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Theorem logfac2 20383
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
StepHypRef Expression
1 flge0nn0 10879 . . 3  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  NN0 )
2 logfac 19881 . . 3  |-  ( ( |_ `  A )  e.  NN0  ->  ( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( log `  n
) )
31, 2syl 17 . 2  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( log `  ( ! `  ( |_ `  A ) ) )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) ( log `  n
) )
4 fzfid 10966 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( 1 ... ( |_ `  A ) )  e.  Fin )
5 fzfid 10966 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  A
) )  e.  Fin )
6 ssrab2 3200 . . . . 5  |-  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  C_  ( 1 ... ( |_ `  A
) )
7 ssfi 7016 . . . . 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 646 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  e.  Fin )
9 flcl 10858 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
109adantr 453 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( |_ `  A
)  e.  ZZ )
11 fznn 10783 . . . . . . . 8  |-  ( ( |_ `  A )  e.  ZZ  ->  (
k  e.  ( 1 ... ( |_ `  A ) )  <->  ( k  e.  NN  /\  k  <_ 
( |_ `  A
) ) ) )
1210, 11syl 17 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( k  e.  ( 1 ... ( |_
`  A ) )  <-> 
( k  e.  NN  /\  k  <_  ( |_ `  A ) ) ) )
1312anbi1d 688 . . . . . 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 9686 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  k  e.  RR )
1514ad2antlr 710 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  e.  RR )
16 elfznn 10750 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
1716ad2antrl 711 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  n  e.  NN )
1817nnred 9694 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  n  e.  RR )
19 reflcl 10859 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
2019ad3antrrr 713 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  ( |_ `  A )  e.  RR )
21 simprr 736 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  ||  n
)
22 nnz 9977 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  k  e.  ZZ )
2322ad2antlr 710 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  e.  ZZ )
24 dvdsle 12501 . . . . . . . . . . . 12  |-  ( ( k  e.  ZZ  /\  n  e.  NN )  ->  ( k  ||  n  ->  k  <_  n )
)
2523, 17, 24syl2anc 645 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  ( k  ||  n  ->  k  <_  n
) )
2621, 25mpd 16 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  <_  n
)
27 elfzle2 10731 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  <_  ( |_ `  A
) )
2827ad2antrl 711 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  n  <_  ( |_ `  A ) )
2915, 18, 20, 26, 28letrd 8906 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  NN )  /\  (
n  e.  ( 1 ... ( |_ `  A ) )  /\  k  ||  n ) )  ->  k  <_  ( |_ `  A ) )
3029expl 604 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( ( k  e.  NN  /\  ( n  e.  ( 1 ... ( |_ `  A
) )  /\  k  ||  n ) )  -> 
k  <_  ( |_ `  A ) ) )
3130pm4.71rd 619 . . . . . . 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 775 . . . . . . 7  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( k  e.  NN  /\  k  ||  n ) )  <->  ( k  e.  NN  /\  ( n  e.  ( 1 ... ( |_ `  A
) )  /\  k  ||  n ) ) )
33 anass 633 . . . . . . . 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 775 . . . . . . . 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 242 . . . . . . 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 281 . . . . . 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 249 . . . . 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 3967 . . . . . . 7  |-  ( x  =  n  ->  (
k  ||  x  <->  k  ||  n ) )
3938elrab 2874 . . . . . 6  |-  ( n  e.  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x } 
<->  ( n  e.  ( 1 ... ( |_
`  A ) )  /\  k  ||  n
) )
4039anbi2i 678 . . . . 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 3966 . . . . . . 7  |-  ( x  =  k  ->  (
x  ||  n  <->  k  ||  n ) )
4241elrab 2874 . . . . . 6  |-  ( k  e.  { x  e.  NN  |  x  ||  n }  <->  ( k  e.  NN  /\  k  ||  n ) )
4342anbi2i 678 . . . . 5  |-  ( ( n  e.  ( 1 ... ( |_ `  A ) )  /\  k  e.  { x  e.  NN  |  x  ||  n } )  <->  ( n  e.  ( 1 ... ( |_ `  A ) )  /\  ( k  e.  NN  /\  k  ||  n ) ) )
4437, 40, 433bitr4g 281 . . . 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 10750 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  k  e.  NN )
4645adantl 454 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  k  e.  NN )
47 vmacl 20283 . . . . . . 7  |-  ( k  e.  NN  ->  (Λ `  k )  e.  RR )
4846, 47syl 17 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  (Λ `  k )  e.  RR )
4948recnd 8794 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  (Λ `  k )  e.  CC )
5049adantrr 700 . . . 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 12167 . . 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 12182 . . . . . 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 645 . . . . 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 10966 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  k ) ) )  e.  Fin )
55 simpll 733 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
56 eqid 2256 . . . . . . . . . 10  |-  ( m  e.  ( 1 ... ( |_ `  ( A  /  k ) ) )  |->  ( k  x.  m ) )  =  ( m  e.  ( 1 ... ( |_
`  ( A  / 
k ) ) ) 
|->  ( k  x.  m
) )
