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Theorem pcqcl 13150
Description: Closure of the general prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)
Assertion
Ref Expression
pcqcl  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  -> 
( P  pCnt  N
)  e.  ZZ )

Proof of Theorem pcqcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 733 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  N  e.  QQ )
2 elq 10501 . . 3  |-  ( N  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  /  y ) )
31, 2sylib 189 . 2  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  / 
y ) )
4 nncn 9933 . . . . . . . . . . . 12  |-  ( y  e.  NN  ->  y  e.  CC )
5 nnne0 9957 . . . . . . . . . . . 12  |-  ( y  e.  NN  ->  y  =/=  0 )
64, 5div0d 9714 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  (
0  /  y )  =  0 )
76ad2antll 710 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( 0  /  y )  =  0 )
8 oveq1 6020 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
x  /  y )  =  ( 0  / 
y ) )
98eqeq1d 2388 . . . . . . . . . 10  |-  ( x  =  0  ->  (
( x  /  y
)  =  0  <->  (
0  /  y )  =  0 ) )
107, 9syl5ibrcom 214 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( x  =  0  ->  (
x  /  y )  =  0 ) )
1110necon3d 2581 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( (
x  /  y )  =/=  0  ->  x  =/=  0 ) )
12 an32 774 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  x  =/=  0
)  <->  ( ( x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN ) )
13 pcdiv 13146 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  (
x  /  y ) )  =  ( ( P  pCnt  x )  -  ( P  pCnt  y ) ) )
14 pczcl 13142 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  e.  NN0 )
1514nn0zd 10298 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  e.  ZZ )
16153adant3 977 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  x
)  e.  ZZ )
17 nnz 10228 . . . . . . . . . . . . . . . 16  |-  ( y  e.  NN  ->  y  e.  ZZ )
1817, 5jca 519 . . . . . . . . . . . . . . 15  |-  ( y  e.  NN  ->  (
y  e.  ZZ  /\  y  =/=  0 ) )
19 pczcl 13142 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  (
y  e.  ZZ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  NN0 )
2019nn0zd 10298 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  (
y  e.  ZZ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  ZZ )
2118, 20sylan2 461 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  y  e.  NN )  ->  ( P  pCnt  y )  e.  ZZ )
22213adant2 976 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  y
)  e.  ZZ )
2316, 22zsubcld 10305 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( ( P  pCnt  x )  -  ( P 
pCnt  y ) )  e.  ZZ )
2413, 23eqeltrd 2454 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  (
x  /  y ) )  e.  ZZ )
25243expb 1154 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  (
( x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN ) )  ->  ( P  pCnt  ( x  / 
y ) )  e.  ZZ )
2612, 25sylan2b 462 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  (
( x  e.  ZZ  /\  y  e.  NN )  /\  x  =/=  0
) )  ->  ( P  pCnt  ( x  / 
y ) )  e.  ZZ )
2726expr 599 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( x  =/=  0  ->  ( P 
pCnt  ( x  / 
y ) )  e.  ZZ ) )
2811, 27syld 42 . . . . . . 7  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( (
x  /  y )  =/=  0  ->  ( P  pCnt  ( x  / 
y ) )  e.  ZZ ) )
29 neeq1 2551 . . . . . . . 8  |-  ( N  =  ( x  / 
y )  ->  ( N  =/=  0  <->  ( x  /  y )  =/=  0 ) )
30 oveq2 6021 . . . . . . . . 9  |-  ( N  =  ( x  / 
y )  ->  ( P  pCnt  N )  =  ( P  pCnt  (
x  /  y ) ) )
3130eleq1d 2446 . . . . . . . 8  |-  ( N  =  ( x  / 
y )  ->  (
( P  pCnt  N
)  e.  ZZ  <->  ( P  pCnt  ( x  /  y
) )  e.  ZZ ) )
3229, 31imbi12d 312 . . . . . . 7  |-  ( N  =  ( x  / 
y )  ->  (
( N  =/=  0  ->  ( P  pCnt  N
)  e.  ZZ )  <-> 
( ( x  / 
y )  =/=  0  ->  ( P  pCnt  (
x  /  y ) )  e.  ZZ ) ) )
3328, 32syl5ibrcom 214 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( N  =  ( x  / 
y )  ->  ( N  =/=  0  ->  ( P  pCnt  N )  e.  ZZ ) ) )
3433com23 74 . . . . 5  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( N  =/=  0  ->  ( N  =  ( x  / 
y )  ->  ( P  pCnt  N )  e.  ZZ ) ) )
3534impancom 428 . . . 4  |-  ( ( P  e.  Prime  /\  N  =/=  0 )  ->  (
( x  e.  ZZ  /\  y  e.  NN )  ->  ( N  =  ( x  /  y
)  ->  ( P  pCnt  N )  e.  ZZ ) ) )
3635adantrl 697 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  -> 
( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( N  =  ( x  /  y )  -> 
( P  pCnt  N
)  e.  ZZ ) ) )
3736rexlimdvv 2772 . 2  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  -> 
( E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  /  y )  ->  ( P  pCnt  N )  e.  ZZ ) )
383, 37mpd 15 1  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  -> 
( P  pCnt  N
)  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   E.wrex 2643  (class class class)co 6013   0cc0 8916    - cmin 9216    / cdiv 9602   NNcn 9925   ZZcz 10207   QQcq 10499   Primecprime 12999    pCnt cpc 13130
This theorem is referenced by:  pcqdiv  13151  pcexp  13153  pcxcl  13154  pcadd  13178  qexpz  13190  expnprm  13191  padicabv  21184  padicabvf  21185  padicabvcxp  21186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-q 10500  df-rp 10538  df-fl 11122  df-mod 11171  df-seq 11244  df-exp 11303  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-dvds 12773  df-gcd 12927  df-prm 13000  df-pc 13131
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