MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-vma Unicode version

Definition df-vma 20335
Description: Define the von Mangoldt function, which gives the logarithm of the prime at a prime power, and is zero elsewhere. (Contributed by Mario Carneiro, 7-Apr-2016.)
Assertion
Ref Expression
df-vma  |- Λ  =  ( x  e.  NN  |->  [_ { p  e.  Prime  |  p  ||  x }  /  s ]_ if ( ( # `  s
)  =  1 ,  ( log `  U. s ) ,  0 ) )
Distinct variable group:    s, p, x

Detailed syntax breakdown of Definition df-vma
StepHypRef Expression
1 cvma 20329 . 2  class Λ
2 vx . . 3  set  x
3 cn 9746 . . 3  class  NN
4 vs . . . 4  set  s
5 vp . . . . . . 7  set  p
65cv 1622 . . . . . 6  class  p
72cv 1622 . . . . . 6  class  x
8 cdivides 12531 . . . . . 6  class  ||
96, 7, 8wbr 4023 . . . . 5  wff  p  ||  x
10 cprime 12758 . . . . 5  class  Prime
119, 5, 10crab 2547 . . . 4  class  { p  e.  Prime  |  p  ||  x }
124cv 1622 . . . . . . 7  class  s
13 chash 11337 . . . . . . 7  class  #
1412, 13cfv 5255 . . . . . 6  class  ( # `  s )
15 c1 8738 . . . . . 6  class  1
1614, 15wceq 1623 . . . . 5  wff  ( # `  s )  =  1
1712cuni 3827 . . . . . 6  class  U. s
18 clog 19912 . . . . . 6  class  log
1917, 18cfv 5255 . . . . 5  class  ( log `  U. s )
20 cc0 8737 . . . . 5  class  0
2116, 19, 20cif 3565 . . . 4  class  if ( ( # `  s
)  =  1 ,  ( log `  U. s ) ,  0 )
224, 11, 21csb 3081 . . 3  class  [_ {
p  e.  Prime  |  p 
||  x }  / 
s ]_ if ( (
# `  s )  =  1 ,  ( log `  U. s
) ,  0 )
232, 3, 22cmpt 4077 . 2  class  ( x  e.  NN  |->  [_ {
p  e.  Prime  |  p 
||  x }  / 
s ]_ if ( (
# `  s )  =  1 ,  ( log `  U. s
) ,  0 ) )
241, 23wceq 1623 1  wff Λ  =  ( x  e.  NN  |->  [_ { p  e.  Prime  |  p  ||  x }  /  s ]_ if ( ( # `  s
)  =  1 ,  ( log `  U. s ) ,  0 ) )
Colors of variables: wff set class
This definition is referenced by:  vmaval  20351  vmaf  20357
  Copyright terms: Public domain W3C validator