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Theorem pcval 12913
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 2304 . . . . 5  |-  ( ( p  =  P  /\  r  =  N )  ->  ( r  =  0  <-> 
N  =  0 ) )
3 eqeq1 2302 . . . . . . . 8  |-  ( r  =  N  ->  (
r  =  ( x  /  y )  <->  N  =  ( x  /  y
) ) )
4 oveq1 5881 . . . . . . . . . . . . . 14  |-  ( p  =  P  ->  (
p ^ n )  =  ( P ^
n ) )
54breq1d 4049 . . . . . . . . . . . . 13  |-  ( p  =  P  ->  (
( p ^ n
)  ||  x  <->  ( P ^ n )  ||  x ) )
65rabbidv 2793 . . . . . . . . . . . 12  |-  ( p  =  P  ->  { n  e.  NN0  |  ( p ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  x }
)
76supeq1d 7215 . . . . . . . . . . 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 2346 . . . . . . . . . 10  |-  ( p  =  P  ->  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  x } ,  RR ,  <  )  =  S )
104breq1d 4049 . . . . . . . . . . . . 13  |-  ( p  =  P  ->  (
( p ^ n
)  ||  y  <->  ( P ^ n )  ||  y ) )
1110rabbidv 2793 . . . . . . . . . . . 12  |-  ( p  =  P  ->  { n  e.  NN0  |  ( p ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  y }
)
1211supeq1d 7215 . . . . . . . . . . 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 2346 . . . . . . . . . 10  |-  ( p  =  P  ->  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )  =  T )
159, 14oveq12d 5892 . . . . . . . . 9  |-  ( p  =  P  ->  ( sup ( { n  e. 
NN0  |  ( p ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( p ^ n
)  ||  y } ,  RR ,  <  )
)  =  ( S  -  T ) )
1615eqeq2d 2307 . . . . . . . 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 2599 . . . . . 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 5256 . . . . 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 3598 . . . 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 12906 . . . 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 10471 . . . . . 6  |-  +oo  e.  RR*
2322elexi 2810 . . . . 5  |-  +oo  e.  _V
24 iotaex 5252 . . . . 5  |-  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( S  -  T
) ) )  e. 
_V
2523, 24ifex 3636 . . . 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 5994 . . 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 3586 . . 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 2348 . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560   ifcif 3578   class class class wbr 4039   iotacio 5233  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753    +oocpnf 8880   RR*cxr 8882    < clt 8883    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   QQcq 10332   ^cexp 11120    || cdivides 12547   Primecprime 12774    pCnt cpc 12905
This theorem is referenced by:  pczpre  12916  pcdiv  12921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-sup 7210  df-pnf 8885  df-xr 8887  df-pc 12906
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