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Theorem pcval 13218
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 448 . . . . . 6  |-  ( ( p  =  P  /\  r  =  N )  ->  r  =  N )
21eqeq1d 2444 . . . . 5  |-  ( ( p  =  P  /\  r  =  N )  ->  ( r  =  0  <-> 
N  =  0 ) )
3 eqeq1 2442 . . . . . . . 8  |-  ( r  =  N  ->  (
r  =  ( x  /  y )  <->  N  =  ( x  /  y
) ) )
4 oveq1 6088 . . . . . . . . . . . . . 14  |-  ( p  =  P  ->  (
p ^ n )  =  ( P ^
n ) )
54breq1d 4222 . . . . . . . . . . . . 13  |-  ( p  =  P  ->  (
( p ^ n
)  ||  x  <->  ( P ^ n )  ||  x ) )
65rabbidv 2948 . . . . . . . . . . . 12  |-  ( p  =  P  ->  { n  e.  NN0  |  ( p ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  x }
)
76supeq1d 7451 . . . . . . . . . . 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 2486 . . . . . . . . . 10  |-  ( p  =  P  ->  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  x } ,  RR ,  <  )  =  S )
104breq1d 4222 . . . . . . . . . . . . 13  |-  ( p  =  P  ->  (
( p ^ n
)  ||  y  <->  ( P ^ n )  ||  y ) )
1110rabbidv 2948 . . . . . . . . . . . 12  |-  ( p  =  P  ->  { n  e.  NN0  |  ( p ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  y }
)
1211supeq1d 7451 . . . . . . . . . . 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 2486 . . . . . . . . . 10  |-  ( p  =  P  ->  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )  =  T )
159, 14oveq12d 6099 . . . . . . . . 9  |-  ( p  =  P  ->  ( sup ( { n  e. 
NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
)  =  ( S  -  T ) )
1615eqeq2d 2447 . . . . . . . 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 845 . . . . . . 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 2748 . . . . . 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 5439 . . . . 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 3759 . . . 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 13211 . . . 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 10713 . . . . . 6  |-  +oo  e.  RR*
2322elexi 2965 . . . . 5  |-  +oo  e.  _V
24 iotaex 5435 . . . . 5  |-  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) )  e. 
_V
2523, 24ifex 3797 . . . 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 6204 . . 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 3747 . . 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 2488 . 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 629 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 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   {crab 2709   ifcif 3739   class class class wbr 4212   iotacio 5416  (class class class)co 6081   supcsup 7445   RRcr 8989   0cc0 8990    +oocpnf 9117   RR*cxr 9119    < clt 9120    - cmin 9291    / cdiv 9677   NNcn 10000   NN0cn0 10221   ZZcz 10282   QQcq 10574   ^cexp 11382    || cdivides 12852   Primecprime 13079    pCnt cpc 13210
This theorem is referenced by:  pczpre  13221  pcdiv  13226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-sup 7446  df-pnf 9122  df-xr 9124  df-pc 13211
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