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Theorem pcval 13909
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 2449 . . . . 5  |-  ( ( p  =  P  /\  r  =  N )  ->  ( r  =  0  <-> 
N  =  0 ) )
3 eqeq1 2447 . . . . . . . 8  |-  ( r  =  N  ->  (
r  =  ( x  /  y )  <->  N  =  ( x  /  y
) ) )
4 oveq1 6096 . . . . . . . . . . . . . 14  |-  ( p  =  P  ->  (
p ^ n )  =  ( P ^
n ) )
54breq1d 4300 . . . . . . . . . . . . 13  |-  ( p  =  P  ->  (
( p ^ n
)  ||  x  <->  ( P ^ n )  ||  x ) )
65rabbidv 2962 . . . . . . . . . . . 12  |-  ( p  =  P  ->  { n  e.  NN0  |  ( p ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  x }
)
76supeq1d 7694 . . . . . . . . . . 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 2491 . . . . . . . . . 10  |-  ( p  =  P  ->  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  x } ,  RR ,  <  )  =  S )
104breq1d 4300 . . . . . . . . . . . . 13  |-  ( p  =  P  ->  (
( p ^ n
)  ||  y  <->  ( P ^ n )  ||  y ) )
1110rabbidv 2962 . . . . . . . . . . . 12  |-  ( p  =  P  ->  { n  e.  NN0  |  ( p ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  y }
)
1211supeq1d 7694 . . . . . . . . . . 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 2491 . . . . . . . . . 10  |-  ( p  =  P  ->  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )  =  T )
159, 14oveq12d 6107 . . . . . . . . 9  |-  ( p  =  P  ->  ( sup ( { n  e. 
NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
)  =  ( S  -  T ) )
1615eqeq2d 2452 . . . . . . . 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 869 . . . . . . 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 2756 . . . . . 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 5400 . . . . 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 3812 . . . 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 13902 . . . 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 11091 . . . . 5  |- +oo  e.  _V
23 iotaex 5396 . . . . 5  |-  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) )  e. 
_V
2422, 23ifex 3856 . . . 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 6219 . . 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 3799 . . 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 2493 . 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 1369    e. wcel 1756    =/= wne 2604   E.wrex 2714   {crab 2717   ifcif 3789   class class class wbr 4290   iotacio 5377  (class class class)co 6089   supcsup 7688   RRcr 9279   0cc0 9280   +oocpnf 9413    < clt 9416    - cmin 9593    / cdiv 9991   NNcn 10320   NN0cn0 10577   ZZcz 10644   QQcq 10951   ^cexp 11863    || cdivides 13533   Primecprime 13761    pCnt cpc 13901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-sup 7689  df-pnf 9418  df-xr 9420  df-pc 13902
This theorem is referenced by:  pczpre  13912  pcdiv  13917
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