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Theorem mulog2sumlem3 20738
Description: Lemma for mulog2sum 20739. (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 9861 . . . . . 6  |-  2  e.  CC
21a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  2  e.  CC )
3 fzfid 11082 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
4 elfznn 10866 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
54adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
6 mucl 20432 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
mmu `  n )  e.  ZZ )
75, 6syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  ZZ )
87zred 10164 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  RR )
98, 5nndivred 9839 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  RR )
109recnd 8906 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  CC )
11 simpr 447 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
124nnrpd 10436 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
13 rpdivcl 10423 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  /  n )  e.  RR+ )
1411, 12, 13syl2an 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR+ )
1514relogcld 20027 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  RR )
1615recnd 8906 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  CC )
1716sqcld 11290 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  ( x  /  n ) ) ^
2 )  e.  CC )
1817halfcld 10003 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( log `  (
x  /  n ) ) ^ 2 )  /  2 )  e.  CC )
1910, 18mulcld 8900 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 ) )  e.  CC )
203, 19fsumcl 12253 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) )  e.  CC )
21 relogcl 19985 . . . . . . 7  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2221adantl 452 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
2322recnd 8906 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
242, 20, 23subdid 9280 . . . 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 12293 . . . . . 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 10 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  2  e.  CC )
2726, 10, 18mul12d 9066 . . . . . . . 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 9874 . . . . . . . . . . 11  |-  2  =/=  0
2928a1i 10 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  2  =/=  0 )
3017, 26, 29divcan2d 9583 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 ) )  =  ( ( log `  (
x  /  n ) ) ^ 2 ) )
3130oveq2d 5916 . . . . . . . 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 2348 . . . . . . 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 12223 . . . . . 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 2348 . . . . 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 5915 . . . 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 2348 . . 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 4143 . 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 9202 . . 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 10411 . . . . 5  |-  RR+  C_  RR
40 o1const 12140 . . . . 5  |-  ( (
RR+  C_  RR  /\  2  e.  CC )  ->  (
x  e.  RR+  |->  2 )  e.  O ( 1 ) )
4139, 1, 40mp2an 653 . . . 4  |-  ( x  e.  RR+  |->  2 )  e.  O ( 1 )
4241a1i 10 . . 3  |-  ( ph  ->  ( x  e.  RR+  |->  2 )  e.  O
( 1 ) )
43 emre 20352 . . . . . . . . . . . . 13  |-  gamma  e.  RR
4443recni 8894 . . . . . . . . . . . 12  |-  gamma  e.  CC
45 mulcl 8866 . . . . . . . . . . . 12  |-  ( (
gamma  e.  CC  /\  ( log `  ( x  /  n ) )  e.  CC )  ->  ( gamma  x.  ( log `  (
x  /  n ) ) )  e.  CC )
4644, 16, 45sylancr 644 . . . . . . . . . . 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 12024 . . . . . . . . . . . . 13  |-  ( F  ~~> r  L  ->  L  e.  CC )
4947, 48syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  CC )
5049ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  L  e.  CC )
5146, 50subcld 9202 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
)  e.  CC )
5218, 51addcld 8899 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) )  e.  CC )
5310, 52mulcld 8900 . . . . . . . 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 12253 . . . . . . 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 8900 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) )  e.  CC )
563, 55fsumcl 12253 . . . . . . 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 9234 . . . . . 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 12297 . . . . . . . 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 9280 . . . . . . . . . 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 9203 . . . . . . . . . . 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 5916 . . . . . . . . . 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 2350 . . . . . . . . 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 12223 . . . . . . . 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 2350 . . . . . . 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 5915 . . . . . 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 2348 . . . . 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 4143 . . . 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 9202 . . . . 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 2316 . . . . . 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 2316 . . . . . 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 20737 . . . . 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 10 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  gamma  e.  CC )
7410, 16mulcld 8900 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) )  e.  CC )
753, 73, 74fsummulc2 12293 . . . . . . . . 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 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  L  e.  CC )
