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Theorem pcval 12988
Description: The value of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)
Hypotheses
Ref Expression
pcval.1  |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )
pcval.2  |-  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )
Assertion
Ref Expression
pcval  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  -> 
( P  pCnt  N
)  =  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) ) )
Distinct variable groups:    x, n, y, z, N    P, n, x, y, z    z, S   
z, T
Allowed substitution hints:    S( x, y, n)    T( x, y, n)

Proof of Theorem pcval
Dummy variables  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . . . 6  |-  ( ( p  =  P  /\  r  =  N )  ->  r  =  N )
21eqeq1d 2366 . . . . 5  |-  ( ( p  =  P  /\  r  =  N )  ->  ( r  =  0  <-> 
N  =  0 ) )
3 eqeq1 2364 . . . . . . . 8  |-  ( r  =  N  ->  (
r  =  ( x  /  y )  <->  N  =  ( x  /  y
) ) )
4 oveq1 5949 . . . . . . . . . . . . . 14  |-  ( p  =  P  ->  (
p ^ n )  =  ( P ^
n ) )
54breq1d 4112 . . . . . . . . . . . . 13  |-  ( p  =  P  ->  (
( p ^ n
)  ||  x  <->  ( P ^ n )  ||  x ) )
65rabbidv 2856 . . . . . . . . . . . 12  |-  ( p  =  P  ->  { n  e.  NN0  |  ( p ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  x }
)
76supeq1d 7286 . . . . . . . . . . 11  |-  ( p  =  P  ->  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  ) )
8 pcval.1 . . . . . . . . . . 11  |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )
97, 8syl6eqr 2408 . . . . . . . . . 10  |-  ( p  =  P  ->  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  x } ,  RR ,  <  )  =  S )
104breq1d 4112 . . . . . . . . . . . . 13  |-  ( p  =  P  ->  (
( p ^ n
)  ||  y  <->  ( P ^ n )  ||  y ) )
1110rabbidv 2856 . . . . . . . . . . . 12  |-  ( p  =  P  ->  { n  e.  NN0  |  ( p ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  y }
)
1211supeq1d 7286 . . . . . . . . . . 11  |-  ( p  =  P  ->  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) )
13 pcval.2 . . . . . . . . . . 11  |-  T  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )
1412, 13syl6eqr 2408 . . . . . . . . . 10  |-  ( p  =  P  ->  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )  =  T )
159, 14oveq12d 5960 . . . . . . . . 9  |-  ( p  =  P  ->  ( sup ( { n  e. 
NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
)  =  ( S  -  T ) )
1615eqeq2d 2369 . . . . . . . 8  |-  ( p  =  P  ->  (
z  =  ( sup ( { n  e. 
NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
)  <->  z  =  ( S  -  T ) ) )
173, 16bi2anan9r 844 . . . . . . 7  |-  ( ( p  =  P  /\  r  =  N )  ->  ( ( r  =  ( x  /  y
)  /\  z  =  ( sup ( { n  e.  NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
) )  <->  ( N  =  ( x  / 
y )  /\  z  =  ( S  -  T ) ) ) )
18172rexbidv 2662 . . . . . 6  |-  ( ( p  =  P  /\  r  =  N )  ->  ( 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 ,  <  )
) )  <->  E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) ) )
1918iotabidv 5319 . . . . 5  |-  ( ( p  =  P  /\  r  =  N )  ->  ( 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 ,  <  ) ) ) )  =  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) ) )
202, 19ifbieq2d 3661 . . . 4  |-  ( ( p  =  P  /\  r  =  N )  ->  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 ,  <  )
) ) ) )  =  if ( N  =  0 ,  +oo ,  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) ) ) )
21 df-pc 12981 . . . 4  |-  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 ,  <  )
) ) ) ) )
22 pnfxr 10544 . . . . . 6  |-  +oo  e.  RR*
2322elexi 2873 . . . . 5  |-  +oo  e.  _V
24 iotaex 5315 . . . . 5  |-  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) )  e. 
_V
2523, 24ifex 3699 . . . 4  |-  if ( N  =  0 , 
+oo ,  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) ) )  e.  _V
2620, 21, 25ovmpt2a 6062 . . 3  |-  ( ( P  e.  Prime  /\  N  e.  QQ )  ->  ( P  pCnt  N )  =  if ( N  =  0 ,  +oo , 
( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) ) ) )
27 ifnefalse 3649 . . 3  |-  ( N  =/=  0  ->  if ( N  =  0 ,  +oo ,  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) ) )  =  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y )  /\  z  =  ( S  -  T ) ) ) )
2826, 27sylan9eq 2410 . 2  |-  ( ( ( P  e.  Prime  /\  N  e.  QQ )  /\  N  =/=  0
)  ->  ( P  pCnt  N )  =  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) ) )
2928anasss 628 1  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  -> 
( P  pCnt  N
)  =  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620   {crab 2623   ifcif 3641   class class class wbr 4102   iotacio 5296  (class class class)co 5942   supcsup 7280   RRcr 8823   0cc0 8824    +oocpnf 8951   RR*cxr 8953    < clt 8954    - cmin 9124    / cdiv 9510   NNcn 9833   NN0cn0 10054   ZZcz 10113   QQcq 10405   ^cexp 11194    || cdivides 12622   Primecprime 12849    pCnt cpc 12980
This theorem is referenced by:  pczpre  12991  pcdiv  12996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-iota 5298  df-fun 5336  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-sup 7281  df-pnf 8956  df-xr 8958  df-pc 12981
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