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Theorem logfac2 20456
Description: Another expression for the logarithm of a factorial, in terms of the von Mangoldt function. Equation 9.2.7 of [Shapiro], p. 329. (Contributed by Mario Carneiro, 15-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
Assertion
Ref Expression
logfac2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ k  e.  ( 1 ... ( |_
`  A ) ) ( (Λ `  k
)  x.  ( |_
`  ( A  / 
k ) ) ) )
Distinct variable group:    A, k

Proof of Theorem logfac2
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flge0nn0 10948 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
2 logfac 19954 . . 3  |-  ( ( |_ `  A )  e.  NN0  ->  ( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( log `  n
) )
31, 2syl 15 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) ( log `  n
) )
4 fzfid 11035 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 1 ... ( |_ `  A ) )  e.  Fin )
5 fzfid 11035 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  A
) )  e.  Fin )
6 ssrab2 3258 . . . . 5  |-  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  C_  ( 1 ... ( |_ `  A
) )
7 ssfi 7083 . . . . 5  |-  ( ( ( 1 ... ( |_ `  A ) )  e.  Fin  /\  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  C_  ( 1 ... ( |_ `  A ) ) )  ->  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  e.  Fin )
85, 6, 7sylancl 643 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  e.  Fin )
9 flcl 10927 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
109adantr 451 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  ZZ )
11 fznn 10852 . . . . . . . 8  |-  ( ( |_ `  A )  e.  ZZ  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  ( k  e.  NN  /\  k  <_ 
( |_ `  A
) ) ) )
1210, 11syl 15 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( k  e.  ( 1 ... ( |_
`  A ) )  <-> 
( k  e.  NN  /\  k  <_  ( |_ `  A ) ) ) )
1312anbi1d 685 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( k  e.  ( 1 ... ( |_ `  A ) )  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n
) )  <->  ( (
k  e.  NN  /\  k  <_  ( |_ `  A ) )  /\  ( n  e.  (
1 ... ( |_ `  A ) )  /\  k  ||  n ) ) ) )
14 nnre 9753 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR )
1514ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  e.  RR )
16 elfznn 10819 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
1716ad2antrl 708 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  n  e.  NN )
1817nnred 9761 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  n  e.  RR )
19 reflcl 10928 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
2019ad3antrrr 710 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  ( |_ `  A )  e.  RR )
21 simprr 733 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  ||  n
)
22 nnz 10045 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  k  e.  ZZ )
2322ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  e.  ZZ )
24 dvdsle 12574 . . . . . . . . . . . 12  |-  ( ( k  e.  ZZ  /\  n  e.  NN )  ->  ( k  ||  n  ->  k  <_  n )
)
2523, 17, 24syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  ( k  ||  n  ->  k  <_  n
) )
2621, 25mpd 14 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  <_  n
)
27 elfzle2 10800 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  <_  ( |_ `  A
) )
2827ad2antrl 708 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  n  <_  ( |_ `  A ) )
2915, 18, 20, 26, 28letrd 8973 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  <_  ( |_ `  A ) )
3029expl 601 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( k  e.  NN  /\  ( n  e.  ( 1 ... ( |_ `  A
) )  /\  k  ||  n ) )  -> 
k  <_  ( |_ `  A ) ) )
3130pm4.71rd 616 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( k  e.  NN  /\  ( n  e.  ( 1 ... ( |_ `  A
) )  /\  k  ||  n ) )  <->  ( k  <_  ( |_ `  A
)  /\  ( k  e.  NN  /\  ( n  e.  ( 1 ... ( |_ `  A
) )  /\  k  ||  n ) ) ) ) )
32 an12 772 . . . . . . 7  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( k  e.  NN  /\  k  ||  n ) )  <->  ( k  e.  NN  /\  ( n  e.  ( 1 ... ( |_ `  A
) )  /\  k  ||  n ) ) )
33 anass 630 . . . . . . . 8  |-  ( ( ( k  e.  NN  /\  k  <_  ( |_ `  A ) )  /\  ( n  e.  (
1 ... ( |_ `  A ) )  /\  k  ||  n ) )  <-> 
( k  e.  NN  /\  ( k  <_  ( |_ `  A )  /\  ( n  e.  (
1 ... ( |_ `  A ) )  /\  k  ||  n ) ) ) )
34 an12 772 . . . . . . . 8  |-  ( ( k  e.  NN  /\  ( k  <_  ( |_ `  A )  /\  ( n  e.  (
1 ... ( |_ `  A ) )  /\  k  ||  n ) ) )  <->  ( k  <_ 