5755, 46, 56dvdsflf1o 20354 . . . . . . . . 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 6813 . . . . . . . . 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 645 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  k ) ) )  ~~  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x } )
60 hasheni 11278 . . . . . . . 8  |-  ( ( 1 ... ( |_
`  ( A  / 
k ) ) ) 
~~  { x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }  ->  ( # `  (
1 ... ( |_ `  ( A  /  k
) ) ) )  =  ( # `  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
) )
6159, 60syl 17 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( # `  (
1 ... ( |_ `  ( A  /  k
) ) ) )  =  ( # `  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
) )
62 simpl 445 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  0  <_  A )  ->  A  e.  RR )
63 nndivre 9714 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  k  e.  NN )  ->  ( A  /  k
)  e.  RR )
6462, 45, 63syl2an 465 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( A  / 
k )  e.  RR )
65 nngt0 9708 . . . . . . . . . . . 12  |-  ( k  e.  NN  ->  0  <  k )
6614, 65jca 520 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
6745, 66syl 17 . . . . . . . . . 10  |-  ( k  e.  ( 1 ... ( |_ `  A
) )  ->  (
k  e.  RR  /\  0  <  k ) )
68 divge0 9558 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( k  e.  RR  /\  0  <  k ) )  ->  0  <_  ( A  /  k ) )
6967, 68sylan2 462 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  0  <_  ( A  /  k ) )
70 flge0nn0 10879 . . . . . . . . 9  |-  ( ( ( A  /  k
)  e.  RR  /\  0  <_  ( A  / 
k ) )  -> 
( |_ `  ( A  /  k ) )  e.  NN0 )
7164, 69, 70syl2anc 645 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  NN0 )
72 hashfz1 11276 . . . . . . . 8  |-  ( ( |_ `  ( A  /  k ) )  e.  NN0  ->  ( # `  ( 1 ... ( |_ `  ( A  / 
k ) ) ) )  =  ( |_
`  ( A  / 
k ) ) )
7371, 72syl 17 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( # `  (
1 ... ( |_ `  ( A  /  k
) ) ) )  =  ( |_ `  ( A  /  k
) ) )
7461, 73eqtr3d 2290 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( # `  {
x  e.  ( 1 ... ( |_ `  A ) )  |  k  ||  x }
)  =  ( |_
`  ( A  / 
k ) ) )
7574oveq1d 5772 . . . . 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 10861 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  ZZ )
7776zcnd 10050 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( |_ `  ( A  /  k
) )  e.  CC )
7877, 49mulcomd 8789 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  k  e.  (
1 ... ( |_ `  A ) ) )  ->  ( ( |_
`  ( A  / 
k ) )  x.  (Λ `  k )
)  =  ( (Λ `  k )  x.  ( |_ `  ( A  / 
k ) ) ) )
7953, 75, 783eqtrd 2292 . . . 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 12106 . . 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 454 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  n  e.  (
1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
82 vmasum 20382 . . . . 5  |-  ( n  e.  NN  ->  sum_ k  e.  { x  e.  NN  |  x  ||  n } 
(Λ `  k )  =  ( log `  n
) )
8381, 82syl 17 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  n  e.  (
1 ... ( |_ `  A ) ) )  ->  sum_ k  e.  {
x  e.  NN  |  x  ||  n }  (Λ `  k )  =  ( log `  n ) )
8483sumeq2dv 12106 . . 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 2296 . 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 2291 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   {crab 2519    C_ wss 3094   class class class wbr 3963    e. cmpt 4017   -1-1-onto->wf1o 4637   ` cfv 4638  (class class class)co 5757    ~~ cen 6793   Fincfn 6796   CCcc 8668   RRcr 8669   0cc0 8670   1c1 8671    x. cmul 8675    < clt 8800    <_ cle 8801    / cdiv 9356   NNcn 9679   NN0cn0 9897   ZZcz 9956   ...cfz 10713   |_cfl 10855   !cfa 11219   #chash 11268   sum_csu 12088    || cdivides 12458   logclog 19839  Λcvma 20256
This theorem is referenced by:  vmadivsum  20558
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-er 6593  df-map 6707  df-pm 6708  df-ixp 6751  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-fi 7098  df-sup 7127  df-oi 7158  df-card 7505  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-q 10249  df-rp 10287  df-xneg 10384  df-xadd 10385  df-xmul 10386  df-ioo 10591  df-ioc 10592  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-fac 11220  df-bc 11247  df-hash 11269  df-shft 11492  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-limsup 11875  df-clim 11892  df-rlim 11893  df-sum 12089  df-ef 12276  df-sin 12278  df-cos 12279  df-pi 12281  df-divides 12459  df-gcd 12613  df-prime 12686  df-pc 12817  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-starv 13150  df-sca 13151  df-vsca 13152  df-tset 13154  df-ple 13155  df-ds 13157  df-hom 13159  df-cco 13160  df-rest 13254  df-topn 13255  df-topgen 13271  df-pt 13272  df-prds 13275  df-xrs 13330  df-0g 13331  df-gsum 13332  df-qtop 13337  df-imas 13338  df-xps 13340  df-mre 13415  df-mrc 13416  df-acs 13418  df-mnd 14294  df-submnd 14343  df-mulg 14419  df-cntz 14720  df-cmn 15018  df-xmet 16300  df-met 16301  df-bl 16302  df-mopn 16303  df-cnfld 16305  df-top 16563  df-bases 16565  df-topon 16566  df-topsp 16567  df-cld 16683  df-ntr 16684  df-cls 16685  df-nei 16762  df-lp 16795  df-perf 16796  df-cn 16884  df-cnp 16885  df-haus 16970  df-tx 17184  df-hmeo 17373  df-fbas 17447  df-fg 17448  df-fil 17468  df-fm 17560  df-flim 17561  df-flf 17562  df-xms 17812  df-ms 17813  df-tms 17814  df-cncf 18309  df-limc 19143  df-dv 19144  df-log 19841  df-vma 20262
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