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Definition df-pc 12890
Description: Define the prime count function, which returns the largest exponent of a given prime (or other natural number) that divides the number. For rational numbers, it returns negative values according to the power of a prime in the denominator. (Contributed by Mario Carneiro, 23-Feb-2014.)
Assertion
Ref Expression
df-pc  |-  pCnt  =  ( p  e.  Prime ,  r  e.  QQ  |->  if ( r  =  0 ,  +oo ,  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( r  =  ( x  /  y
)  /\  z  =  ( sup ( { n  e.  NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
) ) ) ) )
Distinct variable group:    r, p, z, x, y, n

Detailed syntax breakdown of Definition df-pc
StepHypRef Expression
1 cpc 12889 . 2  class  pCnt
2 vp . . 3  set  p
3 vr . . 3  set  r
4 cprime 12758 . . 3  class  Prime
5 cq 10316 . . 3  class  QQ
63cv 1622 . . . . 5  class  r
7 cc0 8737 . . . . 5  class  0
86, 7wceq 1623 . . . 4  wff  r  =  0
9 cpnf 8864 . . . 4  class  +oo
10 vx . . . . . . . . . . 11  set  x
1110cv 1622 . . . . . . . . . 10  class  x
12 vy . . . . . . . . . . 11  set  y
1312cv 1622 . . . . . . . . . 10  class  y
14 cdiv 9423 . . . . . . . . . 10  class  /
1511, 13, 14co 5858 . . . . . . . . 9  class  ( x  /  y )
166, 15wceq 1623 . . . . . . . 8  wff  r  =  ( x  /  y
)
17 vz . . . . . . . . . 10  set  z
1817cv 1622 . . . . . . . . 9  class  z
192cv 1622 . . . . . . . . . . . . . 14  class  p
20 vn . . . . . . . . . . . . . . 15  set  n
2120cv 1622 . . . . . . . . . . . . . 14  class  n
22 cexp 11104 . . . . . . . . . . . . . 14  class  ^
2319, 21, 22co 5858 . . . . . . . . . . . . 13  class  ( p ^ n )
24 cdivides 12531 . . . . . . . . . . . . 13  class  ||
2523, 11, 24wbr 4023 . . . . . . . . . . . 12  wff  ( p ^ n )  ||  x
26 cn0 9965 . . . . . . . . . . . 12  class  NN0
2725, 20, 26crab 2547 . . . . . . . . . . 11  class  { n  e.  NN0  |  ( p ^ n )  ||  x }
28 cr 8736 . . . . . . . . . . 11  class  RR
29 clt 8867 . . . . . . . . . . 11  class  <
3027, 28, 29csup 7193 . . . . . . . . . 10  class  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  x } ,  RR ,  <  )
3123, 13, 24wbr 4023 . . . . . . . . . . . 12  wff  ( p ^ n )  ||  y
3231, 20, 26crab 2547 . . . . . . . . . . 11  class  { n  e.  NN0  |  ( p ^ n )  ||  y }
3332, 28, 29csup 7193 . . . . . . . . . 10  class  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
34 cmin 9037 . . . . . . . . . 10  class  -
3530, 33, 34co 5858 . . . . . . . . 9  class  ( sup ( { n  e. 
NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
)
3618, 35wceq 1623 . . . . . . . 8  wff  z  =  ( sup ( { n  e.  NN0  | 
( p ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n )  ||  y } ,  RR ,  <  ) )
3716, 36wa 358 . . . . . . 7  wff  ( r  =  ( x  / 
y )  /\  z  =  ( sup ( { n  e.  NN0  |  ( p ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n )  ||  y } ,  RR ,  <  ) ) )
38 cn 9746 . . . . . . 7  class  NN
3937, 12, 38wrex 2544 . . . . . 6  wff  E. y  e.  NN  ( r  =  ( x  /  y
)  /\  z  =  ( sup ( { n  e.  NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
) )
40 cz 10024 . . . . . 6  class  ZZ
4139, 10, 40wrex 2544 . . . . 5  wff  E. x  e.  ZZ  E. y  e.  NN  ( r  =  ( x  /  y
)  /\  z  =  ( sup ( { n  e.  NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
) )
4241, 17cio 5217 . . . 4  class  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( r  =  ( x  /  y
)  /\  z  =  ( sup ( { n  e.  NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
) ) )
438, 9, 42cif 3565 . . 3  class  if ( r  =  0 , 
+oo ,  ( iota z E. x  e.  ZZ  E. y  e.  NN  (
r  =  ( x  /  y )  /\  z  =  ( sup ( { n  e.  NN0  |  ( p ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n )  ||  y } ,  RR ,  <  ) ) ) ) )
442, 3, 4, 5, 43cmpt2 5860 . 2  class  ( p  e.  Prime ,  r  e.  QQ  |->  if ( r  =  0 ,  +oo ,  ( iota z E. x  e.  ZZ  E. y  e.  NN  (
r  =  ( x  /  y )  /\  z  =  ( sup ( { n  e.  NN0  |  ( p ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n )  ||  y } ,  RR ,  <  ) ) ) ) ) )
451, 44wceq 1623 1  wff  pCnt  =  ( p  e.  Prime ,  r  e.  QQ  |->  if ( r  =  0 ,  +oo ,  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( r  =  ( x  /  y
)  /\  z  =  ( sup ( { n  e.  NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
) ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  pcval  12897  pc0  12907
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