( |_ `  A
)  /\  ( k  e.  NN  /\  ( n  e.  ( 1 ... ( |_ `  A
) )  /\  k  ||  n ) ) ) )
3533, 34bitri 240 . . . . . . 7  |-  ( ( ( k  e.  NN  /\  k  <_  ( |_ `  A ) )  /\  ( n  e.  (
1 ... ( |_ `  A ) )  /\  k  ||  n ) )  <-> 
( k  <_  ( |_ `  A )  /\  ( k  e.  NN  /\  ( n  e.  ( 1 ... ( |_
`  A ) )  /\  k  ||  n
) ) ) )
3631, 32, 353bitr4g 279 . . . . . 6  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( k  e.  NN  /\  k  ||  n ) )  <->  ( (
k  e.  NN  /\  k  <_  ( |_ `  A ) )  /\  ( n  e.  (
1 ... ( |_ `  A ) )  /\  k  ||  n ) ) ) )
3713, 36bitr4d 247 . . . . 5  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( k  e.  ( 1 ... ( |_ `  A ) )  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n
) )  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( k  e.  NN  /\  k  ||  n ) ) ) )
38 breq2 4027 . . . . . . 7  |-  ( x  =  n  ->  (
k  ||  x  <->  k  ||  n ) )
3938elrab 2923 . . . . . 6  |-  ( n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x } 
<->  ( n  e.  ( 1 ... ( |_
`  A ) )  /\  k  ||  n
) )
4039anbi2i 675 . . . . 5  |-  ( ( k  e.  ( 1 ... ( |_ `  A ) )  /\  n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x } )  <->  ( k  e.  ( 1 ... ( |_ `  A ) )  /\  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n
) ) )
41 breq1 4026 . . . . . . 7  |-  ( x  =  k  ->  (
x  ||  n  <->  k  ||  n ) )
4241elrab 2923 . . . . . 6  |-  ( k  e.  { x  e.  NN  |  x  ||  n }  <->  ( k  e.  NN  /\  k  ||  n ) )
4342anbi2i 675 . . . . 5  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  k  e.  { x  e.  NN  |  x  ||  n } )  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( k  e.  NN  /\  k  ||  n ) ) )
4437, 40, 433bitr4g 279 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( k  e.  ( 1 ... ( |_ `  A ) )  /\  n  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
)  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  k  e.  {
x  e.  NN  |  x  ||  n } ) ) )
45 elfznn 10819 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  k  e.  NN )
4645adantl 452 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  k  e.  NN )
47 vmacl 20356 . . . . . . 7  |-  ( k  e.  NN  ->  (Λ `  k )  e.  RR )
4846, 47syl 15 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  (Λ `  k )  e.  RR )
4948recnd 8861 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  (Λ `  k )  e.  CC )
5049adantrr 697 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( k  e.  ( 1 ... ( |_
`  A ) )  /\  n  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
) )  ->  (Λ `  k )  e.  CC )
514, 4, 8, 44, 50fsumcom2 12237 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  sum_ k  e.  ( 1 ... ( |_ `  A ) ) sum_ n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  (Λ `  k )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) )
sum_ k  e.  {
x  e.  NN  |  x  ||  n }  (Λ `  k ) )
52 fsumconst 12252 . . . . . 6  |-  ( ( { x  e.  ( 1 ... ( |_
`  A ) )  |  k  ||  x }  e.  Fin  /\  (Λ `  k )  e.  CC )  ->  sum_ n  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x } 
(Λ `  k )  =  ( ( # `  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
)  x.  (Λ `  k
) ) )
538, 49, 52syl2anc 642 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  sum_ n  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x } 
(Λ `  k )  =  ( ( # `  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
)  x.  (Λ `  k
) ) )
54 fzfid 11035 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  k ) ) )  e.  Fin )
55 simpll 730 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
56 eqid 2283 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... ( |_ `  ( A  /  k ) ) )  |->  ( k  x.  m ) )  =  ( m  e.  ( 1 ... ( |_
`  ( A  / 
k ) ) ) 
|->  ( k  x.  m
) )
5755, 46, 56dvdsflf1o 20427 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( m  e.  ( 1 ... ( |_ `  ( A  / 
k ) ) ) 
|->  ( k  x.  m
) ) : ( 1 ... ( |_
`  ( A  / 
k ) ) ) -1-1-onto-> { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
)
58 f1oeng 6880 . . . . . . . . 9  |-  ( ( ( 1 ... ( |_ `  ( A  / 
k ) ) )  e.  Fin  /\  (
m  e.  ( 1 ... ( |_ `  ( A  /  k
) ) )  |->  ( k  x.  m ) ) : ( 1 ... ( |_ `  ( A  /  k
) ) ) -1-1-onto-> { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x } )  ->  (
1 ... ( |_ `  ( A  /  k
) ) )  ~~  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
)
5954, 57, 58syl2anc 642 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  k ) ) )  ~~  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x } )
