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

Definition df-vma 23569
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 23563 . 2  class Λ
2 vx . . 3  setvar  x
3 cn 10531 . . 3  class  NN
4 vs . . . 4  setvar  s
5 vp . . . . . . 7  setvar  p
65cv 1397 . . . . . 6  class  p
72cv 1397 . . . . . 6  class  x
8 cdvds 14070 . . . . . 6  class  ||
96, 7, 8wbr 4439 . . . . 5  wff  p  ||  x
10 cprime 14301 . . . . 5  class  Prime
119, 5, 10crab 2808 . . . 4  class  { p  e.  Prime  |  p  ||  x }
124cv 1397 . . . . . . 7  class  s
13 chash 12387 . . . . . . 7  class  #
1412, 13cfv 5570 . . . . . 6  class  ( # `  s )
15 c1 9482 . . . . . 6  class  1
1614, 15wceq 1398 . . . . 5  wff  ( # `  s )  =  1
1712cuni 4235 . . . . . 6  class  U. s
18 clog 23108 . . . . . 6  class  log
1917, 18cfv 5570 . . . . 5  class  ( log `  U. s )
20 cc0 9481 . . . . 5  class  0
2116, 19, 20cif 3929 . . . 4  class  if ( ( # `  s
)  =  1 ,  ( log `  U. s ) ,  0 )
224, 11, 21csb 3420 . . 3  class  [_ {
p  e.  Prime  |  p 
||  x }  / 
s ]_ if ( (
# `  s )  =  1 ,  ( log `  U. s
) ,  0 )
232, 3, 22cmpt 4497 . 2  class  ( x  e.  NN  |->  [_ {
p  e.  Prime  |  p 
||  x }  / 
s ]_ if ( (
# `  s )  =  1 ,  ( log `  U. s
) ,  0 ) )
241, 23wceq 1398 1  wff Λ  =  ( x  e.  NN  |->  [_ { p  e.  Prime  |  p  ||  x }  /  s ]_ if ( ( # `  s
)  =  1 ,  ( log `  U. s ) ,  0 ) )
Colors of variables: wff setvar class
This definition is referenced by:  vmaval  23585  vmaf  23591
  Copyright terms: Public domain W3C validator