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Theorem pcval 14227
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 461 . . . . . 6  |-  ( ( p  =  P  /\  r  =  N )  ->  r  =  N )
21eqeq1d 2469 . . . . 5  |-  ( ( p  =  P  /\  r  =  N )  ->  ( r  =  0  <-> 
N  =  0 ) )
3 eqeq1 2471 . . . . . . . 8  |-  ( r  =  N  ->  (
r  =  ( x  /  y )  <->  N  =  ( x  /  y
) ) )
4 oveq1 6291 . . . . . . . . . . . . . 14  |-  ( p  =  P  ->  (
p ^ n )  =  ( P ^
n ) )
54breq1d 4457 . . . . . . . . . . . . 13  |-  ( p  =  P  ->  (
( p ^ n
)  ||  x  <->  ( P ^ n )  ||  x ) )
65rabbidv 3105 . . . . . . . . . . . 12  |-  ( p  =  P  ->  { n  e.  NN0  |  ( p ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  x }
)
76supeq1d 7906 . . . . . . . . . . 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 2526 . . . . . . . . . 10  |-  ( p  =  P  ->  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  x } ,  RR ,  <  )  =  S )
104breq1d 4457 . . . . . . . . . . . . 13  |-  ( p  =  P  ->  (
( p ^ n
)  ||  y  <->  ( P ^ n )  ||  y ) )
1110rabbidv 3105 . . . . . . . . . . . 12  |-  ( p  =  P  ->  { n  e.  NN0  |  ( p ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  y }
)
1211supeq1d 7906 . . . . . . . . . . 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 2526 . . . . . . . . . 10  |-  ( p  =  P  ->  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )  =  T )
159, 14oveq12d 6302 . . . . . . . . 9  |-  ( p  =  P  ->  ( sup ( { n  e. 
NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
)  =  ( S  -  T ) )
1615eqeq2d 2481 . . . . . . . 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 872 . . . . . . 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 2980 . . . . . 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 5572 . . . . 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 3964 . . . 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 14220 . . . 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 pnfex 11322 . . . . 5  |- +oo  e.  _V
23 iotaex 5568 . . . . 5  |-  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) )  e. 
_V
2422, 23ifex 4008 . . . 4  |-  if ( N  =  0 , +oo ,  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) ) )  e.  _V
2520, 21, 24ovmpt2a 6417 . . 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
) ) ) ) )
26 ifnefalse 3951 . . 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 ) ) ) )
2725, 26sylan9eq 2528 . 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
) ) ) )
2827anasss 647 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   {crab 2818   ifcif 3939   class class class wbr 4447   iotacio 5549  (class class class)co 6284   supcsup 7900   RRcr 9491   0cc0 9492   +oocpnf 9625    < clt 9628    - cmin 9805    / cdiv 10206   NNcn 10536   NN0cn0 10795   ZZcz 10864   QQcq 11182   ^cexp 12134    || cdivides 13847   Primecprime 14076    pCnt cpc 14219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-sup 7901  df-pnf 9630  df-xr 9632  df-pc 14220
This theorem is referenced by:  pczpre  14230  pcdiv  14235
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