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Theorem pczpre 14245
Description: Connect the prime count pre-function to the actual prime count function, when restricted to the integers. (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
pczpre.1  |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  N } ,  RR ,  <  )
Assertion
Ref Expression
pczpre  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P  pCnt  N
)  =  S )
Distinct variable groups:    n, N    P, n
Allowed substitution hint:    S( n)

Proof of Theorem pczpre
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zq 11194 . . 3  |-  ( N  e.  ZZ  ->  N  e.  QQ )
2 eqid 2441 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )
3 eqid 2441 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )
42, 3pcval 14242 . . 3  |-  ( ( 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  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) ) )
51, 4sylanr1 652 . 2  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P  pCnt  N
)  =  ( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) ) )
6 simprl 755 . . . 4  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  ZZ )
76zcnd 10972 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  CC )
87div1d 10315 . . . . 5  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( N  /  1
)  =  N )
98eqcomd 2449 . . . 4  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  =  ( N  /  1 ) )
10 prmuz2 14109 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
11 eqid 2441 . . . . . . . 8  |-  1  =  1
12 eqid 2441 . . . . . . . . 9  |-  { n  e.  NN0  |  ( P ^ n )  ||  1 }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  1 }
13 eqid 2441 . . . . . . . . 9  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  1 } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  1 } ,  RR ,  <  )
1412, 13pcpre1 14240 . . . . . . . 8  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  1  =  1 )  ->  sup ( { n  e. 
NN0  |  ( P ^ n )  ||  1 } ,  RR ,  <  )  =  0 )
1510, 11, 14sylancl 662 . . . . . . 7  |-  ( P  e.  Prime  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  1 } ,  RR ,  <  )  =  0 )
1615adantr 465 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  sup ( { n  e. 
NN0  |  ( P ^ n )  ||  1 } ,  RR ,  <  )  =  0 )
1716oveq2d 6294 . . . . 5  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  1 } ,  RR ,  <  )
)  =  ( S  -  0 ) )
18 eqid 2441 . . . . . . . . . 10  |-  { n  e.  NN0  |  ( P ^ n )  ||  N }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  N }
19 pczpre.1 . . . . . . . . . 10  |-  S  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  N } ,  RR ,  <  )
2018, 19pcprecl 14237 . . . . . . . . 9  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  N )
)
2110, 20sylan 471 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  N )
)
2221simpld 459 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  S  e.  NN0 )
2322nn0cnd 10857 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  S  e.  CC )
2423subid1d 9922 . . . . 5  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  -  0 )  =  S )
2517, 24eqtr2d 2483 . . . 4  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  S  =  ( S  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  1 } ,  RR ,  <  ) ) )
26 1nn 10550 . . . . 5  |-  1  e.  NN
27 oveq1 6285 . . . . . . . 8  |-  ( x  =  N  ->  (
x  /  y )  =  ( N  / 
y ) )
2827eqeq2d 2455 . . . . . . 7  |-  ( x  =  N  ->  ( N  =  ( x  /  y )  <->  N  =  ( N  /  y
) ) )
29 breq2 4438 . . . . . . . . . . . 12  |-  ( x  =  N  ->  (
( P ^ n
)  ||  x  <->  ( P ^ n )  ||  N ) )
3029rabbidv 3085 . . . . . . . . . . 11  |-  ( x  =  N  ->  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  N }
)
3130supeq1d 7905 . . . . . . . . . 10  |-  ( x  =  N  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  N } ,  RR ,  <  ) )
3231, 19syl6eqr 2500 . . . . . . . . 9  |-  ( x  =  N  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  S )
3332oveq1d 6293 . . . . . . . 8  |-  ( x  =  N  ->  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
)  =  ( S  -  sup ( { n  e.  NN0  | 
( P ^ n
)  ||  y } ,  RR ,  <  )
) )
3433eqeq2d 2455 . . . . . . 7  |-  ( x  =  N  ->  ( S  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) )  <->  S  =  ( S  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) )
3528, 34anbi12d 710 . . . . . 6  |-  ( x  =  N  ->  (
( N  =  ( x  /  y )  /\  S  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) )  <->  ( N  =  ( N  / 
y )  /\  S  =  ( S  -  sup ( { n  e. 
NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) ) )
36 oveq2 6286 . . . . . . . 8  |-  ( y  =  1  ->  ( N  /  y )  =  ( N  /  1
) )
3736eqeq2d 2455 . . . . . . 7  |-  ( y  =  1  ->  ( N  =  ( N  /  y )  <->  N  =  ( N  /  1
) ) )
38 breq2 4438 . . . . . . . . . . 11  |-  ( y  =  1  ->  (
( P ^ n
)  ||  y  <->  ( P ^ n )  ||  1 ) )
3938rabbidv 3085 . . . . . . . . . 10  |-  ( y  =  1  ->  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  1 }
)
4039supeq1d 7905 . . . . . . . . 9  |-  ( y  =  1  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  1 } ,  RR ,  <  ) )
4140oveq2d 6294 . . . . . . . 8  |-  ( y  =  1  ->  ( S  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
)  =  ( S  -  sup ( { n  e.  NN0  | 
( P ^ n
)  ||  1 } ,  RR ,  <  )
) )
4241eqeq2d 2455 . . . . . . 7  |-  ( y  =  1  ->  ( S  =  ( S  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) )  <->  S  =  ( S  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  1 } ,  RR ,  <  )
) ) )
4337, 42anbi12d 710 . . . . . 6  |-  ( y  =  1  ->  (
( N  =  ( N  /  y )  /\  S  =  ( S  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) )  <->  ( N  =  ( N  / 
1 )  /\  S  =  ( S  -  sup ( { n  e. 
NN0  |  ( P ^ n )  ||  1 } ,  RR ,  <  ) ) ) ) )
4435, 43rspc2ev 3205 . . . . 5  |-  ( ( N  e.  ZZ  /\  1  e.  NN  /\  ( N  =  ( N  /  1 )  /\  S  =  ( S  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  1 } ,  RR ,  <  ) ) ) )  ->  E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y )  /\  S  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) )
4526, 44mp3an2 1311 . . . 4  |-  ( ( N  e.  ZZ  /\  ( N  =  ( N  /  1 )  /\  S  =  ( S  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  1 } ,  RR ,  <  ) ) ) )  ->  E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y )  /\  S  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) )
466, 9, 25, 45syl12anc 1225 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y )  /\  S  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) )
47 ltso 9665 . . . . . 6  |-  <  Or  RR
4847supex 7922 . . . . 5  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  N } ,  RR ,  <  )  e.  _V
4919, 48eqeltri 2525 . . . 4  |-  S  e. 
_V
502, 3pceu 14244 . . . . 5  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  E! z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) )
511, 50sylanr1 652 . . . 4  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  E! z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) )
52 eqeq1 2445 . . . . . . 7  |-  ( z  =  S  ->  (
z  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
)  <->  S  =  ( sup ( { n  e. 
NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) )
5352anbi2d 703 . . . . . 6  |-  ( z  =  S  ->  (
( N  =  ( x  /  y )  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) )  <->  ( N  =  ( x  / 
y )  /\  S  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) ) )
54532rexbidv 2959 . . . . 5  |-  ( z  =  S  ->  ( E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( 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
)  /\  S  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) ) )
5554iota2 5564 . . . 4  |-  ( ( S  e.  _V  /\  E! z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( 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
)  /\  S  =  ( 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  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) )  =  S ) )
5649, 51, 55sylancr 663 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  S  =  ( 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  =  ( sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  ) ) ) )  =  S ) )
5746, 56mpbid 210 . 2  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( iota z E. x  e.  ZZ  E. y  e.  NN  ( N  =  ( x  /  y
)  /\  z  =  ( sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
) ) )  =  S )
585, 57eqtrd 2482 1  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( P  pCnt  N
)  =  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   E!weu 2266    =/= wne 2636   E.wrex 2792   {crab 2795   _Vcvv 3093   class class class wbr 4434   iotacio 5536   ` cfv 5575  (class class class)co 6278   supcsup 7899   RRcr 9491   0cc0 9492   1c1 9493    < clt 9628    - cmin 9807    / cdiv 10209   NNcn 10539   2c2 10588   NN0cn0 10798   ZZcz 10867   ZZ>=cuz 11087   QQcq 11188   ^cexp 12142    || cdvds 13860   Primecprime 14091    pCnt cpc 14234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-recs 7041  df-rdg 7075  df-1o 7129  df-2o 7130  df-oadd 7133  df-er 7310  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-sup 7900  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-div 10210  df-nn 10540  df-2 10597  df-3 10598  df-n0 10799  df-z 10868  df-uz 11088  df-q 11189  df-rp 11227  df-fl 11905  df-mod 11973  df-seq 12084  df-exp 12143  df-cj 12908  df-re 12909  df-im 12910  df-sqrt 13044  df-abs 13045  df-dvds 13861  df-gcd 14019  df-prm 14092  df-pc 14235
This theorem is referenced by:  pczcl  14246  pcmul  14249  pcdiv  14250  pc1  14253  pczdvds  14260  pczndvds  14262  pczndvds2  14264  pcneg  14271
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