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Theorem mulog2sumlem3 20685
Description: Lemma for mulog2sum 20686. (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 9816 . . . . . 6  |-  2  e.  CC
21a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  2  e.  CC )
3 fzfid 11035 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
4 elfznn 10819 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
54adantl 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
6 mucl 20379 . . . . . . . . . . 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 10117 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  RR )
98, 5nndivred 9794 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  RR )
109recnd 8861 . . . . . . 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 10389 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
13 rpdivcl 10376 . . . . . . . . . . . 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 19974 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  RR )
1615recnd 8861 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  CC )
1716sqcld 11243 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  ( x  /  n ) ) ^
2 )  e.  CC )
1817halfcld 9956 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( log `  (
x  /  n ) ) ^ 2 )  /  2 )  e.  CC )
1910, 18mulcld 8855 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 ) )  e.  CC )
203, 19fsumcl 12206 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) )  e.  CC )
21 relogcl 19932 . . . . . . 7  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2221adantl 452 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
2322recnd 8861 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
242, 20, 23subdid 9235 . . . 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 12246 . . . . . 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 9021 . . . . . . . 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 9829 . . . . . . . . . . 11  |-  2  =/=  0
2928a1i 10 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  2  =/=  0 )
3017, 26, 29divcan2d 9538 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 ) )  =  ( ( log `  (
x  /  n ) ) ^ 2 ) )
3130oveq2d 5874 . . . . . . . 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 2315 . . . . . . 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 12176 . . . . . 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 2315 . . . . 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 5873 . . . 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 2315 . . 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 4106 . 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 9157 . . 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 10364 . . . . 5  |-  RR+  C_  RR
40 o1const 12093 . . . . 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 20299 . . . . . . . . . . . . 13  |-  gamma  e.  RR
4443recni 8849 . . . . . . . . . . . 12  |-  gamma  e.  CC
45 mulcl 8821 . . . . . . . . . . . 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 11977 . . . . . . . . . . . . 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 9157 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
)  e.  CC )
5218, 51addcld 8854 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) )  e.  CC )
5310, 52mulcld 8855 . . . . . . . 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 12206 . . . . . . 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 8855 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) )  e.  CC )
563, 55fsumcl 12206 . . . . . . 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 9189 . . . . . 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 12250 . . . . . . . 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 9235 . . . . . . . . . 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 9158 . . . . . . . . . . 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 5874 . . . . . . . . . 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 2317 . . . . . . . . 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 12176 . . . . . . . 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 2317 . . . . . . 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 5873 . . . . . 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 2315 . . . . 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 4106 . . . 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 9157 . . . . 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 2283 . . . . . 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 2283 . . . . . 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 20684 . . . . 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 8855 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) )  e.  CC )
753, 73, 74fsummulc2 12246 . . . . . . . . 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 12247 . . . . . . . . 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 5876 . . . . . . . 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 8821 . . . . . . . . . 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 8855 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  L )  e.  CC )
823, 80, 81fsumsub 12250 . . . . . . . 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 9021 . . . . . . . . . . 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 5873 . . . . . . . . . 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 9235 . . . . . . . . . 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 2318 . . . . . . . . 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 12176 . . . . . . . 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 2321 . . . . . . 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 4106 . . . . . 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 12206 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) )  e.  CC )
92 mulcl 8821 . . . . . . . 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 12206 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  e.  CC )
9594, 76mulcld 8855 . . . . . . 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 12093 . . . . . . . . 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 20681 . . . . . . . . 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 12098 . . . . . . 7  |-  ( ph  ->  ( x  e.  RR+  |->  ( gamma  x.  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  e.  O ( 1 ) )
102 mudivsum 20679 . . . . . . . . 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 12093 . . . . . . . . 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 12098 . . . . . . 7  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( mmu `  n )  /  n
)  x.  L ) )  e.  O ( 1 ) )
10793, 95, 101, 106o1sub2 12099 . . . . . 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 2358 . . . . 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 12099 . . . 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 2358 . . 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 12098 . 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 2358 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 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   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    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   ZZcz 10024   RR+crp 10354   ...cfz 10782   |_cfl 10924   ^cexp 11104   abscabs 11719    ~~> r crli 11959   O ( 1 )co1 11960   sum_csu 12158   _eceu 12344   logclog 19912   gammacem 20286   mmucmu 20332
This theorem is referenced by:  mulog2sum  20686
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-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  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-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-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  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-mu 20338
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