60 hasheni 11347 . . . . . . . 8  |-  ( ( 1 ... ( |_
`  ( A  / 
k ) ) ) 
~~  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  ->  ( # `  (
1 ... ( |_ `  ( A  /  k
) ) ) )  =  ( # `  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
) )
6159, 60syl 15 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( # `  (
1 ... ( |_ `  ( A  /  k
) ) ) )  =  ( # `  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
) )
62 simpl 443 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  RR )
63 nndivre 9781 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  k  e.  NN )  ->  ( A  /  k
)  e.  RR )
6462, 45, 63syl2an 463 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( A  / 
k )  e.  RR )
65 nngt0 9775 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  0  <  k )
6614, 65jca 518 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
6745, 66syl 15 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  (
k  e.  RR  /\  0  <  k ) )
68 divge0 9625 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( k  e.  RR  /\  0  <  k ) )  ->  0  <_  ( A  /  k ) )
6967, 68sylan2 460 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  0  <_  ( A  /  k ) )
70 flge0nn0 10948 . . . . . . . . 9  |-  ( ( ( A  /  k
)  e.  RR  /\  0  <_  ( A  / 
k ) )  -> 
( |_ `  ( A  /  k ) )  e.  NN0 )
7164, 69, 70syl2anc 642 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  NN0 )
72 hashfz1 11345 . . . . . . . 8  |-  ( ( |_ `  ( A  /  k ) )  e.  NN0  ->  ( # `  ( 1 ... ( |_ `  ( A  / 
k ) ) ) )  =  ( |_
`  ( A  / 
k ) ) )
7371, 72syl 15 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( # `  (
1 ... ( |_ `  ( A  /  k
) ) ) )  =  ( |_ `  ( A  /  k
) ) )
7461, 73eqtr3d 2317 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( # `  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
)  =  ( |_
`  ( A  / 
k ) ) )
7574oveq1d 5873 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( ( # `  { x  e.  ( 1 ... ( |_
`  A ) )  |  k  ||  x } )  x.  (Λ `  k ) )  =  ( ( |_ `  ( A  /  k
) )  x.  (Λ `  k ) ) )
7664flcld 10930 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  ZZ )
7776zcnd 10118 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  CC )
7877, 49mulcomd 8856 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( ( |_
`  ( A  / 
k ) )  x.  (Λ `  k )
)  =  ( (Λ `  k )  x.  ( |_ `  ( A  / 
k ) ) ) )
7953, 75, 783eqtrd 2319 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  sum_ n  e.  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x } 
(Λ `  k )  =  ( (Λ `  k
)  x.  ( |_
`  ( A  / 
k ) ) ) )
8079sumeq2dv 12176 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  sum_ k  e.  ( 1 ... ( |_ `  A ) ) sum_ n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  (Λ `  k )  =  sum_ k  e.  ( 1 ... ( |_
`  A ) ) ( (Λ `  k
)  x.  ( |_
`  ( A  / 
k ) ) ) )
8116adantl 452 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  n  e.  (
1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
82 vmasum 20455 . . . . 5  |-  ( n  e.  NN  ->  sum_ k  e.  { x  e.  NN  |  x  ||  n } 
(Λ `  k )  =  ( log `  n
) )
8381, 82syl 15 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  n  e.  (
1 ... ( |_ `  A ) ) )  ->  sum_ k  e.  {
x  e.  NN  |  x  ||  n }  (Λ `  k )  =  ( log `  n ) )
8483sumeq2dv 12176 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  sum_ n  e.  ( 1 ... ( |_ `  A ) ) sum_ k  e.  { x  e.  NN  |  x  ||  n }  (Λ `  k
)  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( log `  n
) )
8551, 80, 843eqtr3d 2323 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  sum_ k  e.  ( 1 ... ( |_ `  A ) ) ( (Λ `  k )  x.  ( |_ `  ( A  /  k ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( log `  n
) )
863, 85eqtr4d 2318 1  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ k  e.  ( 1 ... ( |_
`  A ) ) ( (Λ `  k
)  x.  ( |_
`  ( A  / 
k ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    ~~ cen 6860   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ...cfz 10782   |_cfl 10924   !cfa 11288   #chash 11337   sum_csu 12158    || cdivides 12531   logclog 19912  Λcvma 20329
This theorem is referenced by:  vmadivsum  20631
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-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-sum 12159  df-ef 12349  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-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-vma 20335
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