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Theorem logfac2 22536
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 11658 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
2 logfac 22029 . . 3  |-  ( ( |_ `  A )  e.  NN0  ->  ( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( log `  n
) )
31, 2syl 16 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) ( log `  n
) )
4 fzfid 11787 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 1 ... ( |_ `  A ) )  e.  Fin )
5 fzfid 11787 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  A
) )  e.  Fin )
6 ssrab2 3432 . . . . 5  |-  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  C_  ( 1 ... ( |_ `  A
) )
7 ssfi 7525 . . . . 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 662 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  e.  Fin )
9 flcl 11637 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
109adantr 465 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  ZZ )
11 fznn 11518 . . . . . . . 8  |-  ( ( |_ `  A )  e.  ZZ  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  ( k  e.  NN  /\  k  <_ 
( |_ `  A
) ) ) )
1210, 11syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( k  e.  ( 1 ... ( |_
`  A ) )  <-> 
( k  e.  NN  /\  k  <_  ( |_ `  A ) ) ) )
1312anbi1d 704 . . . . . 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 10321 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR )
1514ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  e.  RR )
16 elfznn 11470 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
1716ad2antrl 727 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  n  e.  NN )
1817nnred 10329 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  n  e.  RR )
19 reflcl 11638 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
2019ad3antrrr 729 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  ( |_ `  A )  e.  RR )
21 simprr 756 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  ||  n
)
22 nnz 10660 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  k  e.  ZZ )
2322ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  e.  ZZ )
24 dvdsle 13570 . . . . . . . . . . . 12  |-  ( ( k  e.  ZZ  /\  n  e.  NN )  ->  ( k  ||  n  ->  k  <_  n )
)
2523, 17, 24syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  ( k  ||  n  ->  k  <_  n
) )
2621, 25mpd 15 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  <_  n
)
27 elfzle2 11447 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  <_  ( |_ `  A
) )
2827ad2antrl 727 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  n  <_  ( |_ `  A ) )
2915, 18, 20, 26, 28letrd 9520 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  <_  ( |_ `  A ) )
3029expl 618 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( k  e.  NN  /\  ( n  e.  ( 1 ... ( |_ `  A
) )  /\  k  ||  n ) )  -> 
k  <_  ( |_ `  A ) ) )
3130pm4.71rd 635 . . . . . . 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 795 . . . . . . 7  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( k  e.  NN  /\  k  ||  n ) )  <->  ( k  e.  NN  /\  ( n  e.  ( 1 ... ( |_ `  A
) )  /\  k  ||  n ) ) )
33 anass 649 . . . . . . . 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 795 . . . . . . . 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 249 . . . . . . 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 288 . . . . . 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 256 . . . . 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 4291 . . . . . . 7  |-  ( x  =  n  ->  (
k  ||  x  <->  k  ||  n ) )
3938elrab 3112 . . . . . 6  |-  ( n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x } 
<->  ( n  e.  ( 1 ... ( |_
`  A ) )  /\  k  ||  n
) )
4039anbi2i 694 . . . . 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 4290 . . . . . . 7  |-  ( x  =  k  ->  (
x  ||  n  <->  k  ||  n ) )
4241elrab 3112 . . . . . 6  |-  ( k  e.  { x  e.  NN  |  x  ||  n }  <->  ( k  e.  NN  /\  k  ||  n ) )
4342anbi2i 694 . . . . 5  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  k  e.  { x  e.  NN  |  x  ||  n } )  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( k  e.  NN  /\  k  ||  n ) ) )
4437, 40, 433bitr4g 288 . . . 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 11470 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  k  e.  NN )
4645adantl 466 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  k  e.  NN )
47 vmacl 22436 . . . . . . 7  |-  ( k  e.  NN  ->  (Λ `  k )  e.  RR )
4846, 47syl 16 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  (Λ `  k )  e.  RR )
4948recnd 9404 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  (Λ `  k )  e.  CC )
5049adantrr 716 . . . 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 13233 . . 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 13249 . . . . . 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 661 . . . . 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 11787 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  k ) ) )  e.  Fin )
