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Theorem logfac2 20954
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 11180 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
2 logfac 20448 . . 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 11267 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 1 ... ( |_ `  A ) )  e.  Fin )
5 fzfid 11267 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  A
) )  e.  Fin )
6 ssrab2 3388 . . . . 5  |-  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  C_  ( 1 ... ( |_ `  A
) )
7 ssfi 7288 . . . . 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 644 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  e.  Fin )
9 flcl 11159 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
109adantr 452 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  ZZ )
11 fznn 11070 . . . . . . . 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 686 . . . . . 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 9963 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR )
1514ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  e.  RR )
16 elfznn 11036 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
1716ad2antrl 709 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  n  e.  NN )
1817nnred 9971 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  n  e.  RR )
19 reflcl 11160 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
2019ad3antrrr 711 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  ( |_ `  A )  e.  RR )
21 simprr 734 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  ||  n
)
22 nnz 10259 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  k  e.  ZZ )
2322ad2antlr 708 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  e.  ZZ )
24 dvdsle 12850 . . . . . . . . . . . 12  |-  ( ( k  e.  ZZ  /\  n  e.  NN )  ->  ( k  ||  n  ->  k  <_  n )
)
2523, 17, 24syl2anc 643 . . . . . . . . . . 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 11017 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  <_  ( |_ `  A
) )
2827ad2antrl 709 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  n  <_  ( |_ `  A ) )
2915, 18, 20, 26, 28letrd 9183 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  <_  ( |_ `  A ) )
3029expl 602 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( k  e.  NN  /\  ( n  e.  ( 1 ... ( |_ `  A
) )  /\  k  ||  n ) )  -> 
k  <_  ( |_ `  A ) ) )
3130pm4.71rd 617 . . . . . . 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 773 . . . . . . 7  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( k  e.  NN  /\  k  ||  n ) )  <->  ( k  e.  NN  /\  ( n  e.  ( 1 ... ( |_ `  A
) )  /\  k  ||  n ) ) )
33 anass 631 . . . . . . . 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 773 . . . . . . . 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 241 . . . . . . 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 280 . . . . . 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 248 . . . . 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 4176 . . . . . . 7  |-  ( x  =  n  ->  (
k  ||  x  <->  k  ||  n ) )
3938elrab 3052 . . . . . 6  |-  ( n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x } 
<->  ( n  e.  ( 1 ... ( |_
`  A ) )  /\  k  ||  n
) )
4039anbi2i 676 . . . . 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 4175 . . . . . . 7  |-  ( x  =  k  ->  (
x  ||  n  <->  k  ||  n ) )
4241elrab 3052 . . . . . 6  |-  ( k  e.  { x  e.  NN  |  x  ||  n }  <->  ( k  e.  NN  /\  k  ||  n ) )
4342anbi2i 676 . . . . 5  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  k  e.  { x  e.  NN  |  x  ||  n } )  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( k  e.  NN  /\  k  ||  n ) ) )
4437, 40, 433bitr4g 280 . . . 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 11036 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  k  e.  NN )
4645adantl 453 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  k  e.  NN )
47 vmacl 20854 . . . . . . 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 9070 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  (Λ `  k )  e.  CC )
5049adantrr 698 . . . 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 12513 . . 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 12528 . . . . . 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 643 . . . . 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 11267 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  k ) ) )  e.  Fin )
55 simpll 731 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
56 eqid 2404 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... ( |_ `  ( A  /  k ) ) )  |->  ( k  x.  m ) )  =  ( m  e.  ( 1 ... ( |_
`  ( A  / 
k ) ) ) 
|->  ( k  x.  m
) )
5755, 46, 56dvdsflf1o 20925 . . . . . . . . 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 7085 . . . . . . . . 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 643 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  k ) ) )  ~~  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x } )
60 hasheni 11587 . . . . . . . 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 444 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  RR )
63 nndivre 9991 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  k  e.  NN )  ->  ( A  /  k
)  e.  RR )
6462, 45, 63syl2an 464 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( A  / 
k )  e.  RR )
65 nngt0 9985 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  0  <  k )
6614, 65jca 519 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
6745, 66syl 16 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  (
k  e.  RR  /\  0  <  k ) )
68 divge0 9835 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( k  e.  RR  /\  0  <  k ) )  ->  0  <_  ( A  /  k ) )
6967, 68sylan2 461 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  0  <_  ( A  /  k ) )
70 flge0nn0 11180 . . . . . . . . 9  |-  ( ( ( A  /  k
)  e.  RR  /\  0  <_  ( A  / 
k ) )  -> 
( |_ `  ( A  /  k ) )  e.  NN0 )
7164, 69, 70syl2anc 643 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  NN0 )
72 hashfz1 11585 . . . . . . . 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 2438 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( # `  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
)  =  ( |_
`  ( A  / 
k ) ) )
7574oveq1d 6055 . . . . 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 11162 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  ZZ )
7776zcnd 10332 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  CC )
7877, 49mulcomd 9065 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( ( |_
`  ( A  / 
k ) )  x.  (Λ `  k )
)  =  ( (Λ `  k )  x.  ( |_ `  ( A  / 
k ) ) ) )
7953, 75, 783eqtrd 2440 . . . 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 12452 . . 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 453 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  n  e.  (
1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
82 vmasum 20953 . . . . 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 12452 . . 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 2444 . 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 2439 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2670    C_ wss 3280   class class class wbr 4172    e. cmpt 4226   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040    ~~ cen 7065   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951    < clt 9076    <_ cle 9077    / cdiv 9633   NNcn 9956   NN0cn0 10177   ZZcz 10238   ...cfz 10999   |_cfl 11156   !cfa 11521   #chash 11573   sum_csu 12434    || cdivides 12807   logclog 20405  Λcvma 20827
This theorem is referenced by:  vmadivsum  21129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-vma 20833
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