MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulog2sumlem3 Structured version   Unicode version

Theorem mulog2sumlem3 22919
Description: Lemma for mulog2sum 22920. (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 10504 . . . . . 6  |-  2  e.  CC
21a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  2  e.  CC )
3 fzfid 11913 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
4 elfznn 11596 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
54adantl 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
6 mucl 22613 . . . . . . . . . . 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 10859 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  RR )
98, 5nndivred 10482 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  RR )
109recnd 9524 . . . . . . 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 11138 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
13 rpdivcl 11125 . . . . . . . . . . . 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 22206 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  RR )
1615recnd 9524 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  CC )
1716sqcld 12124 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  ( x  /  n ) ) ^
2 )  e.  CC )
1817halfcld 10681 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( log `  (
x  /  n ) ) ^ 2 )  /  2 )  e.  CC )
1910, 18mulcld 9518 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 ) )  e.  CC )
203, 19fsumcl 13329 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( ( log `  (
x  /  n ) ) ^ 2 )  /  2 ) )  e.  CC )
21 relogcl 22161 . . . . . . 7  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2221adantl 466 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  RR )
2322recnd 9524 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( log `  x )  e.  CC )
242, 20, 23subdid 9912 . . . 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 13370 . . . . . 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 9690 . . . . . . . 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 10526 . . . . . . . . . . 11  |-  2  =/=  0
2928a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  2  =/=  0 )
3017, 26, 29divcan2d 10221 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  / 
2 ) )  =  ( ( log `  (
x  /  n ) ) ^ 2 ) )
3130oveq2d 6217 . . . . . . . 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 2495 . . . . . . 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 13299 . . . . . 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 2495 . . . . 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 6216 . . . 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 2495 . . 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 4487 . 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 9831 . . 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 11113 . . . . 5  |-  RR+  C_  RR
40 o1const 13216 . . . . 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 22533 . . . . . . . . . . . . 13  |-  gamma  e.  RR
4443recni 9510 . . . . . . . . . . . 12  |-  gamma  e.  CC
45 mulcl 9478 . . . . . . . . . . . 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 13100 . . . . . . . . . . . . 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 9831 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
)  e.  CC )
5218, 51addcld 9517 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( ( log `  (
x  /  n ) ) ^ 2 )  /  2 )  +  ( ( gamma  x.  ( log `  ( x  /  n ) ) )  -  L ) )  e.  CC )
5310, 52mulcld 9518 . . . . . . . 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 13329 . . . . . . 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 9518 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( gamma  x.  ( log `  (
x  /  n ) ) )  -  L
) )  e.  CC )
563, 55fsumcl 13329 . . . . . . 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 9863 . . . . . 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 13374 . . . . . . . 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 9912 . . . . . . . . . 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 9832 . . . . . . . . . . 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 6217 . . . . . . . . . 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 2497 . . . . . . . . 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 13299 . . . . . . . 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 2497 . . . . . . 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 6216 . . . . . 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 2495 . . . . 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 4487 . . . 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 9831 . . . . 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 2454 . . . . . 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 2454 . . . . . 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 22918 . . . . 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 9518 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) )  e.  CC )
753, 73, 74fsummulc2 13370 . . . . . . . . 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 13371 . . . . . . . . 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 6219 . . . . . . . 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 9478 . . . . . . . . . 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 9518 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  L )  e.  CC )
823, 80, 81fsumsub 13374 . . . . . . . 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 9690 . . . . . . . . . . 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 6216 . . . . . . . . . 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 9912 . . . . . . . . . 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 2498 . . . . . . . . 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 13299 . . . . . . . 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 2501 . . . . . . 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 4487 . . . . . 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 13329 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) )  e.  CC )
92 mulcl 9478 . . . . . . . 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 13329 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  e.  CC )
9594, 76mulcld 9518 . . . . . . 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 13216 . . . . . . . . 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 22915 . . . . . . . . 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 13221 . . . . . . 7  |-  ( ph  ->  ( x  e.  RR+  |->  ( gamma  x.  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  e.  O(1) )
102 mudivsum 22913 . . . . . . . . 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 13216 . . . . . . . . 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 13221 . . . . . . 7  |-  ( ph  ->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( mmu `  n )  /  n
)  x.  L ) )  e.  O(1) )
10793, 95, 101, 106o1sub2 13222 . . . . . 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 2543 . . . . 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 13222 . . . 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 2543 . . 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 13221 . 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 2543 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 1370    e. wcel 1758    =/= wne 2648    C_ wss 3437   class class class wbr 4401    |-> cmpt 4459   ` cfv 5527  (class class class)co 6201   CCcc 9392   RRcr 9393   0cc0 9394   1c1 9395    + caddc 9397    x. cmul 9399    - cmin 9707    / cdiv 10105   NNcn 10434   2c2 10483   ZZcz 10758   RR+crp 11103   ...cfz 11555   |_cfl 11758   ^cexp 11983   abscabs 12842    ~~> r crli 13082   O(1)co1 13083   sum_csu 13282   _eceu 13467   logclog 22140   gammacem 22519   mmucmu 22566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472  ax-addf 9473  ax-mulf 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-disj 4372  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-ixp 7375  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-fi 7773  df-sup 7803  df-oi 7836  df-card 8221  df-cda 8449  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-q 11066  df-rp 11104  df-xneg 11201  df-xadd 11202  df-xmul 11203  df-ioo 11416  df-ioc 11417  df-ico 11418  df-icc 11419  df-fz 11556  df-fzo 11667  df-fl 11760  df-mod 11827  df-seq 11925  df-exp 11984  df-fac 12170  df-bc 12197  df-hash 12222  df-shft 12675  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-limsup 13068  df-clim 13085  df-rlim 13086  df-o1 13087  df-lo1 13088  df-sum 13283  df-ef 13472  df-e 13473  df-sin 13474  df-cos 13475  df-pi 13477  df-dvds 13655  df-gcd 13810  df-prm 13883  df-pc 14023  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-starv 14373  df-sca 14374  df-vsca 14375  df-ip 14376  df-tset 14377  df-ple 14378  df-ds 14380  df-unif 14381  df-hom 14382  df-cco 14383  df-rest 14481  df-topn 14482  df-0g 14500  df-gsum 14501  df-topgen 14502  df-pt 14503  df-prds 14506  df-xrs 14560  df-qtop 14565  df-imas 14566  df-xps 14568  df-mre 14644  df-mrc 14645  df-acs 14647  df-mnd 15535  df-submnd 15585  df-mulg 15668  df-cntz 15955  df-cmn 16401  df-psmet 17935  df-xmet 17936  df-met 17937  df-bl 17938  df-mopn 17939  df-fbas 17940  df-fg 17941  df-cnfld 17945  df-top 18636  df-bases 18638  df-topon 18639  df-topsp 18640  df-cld 18756  df-ntr 18757  df-cls 18758  df-nei 18835  df-lp 18873  df-perf 18874  df-cn 18964  df-cnp 18965  df-haus 19052  df-cmp 19123  df-tx 19268  df-hmeo 19461  df-fil 19552  df-fm 19644  df-flim 19645  df-flf 19646  df-xms 20028  df-ms 20029  df-tms 20030  df-cncf 20587  df-limc 21475  df-dv 21476  df-log 22142  df-cxp 22143  df-em 22520  df-mu 22572
This theorem is referenced by:  mulog2sum  22920
  Copyright terms: Public domain W3C validator