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Theorem pczpre 14373
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 11107 . . 3  |-  ( N  e.  ZZ  ->  N  e.  QQ )
2 eqid 2382 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  x } ,  RR ,  <  )
3 eqid 2382 . . . 4  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  y } ,  RR ,  <  )
42, 3pcval 14370 . . 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 650 . 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 754 . . . 4  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  ZZ )
76zcnd 10885 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  e.  CC )
87div1d 10229 . . . . 5  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( N  /  1
)  =  N )
98eqcomd 2390 . . . 4  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  N  =  ( N  /  1 ) )
10 prmuz2 14237 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
11 eqid 2382 . . . . . . . 8  |-  1  =  1
12 eqid 2382 . . . . . . . . 9  |-  { n  e.  NN0  |  ( P ^ n )  ||  1 }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  1 }
13 eqid 2382 . . . . . . . . 9  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  1 } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  1 } ,  RR ,  <  )
1412, 13pcpre1 14368 . . . . . . . 8  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  1  =  1 )  ->  sup ( { n  e. 
NN0  |  ( P ^ n )  ||  1 } ,  RR ,  <  )  =  0 )
1510, 11, 14sylancl 660 . . . . . . 7  |-  ( P  e.  Prime  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  1 } ,  RR ,  <  )  =  0 )
1615adantr 463 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  sup ( { n  e. 
NN0  |  ( P ^ n )  ||  1 } ,  RR ,  <  )  =  0 )
1716oveq2d 6212 . . . . 5  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  1 } ,  RR ,  <  )
)  =  ( S  -  0 ) )
18 eqid 2382 . . . . . . . . . 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 14365 . . . . . . . . 9  |-  ( ( P  e.  ( ZZ>= ` 
2 )  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  N )
)
2110, 20sylan 469 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  e.  NN0  /\  ( P ^ S
)  ||  N )
)
2221simpld 457 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  S  e.  NN0 )
2322nn0cnd 10771 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  S  e.  CC )
2423subid1d 9833 . . . . 5  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  -> 
( S  -  0 )  =  S )
2517, 24eqtr2d 2424 . . . 4  |-  ( ( P  e.  Prime  /\  ( N  e.  ZZ  /\  N  =/=  0 ) )  ->  S  =  ( S  -  sup ( { n  e.  NN0  |  ( P ^ n )  ||  1 } ,  RR ,  <  ) ) )
26 1nn 10463 . . . . 5  |-  1  e.  NN
27 oveq1 6203 . . . . . . . 8  |-  ( x  =  N  ->  (
x  /  y )  =  ( N  / 
y ) )
2827eqeq2d 2396 . . . . . . 7  |-  ( x  =  N  ->  ( N  =  ( x  /  y )  <->  N  =  ( N  /  y
) ) )
29 breq2 4371 . . . . . . . . . . . 12  |-  ( x  =  N  ->  (
( P ^ n
)  ||  x  <->  ( P ^ n )  ||  N ) )
3029rabbidv 3026 . . . . . . . . . . 11  |-  ( x  =  N  ->  { n  e.  NN0  |  ( P ^ n )  ||  x }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  N }
)
3130supeq1d 7820 . . . . . . . . . 10  |-  ( x  =  N  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  N } ,  RR ,  <  ) )
3231, 19syl6eqr 2441 . . . . . . . . 9  |-  ( x  =  N  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  x } ,  RR ,  <  )  =  S )
3332oveq1d 6211 . . . . . . . 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 2396 . . . . . . 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 708 . . . . . 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 6204 . . . . . . . 8  |-  ( y  =  1  ->  ( N  /  y )  =  ( N  /  1
) )
3736eqeq2d 2396 . . . . . . 7  |-  ( y  =  1  ->  ( N  =  ( N  /  y )  <->  N  =  ( N  /  1
) ) )
38 breq2 4371 . . . . . . . . . . 11  |-  ( y  =  1  ->  (
( P ^ n
)  ||  y  <->  ( P ^ n )  ||  1 ) )
3938rabbidv 3026 . . . . . . . . . 10  |-  ( y  =  1  ->  { n  e.  NN0  |  ( P ^ n )  ||  y }  =  {
n  e.  NN0  | 
( P ^ n
)  ||  1 }
)
4039supeq1d 7820 . . . . . . . . 9  |-  ( y  =  1  ->  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )  =  sup ( { n  e.  NN0  |  ( P ^ n )  ||  1 } ,  RR ,  <  ) )
4140oveq2d 6212 . . . . . . . 8  |-  ( y  =  1  ->  ( S  -  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  y } ,  RR ,  <  )
)  =  ( S  -  sup ( { n  e.  NN0  | 
( P ^ n
)  ||  1 } ,  RR ,  <  )
) )
4241eqeq2d 2396 . . . . . . 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 708 . . . . . 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 3146 . . . . 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 1310 . . . 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 1224 . . 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 9576 . . . . . 6  |-  <  Or  RR
4847supex 7837 . . . . 5  |-  sup ( { n  e.  NN0  |  ( P ^ n
)  ||  N } ,  RR ,  <  )  e.  _V
4919, 48eqeltri 2466 . . . 4  |-  S  e. 
_V
502, 3pceu 14372 . . . . 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 650 . . . 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 2386 . . . . . . 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 701 . . . . . 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 2900 . . . . 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 5486 . . . 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 661 . . 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 2423 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 367    = wceq 1399    e. wcel 1826   E!weu 2218    =/= wne 2577   E.wrex 2733   {crab 2736   _Vcvv 3034   class class class wbr 4367   iotacio 5458   ` cfv 5496  (class class class)co 6196   supcsup 7815   RRcr 9402   0cc0 9403   1c1 9404    < clt 9539    - cmin 9718    / cdiv 10123   NNcn 10452   2c2 10502   NN0cn0 10712   ZZcz 10781   ZZ>=cuz 11001   QQcq 11101   ^cexp 12069    || cdvds 13988   Primecprime 14219    pCnt cpc 14362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-q 11102  df-rp 11140  df-fl 11828  df-mod 11897  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-dvds 13989  df-gcd 14147  df-prm 14220  df-pc 14363
This theorem is referenced by:  pczcl  14374  pcmul  14377  pcdiv  14378  pc1  14381  pczdvds  14388  pczndvds  14390  pczndvds2  14392  pcneg  14399
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