773, 76, 10fsummulc1 12294 . . . . . . . . 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 5918 . . . . . . . 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 8866 . . . . . . . . . 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 644 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( gamma  x.  ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  CC )
8110, 50mulcld 8900 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  L )  e.  CC )
823, 80, 81fsumsub 12297 . . . . . . . 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 10 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  gamma  e.  CC )
8483, 10, 16mul12d 9066 . . . . . . . . . . 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 5915 . . . . . . . . . 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 9280 . . . . . . . . . 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 2351 . . . . . . . . 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 12223 . . . . . . . 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 2354 . . . . . . 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 4143 . . . . . 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 12253 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) )  e.  CC )
92 mulcl 8866 . . . . . . . 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 644 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( gamma  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  CC )
943, 10fsumcl 12253 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  e.  CC )
9594, 76mulcld 8900 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( mmu `  n )  /  n )  x.  L )  e.  CC )
9644a1i 10 . . . . . . . . 9  |-  ( ph  -> 
gamma  e.  CC )
97 o1const 12140 . . . . . . . . 9  |-  ( (
RR+  C_  RR  /\  gamma  e.  CC )  ->  (
x  e.  RR+  |->  gamma )  e.  O ( 1 ) )
9839, 96, 97sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  gamma )  e.  O ( 1 ) )
99 mulogsum 20734 . . . . . . . . 9  |-  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  O ( 1 )
10099a1i 10 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  sum_
n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  O ( 1 ) )
10173, 91, 98, 100o1mul2 12145 . . . . . . 7  |-  ( ph  ->  ( x  e.  RR+  |->  ( gamma  x.  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  e.  O ( 1 ) )
102 mudivsum 20732 . . . . . . . . 9  |-  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
) )  e.  O
( 1 )
103102a1i 10 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  sum_
n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n
)  /  n ) )  e.  O ( 1 ) )
104 o1const 12140 . . . . . . . . 9  |-  ( (
RR+  C_  RR  /\  L  e.  CC )  ->  (
x  e.  RR+  |->  L )  e.  O ( 1 ) )
10539, 49, 104sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  L )  e.  O
( 1 ) )
10694, 76, 103, 105o1mul2 12145 . . . . . . 7  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( mmu `  n )  /  n
)  x.  L ) )  e.  O ( 1 ) )
10793, 95, 101, 106o1sub2 12146 . . . . . 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 2391 . . . . 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 12146 . . . 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 2391 . . 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 12145 . 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 2391 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 358    = wceq 1633    e. wcel 1701    =/= wne 2479    C_ wss 3186   class class class wbr 4060    e. cmpt 4114   ` cfv 5292  (class class class)co 5900   CCcc 8780   RRcr 8781   0cc0 8782   1c1 8783    + caddc 8785    x. cmul 8787    - cmin 9082    / cdiv 9468   NNcn 9791   2c2 9840   ZZcz 10071   RR+crp 10401   ...cfz 10829   |_cfl 10971   ^cexp 11151   abscabs 11766    ~~> r crli 12006   O ( 1 )co1 12007   sum_csu 12205   _eceu 12391   logclog 19965   gammacem 20339   mmucmu 20385
This theorem is referenced by:  mulog2sum  20739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861  ax-mulf 8862
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-disj 4031  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-map 6817  df-pm 6818  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-sup 7239  df-oi 7270  df-card 7617  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ioo 10707  df-ioc 10708  df-ico 10709  df-icc 10710  df-fz 10830  df-fzo 10918  df-fl 10972  df-mod 11021  df-seq 11094  df-exp 11152  df-fac 11336  df-bc 11363  df-hash 11385  df-shft 11609  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-limsup 11992  df-clim 12009  df-rlim 12010  df-o1 12011  df-lo1 12012  df-sum 12206  df-ef 12396  df-e 12397  df-sin 12398  df-cos 12399  df-pi 12401  df-dvds 12579  df-gcd 12733  df-prm 12806  df-pc 12937  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-starv 13270  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-hom 13279  df-cco 13280  df-rest 13376  df-topn 13377  df-topgen 13393  df-pt 13394  df-prds 13397  df-xrs 13452  df-0g 13453  df-gsum 13454  df-qtop 13459  df-imas 13460  df-xps 13462  df-mre 13537  df-mrc 13538  df-acs 13540  df-mnd 14416  df-submnd 14465  df-mulg 14541  df-cntz 14842  df-cmn 15140  df-xmet 16425  df-met 16426  df-bl 16427  df-mopn 16428  df-fbas 16429  df-fg 16430  df-cnfld 16433  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-cld 16812  df-ntr 16813  df-cls 16814  df-nei 16891  df-lp 16924  df-perf 16925  df-cn 17013  df-cnp 17014  df-haus 17099  df-cmp 17170  df-tx 17313  df-hmeo 17502  df-fil 17593  df-fm 17685  df-flim 17686  df-flf 17687  df-xms 17937  df-ms 17938  df-tms 17939  df-cncf 18434  df-limc 19269  df-dv 19270  df-log 19967  df-cxp 19968  df-em 20340  df-mu 20391
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