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Theorem logfac2 20458
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 10950 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
2 logfac 19956 . . 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 11037 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 1 ... ( |_ `  A ) )  e.  Fin )
5 fzfid 11037 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  A
) )  e.  Fin )
6 ssrab2 3260 . . . . 5  |-  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  C_  ( 1 ... ( |_ `  A
) )
7 ssfi 7085 . . . . 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 10929 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
109adantr 451 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  ZZ )
11 fznn 10854 . . . . . . . 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 9755 . . . . . . . . . . 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 10821 . . . . . . . . . . . 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 9763 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  n  e.  RR )
19 reflcl 10930 . . . . . . . . . . 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 10047 . . . . . . . . . . . . 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 12576 . . . . . . . . . . . 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 10802 . . . . . . . . . . 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 8975 . . . . . . . . 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 4029 . . . . . . 7  |-  ( x  =  n  ->  (
k  ||  x  <->  k  ||  n ) )
3938elrab 2925 . . . . . 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 4028 . . . . . . 7  |-  ( x  =  k  ->  (
x  ||  n  <->  k  ||  n ) )
4241elrab 2925 . . . . . 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 10821 . . . . . . . 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 20358 . . . . . . 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 8863 . . . . 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 12239 . . 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 12254 . . . . . 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 11037 . . . . . . . . 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 2285 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... ( |_ `  ( A  /  k ) ) )  |->  ( k  x.  m ) )  =  ( m  e.  ( 1 ... ( |_
`  ( A  / 
k ) ) ) 
|->  ( k  x.  m
) )
5755, 46, 56dvdsflf1o 20429 . . . . . . . . 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 6882 . . . . . . . . 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 11349 . . . . . . . 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 9783 . . . . . . . . . 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 9777 . . . . . . . . . . . 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 9627 . . . . . . . . . 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 10950 . . . . . . . . 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 11347 . . . . . . . 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 2319 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( # `  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
)  =  ( |_
`  ( A  / 
k ) ) )
7574oveq1d 5875 . . . . 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 10932 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  ZZ )
7776zcnd 10120 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  CC )
7877, 49mulcomd 8858 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( ( |_
`  ( A  / 
k ) )  x.  (Λ `  k )
)  =  ( (Λ `  k )  x.  ( |_ `  ( A  / 
k ) ) ) )
7953, 75, 783eqtrd 2321 . . . 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 12178 . . 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 20457 . . . . 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 12178 . . 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 2325 . 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 2320 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 1625    e. wcel 1686   {crab 2549    C_ wss 3154   class class class wbr 4025    e. cmpt 4079   -1-1-onto->wf1o 5256   ` cfv 5257  (class class class)co 5860    ~~ cen 6862   Fincfn 6865   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    x. cmul 8744    < clt 8869    <_ cle 8870    / cdiv 9425   NNcn 9748   NN0cn0 9967   ZZcz 10026   ...cfz 10784   |_cfl 10926   !cfa 11290   #chash 11339   sum_csu 12160    || cdivides 12533   logclog 19914  Λcvma 20331
This theorem is referenced by:  vmadivsum  20633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-shft 11564  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-sum 12161  df-ef 12351  df-sin 12353  df-cos 12354  df-pi 12356  df-dvds 12534  df-gcd 12688  df-prm 12761  df-pc 12892  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-limc 19218  df-dv 19219  df-log 19916  df-vma 20337
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