  -> 
gamma  e.  CC )
97 o1const 12405 . . . . . . . . 9  |-  ( (
RR+  C_  RR  /\  gamma  e.  CC )  ->  (
x  e.  RR+  |->  gamma )  e.  O ( 1 ) )
9839, 96, 97sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  gamma )  e.  O ( 1 ) )
99 mulogsum 21218 . . . . . . . . 9  |-  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  O ( 1 )
10099a1i 11 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  sum_
n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  O ( 1 ) )
10173, 91, 98, 100o1mul2 12410 . . . . . . 7  |-  ( ph  ->  ( x  e.  RR+  |->  ( gamma  x.  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  e.  O ( 1 ) )
102 mudivsum 21216 . . . . . . . . 9  |-  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
) )  e.  O
( 1 )
103102a1i 11 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  sum_
n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n
)  /  n ) )  e.  O ( 1 ) )
104 o1const 12405 . . . . . . . . 9  |-  ( (
RR+  C_  RR  /\  L  e.  CC )  ->  (
x  e.  RR+  |->  L )  e.  O ( 1 ) )
10539, 49, 104sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  L )  e.  O
( 1 ) )
10694, 76, 103, 105o1mul2 12410 . . . . . . 7  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( mmu `  n )  /  n
)  x.  L ) )  e.  O ( 1 ) )
10793, 95, 101, 106o1sub2 12411 . . . . . 6  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( gamma  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( mmu `  n )  /  n
)  x.  L ) ) )  e.  O
( 1 ) )
10890, 107eqeltrrd 2510 . . . . 5  |-  ( ph  ->  ( x  e.  RR+  |->  sum_
n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( (
gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) )  e.  O ( 1 ) )
10968, 56, 72, 108o1sub2 12411 . . . 4  |-  ( ph  ->  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) ) )  -  ( log `  x ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( (
gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) ) ) )  e.  O ( 1 ) )
11067, 109eqeltrrd 2510 . . 3  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) )  -  ( log `  x
) ) )  e.  O ( 1 ) )
1112, 38, 42, 110o1mul2 12410 . 2  |-  ( ph  ->  ( x  e.  RR+  |->  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) )  -  ( log `  x
) ) ) )  e.  O ( 1 ) )
11237, 111eqeltrrd 2510 1  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x
) ) ) )  e.  O ( 1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283    / cdiv 9669   NNcn 9992   2c2 10041   ZZcz 10274   RR+crp 10604   ...cfz 11035   |_cfl 11193   ^cexp 11374   abscabs 12031    ~~> r crli 12271   O ( 1 )co1 12272   sum_csu 12471   _eceu 12657   logclog 20444   gammacem 20822   mmucmu 20869
This theorem is referenced by:  mulog2sum  21223
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-disj 4175  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-o1 12276  df-lo1 12277  df-sum 12472  df-ef 12662  df-e 12663  df-sin 12664  df-cos 12665  df-pi 12667  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-cmp 17442  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446  df-cxp 20447  df-em 20823  df-mu 20875
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