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Theorem mulog2sumlem3 22760
Description: Lemma for mulog2sum 22761. (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 10384 . . . . . 6  |-  2  e.  CC
21a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  2  e.  CC )
3 fzfid 11787 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
4 elfznn 11470 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
54adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
6 mucl 22454 . . . . . . . . . . 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 10739 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  RR )
98, 5nndivred 10362 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  RR )
109recnd 9404 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  CC )
11 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
124nnrpd 11018 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
13 rpdivcl 11005 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  /  n )  e.  RR+ )
1411, 12, 13syl2an 477 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR+ )
1514relogcld 22047 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  RR )
1615recnd 9404 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  CC )
1716sqcld 11998 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  ( x  /  n ) ) ^
2 )  e.  CC )
1817halfcld 10561 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( log `  (
x  /  n ) ) ^ 2 )  /  2 )  e.  CC )
1910, 18mulcld 9398 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 ) )  e.  CC )
203, 19fsumcl 13202 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) )  e.  CC )
21 relogcl 22002 . . . . . . 7  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2221adantl 466 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
2322recnd 9404 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
242, 20, 23subdid 9792 . . . 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 13243 . . . . . 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 9570 . . . . . . . 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 10406 . . . . . . . . . . 11  |-  2  =/=  0
2928a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  2  =/=  0 )
3017, 26, 29divcan2d 10101 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 ) )  =  ( ( log `  (
x  /  n ) ) ^ 2 ) )
3130oveq2d 6102 . . . . . . . 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 2470 . . . . . . 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 13172 . . . . . 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 2470 . . . . 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 6101 . . . 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 2470 . . 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 4373 . 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 9711 . . 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 10993 . . . . 5  |-  RR+  C_  RR
40 o1const 13089 . . . . 5  |-  ( (
RR+  C_  RR  /\  2  e.  CC )  ->  (
x  e.  RR+  |->  2 )  e.  O(1) )
4139, 1, 40mp2an 672 . . . 4  |-  ( x  e.  RR+  |->  2 )  e.  O(1)
4241a1i 11 . . 3  |-  ( ph  ->  ( x  e.  RR+  |->  2 )  e.  O(1) )
43 emre 22374 . . . . . . . . . . . . 13  |-  gamma  e.  RR
4443recni 9390 . . . . . . . . . . . 12  |-  gamma  e.  CC
45 mulcl 9358 . . . . . . . . . . . 12  |-  ( (
gamma  e.  CC  /\  ( log `  ( x  /  n ) )  e.  CC )  ->  ( gamma  x.  ( log `  (
x  /  n ) ) )  e.  CC )
4644, 16, 45sylancr 663 . . . . . . . . . . 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 12973 . . . . . . . . . . . . 13  |-  ( F  ~~> r  L  ->  L  e.  CC )
4947, 48syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  L  e.  CC )
5049ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  L  e.  CC )
5146, 50subcld 9711 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
)  e.  CC )
5218, 51addcld 9397 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) )  e.  CC )
5310, 52mulcld 9398 . . . . . . . 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 13202 . . . . . . 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 9398 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) )  e.  CC )
563, 55fsumcl 13202 . . . . . . 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 9743 . . . . . 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 13247 . . . . . . . 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 9792 . . . . . . . . . 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 9712 . . . . . . . . . . 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 6102 . . . . . . . . . 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 2472 . . . . . . . . 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 13172 . . . . . . . 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 2472 . . . . . . 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 6101 . . . . . 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 2470 . . . . 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 4373 . . . 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 9711 . . . . 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 2438 . . . . . 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 2438 . . . . . 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 22759 . . . . 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 9398 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) )  e.  CC )
753, 73, 74fsummulc2 13243 . . . . . . . . 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 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  RR+ )  ->  L  e.  CC )
773, 76, 10fsummulc1 13244 . . . . . . . . 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 6104 . . . . . . . 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 9358 . . . . . . . . . 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 663 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( gamma  x.  ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  CC )
8110, 50mulcld 9398 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  L )  e.  CC )
823, 80, 81fsumsub 13247 . . . . . . . 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 9570 . . . . . . . . . . 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 6101 . . . . . . . . . 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 9792 . . . . . . . . . 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 2473 . . . . . . . . 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 13172 . . . . . . . 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 2476 . . . . . . 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 4373 . . . . . 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 13202 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) )  e.  CC )
92 mulcl 9358 . . . . . . . 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 663 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( gamma  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  CC )
943, 10fsumcl 13202 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  e.  CC )
9594, 76mulcld 9398 . . . . . . 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 13089 . . . . . . . . 9  |-  ( (
RR+  C_  RR  /\  gamma  e.  CC )  ->  (
x  e.  RR+  |->  gamma )  e.  O(1) )
9839, 96, 97sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  gamma )  e.  O(1) )
99 mulogsum 22756 . . . . . . . . 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 13094 . . . . . . 7  |-  ( ph  ->  ( x  e.  RR+  |->  ( gamma  x.  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  e.  O(1) )
102 mudivsum 22754 . . . . . . . . 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 13089 . . . . . . . . 9  |-  ( (
RR+  C_  RR  /\  L  e.  CC )  ->  (
x  e.  RR+  |->  L )  e.  O(1) )
10539, 49, 104sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( x  e.  RR+  |->  L )  e.  O(1) )
10694, 76, 103, 105o1mul2 13094 . . . . . . 7  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( mmu `  n )  /  n
)  x.  L ) )  e.  O(1) )
10793, 95, 101, 106o1sub2 13095 . . . . . 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 2513 . . . . 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 13095 . . . 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 2513 . . 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 13094 . 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 2513 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2601    C_ wss 3323   class class class wbr 4287    e. cmpt 4345   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    - cmin 9587    / cdiv 9985   NNcn 10314   2c2 10363   ZZcz 10638   RR+crp 10983   ...cfz 11429   |_cfl 11632   ^cexp 11857   abscabs 12715    ~~> r crli 12955   O(1)co1 12956   sum_csu 13155   _eceu 13340   logclog 21981   gammacem 22360   mmucmu 22407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-o1 12960  df-lo1 12961  df-sum 13156  df-ef 13345  df-e 13346  df-sin 13347  df-cos 13348  df-pi 13350  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-cmp 18965  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317  df-log 21983  df-cxp 21984  df-em 22361  df-mu 22413
This theorem is referenced by:  mulog2sum  22761
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