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Definition df-phi 13862
Description: Define the Euler phi function, which counts the number of integers less than  n and coprime to it. (Contributed by Mario Carneiro, 23-Feb-2014.)
Assertion
Ref Expression
df-phi  |-  phi  =  ( n  e.  NN  |->  ( # `  { x  e.  ( 1 ... n
)  |  ( x  gcd  n )  =  1 } ) )
Distinct variable group:    x, n

Detailed syntax breakdown of Definition df-phi
StepHypRef Expression
1 cphi 13860 . 2  class  phi
2 vn . . 3  setvar  n
3 cn 10343 . . 3  class  NN
4 vx . . . . . . . 8  setvar  x
54cv 1368 . . . . . . 7  class  x
62cv 1368 . . . . . . 7  class  n
7 cgcd 13711 . . . . . . 7  class  gcd
85, 6, 7co 6112 . . . . . 6  class  ( x  gcd  n )
9 c1 9304 . . . . . 6  class  1
108, 9wceq 1369 . . . . 5  wff  ( x  gcd  n )  =  1
11 cfz 11458 . . . . . 6  class  ...
129, 6, 11co 6112 . . . . 5  class  ( 1 ... n )
1310, 4, 12crab 2740 . . . 4  class  { x  e.  ( 1 ... n
)  |  ( x  gcd  n )  =  1 }
14 chash 12124 . . . 4  class  #
1513, 14cfv 5439 . . 3  class  ( # `  { x  e.  ( 1 ... n )  |  ( x  gcd  n )  =  1 } )
162, 3, 15cmpt 4371 . 2  class  ( n  e.  NN  |->  ( # `  { x  e.  ( 1 ... n )  |  ( x  gcd  n )  =  1 } ) )
171, 16wceq 1369 1  wff  phi  =  ( n  e.  NN  |->  ( # `  { x  e.  ( 1 ... n
)  |  ( x  gcd  n )  =  1 } ) )
Colors of variables: wff setvar class
This definition is referenced by:  phival  13863
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