55 simpll 753 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
56 eqid 2438 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... ( |_ `  ( A  /  k ) ) )  |->  ( k  x.  m ) )  =  ( m  e.  ( 1 ... ( |_
`  ( A  / 
k ) ) ) 
|->  ( k  x.  m
) )
5755, 46, 56dvdsflf1o 22507 . . . . . . . . 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 7320 . . . . . . . . 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 661 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  k ) ) )  ~~  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x } )
60 hasheni 12111 . . . . . . . 8  |-  ( ( 1 ... ( |_
`  ( A  / 
k ) ) ) 
~~  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  ->  ( # `  (
1 ... ( |_ `  ( A  /  k
) ) ) )  =  ( # `  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
) )
6159, 60syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( # `  (
1 ... ( |_ `  ( A  /  k
) ) ) )  =  ( # `  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
) )
62 simpl 457 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  RR )
63 nndivre 10349 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  k  e.  NN )  ->  ( A  /  k
)  e.  RR )
6462, 45, 63syl2an 477 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( A  / 
k )  e.  RR )
65 nngt0 10343 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  0  <  k )
6614, 65jca 532 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
6745, 66syl 16 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  (
k  e.  RR  /\  0  <  k ) )
68 divge0 10190 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( k  e.  RR  /\  0  <  k ) )  ->  0  <_  ( A  /  k ) )
6967, 68sylan2 474 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  0  <_  ( A  /  k ) )
70 flge0nn0 11658 . . . . . . . . 9  |-  ( ( ( A  /  k
)  e.  RR  /\  0  <_  ( A  / 
k ) )  -> 
( |_ `  ( A  /  k ) )  e.  NN0 )
7164, 69, 70syl2anc 661 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  NN0 )
72 hashfz1 12109 . . . . . . . 8  |-  ( ( |_ `  ( A  /  k ) )  e.  NN0  ->  ( # `  ( 1 ... ( |_ `  ( A  / 
k ) ) ) )  =  ( |_
`  ( A  / 
k ) ) )
7371, 72syl 16 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( # `  (
1 ... ( |_ `  ( A  /  k
) ) ) )  =  ( |_ `  ( A  /  k
) ) )
7461, 73eqtr3d 2472 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( # `  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
)  =  ( |_
`  ( A  / 
k ) ) )
7574oveq1d 6101 . . . . 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 11640 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  ZZ )
7776zcnd 10740 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  CC )
7877, 49mulcomd 9399 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( ( |_
`  ( A  / 
k ) )  x.  (Λ `  k )
)  =  ( (Λ `  k )  x.  ( |_ `  ( A  / 
k ) ) ) )
7953, 75, 783eqtrd 2474 . . . 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 13172 . . 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 466 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  n  e.  (
1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
82 vmasum 22535 . . . . 5  |-  ( n  e.  NN  ->  sum_ k  e.  { x  e.  NN  |  x  ||  n } 
(Λ `  k )  =  ( log `  n
) )
8381, 82syl 16 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  n  e.  (
1 ... ( |_ `  A ) ) )  ->  sum_ k  e.  {
x  e.  NN  |  x  ||  n }  (Λ `  k )  =  ( log `  n ) )
8483sumeq2dv 13172 . . 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 2478 . 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 2473 1  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ k  e.  ( 1 ... ( |_
`  A ) ) ( (Λ `  k
)  x.  ( |_
`  ( A  / 
k ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2714    C_ wss 3323   class class class wbr 4287    e. cmpt 4345   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086    ~~ cen 7299   Fincfn 7302   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    x. cmul 9279    < clt 9410    <_ cle 9411    / cdiv 9985   NNcn 10314   NN0cn0 10571   ZZcz 10638   ...cfz 11429   |_cfl 11632   !cfa 12043   #chash 12095   sum_csu 13155    || cdivides 13527   logclog 21986  Λcvma 22409
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-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-sum 13156  df-ef 13345  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 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-fbas 17794  df-fg 17795  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-lp 18720  df-perf 18721  df-cn 18811  df-cnp 18812  df-haus 18899  df-tx 19115  df-hmeo 19308  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-xms 19875  df-ms 19876  df-tms 19877  df-cncf 20434  df-limc 21321  df-dv 21322  df-log 21988  df-vma 22415
This theorem is referenced by:  vmadivsum  22711
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