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Theorem pcqcl 12925
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 732 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  N  e.  QQ )
2 elq 10334 . . 3  |-  ( N  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  /  y ) )
31, 2sylib 188 . 2  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  / 
y ) )
4 nncn 9770 . . . . . . . . . . . 12  |-  ( y  e.  NN  ->  y  e.  CC )
5 nnne0 9794 . . . . . . . . . . . 12  |-  ( y  e.  NN  ->  y  =/=  0 )
64, 5div0d 9551 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  (
0  /  y )  =  0 )
76ad2antll 709 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( 0  /  y )  =  0 )
8 oveq1 5881 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
x  /  y )  =  ( 0  / 
y ) )
98eqeq1d 2304 . . . . . . . . . 10  |-  ( x  =  0  ->  (
( x  /  y
)  =  0  <->  (
0  /  y )  =  0 ) )
107, 9syl5ibrcom 213 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( x  =  0  ->  (
x  /  y )  =  0 ) )
1110necon3d 2497 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( (
x  /  y )  =/=  0  ->  x  =/=  0 ) )
12 an32 773 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  x  =/=  0
)  <->  ( ( x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN ) )
13 pcdiv 12921 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  (
x  /  y ) )  =  ( ( P  pCnt  x )  -  ( P  pCnt  y ) ) )
14 pczcl 12917 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  e.  NN0 )
1514nn0zd 10131 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  e.  ZZ )
16153adant3 975 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  x
)  e.  ZZ )
17 nnz 10061 . . . . . . . . . . . . . . . 16  |-  ( y  e.  NN  ->  y  e.  ZZ )
1817, 5jca 518 . . . . . . . . . . . . . . 15  |-  ( y  e.  NN  ->  (
y  e.  ZZ  /\  y  =/=  0 ) )
19 pczcl 12917 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  (
y  e.  ZZ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  NN0 )
2019nn0zd 10131 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  (
y  e.  ZZ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  ZZ )
2118, 20sylan2 460 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  y  e.  NN )  ->  ( P  pCnt  y )  e.  ZZ )
22213adant2 974 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  y
)  e.  ZZ )
2316, 22zsubcld 10138 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( ( P  pCnt  x )  -  ( P 
pCnt  y ) )  e.  ZZ )
2413, 23eqeltrd 2370 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  (
x  /  y ) )  e.  ZZ )
25243expb 1152 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  (
( x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN ) )  ->  ( P  pCnt  ( x  / 
y ) )  e.  ZZ )
2612, 25sylan2b 461 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  (
( x  e.  ZZ  /\  y  e.  NN )  /\  x  =/=  0
) )  ->  ( P  pCnt  ( x  / 
y ) )  e.  ZZ )
2726expr 598 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( x  =/=  0  ->  ( P 
pCnt  ( x  / 
y ) )  e.  ZZ ) )
2811, 27syld 40 . . . . . . 7  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( (
x  /  y )  =/=  0  ->  ( P  pCnt  ( x  / 
y ) )  e.  ZZ ) )
29 neeq1 2467 . . . . . . . 8  |-  ( N  =  ( x  / 
y )  ->  ( N  =/=  0  <->  ( x  /  y )  =/=  0 ) )
30 oveq2 5882 . . . . . . . . 9  |-  ( N  =  ( x  / 
y )  ->  ( P  pCnt  N )  =  ( P  pCnt  (
x  /  y ) ) )
3130eleq1d 2362 . . . . . . . 8  |-  ( N  =  ( x  / 
y )  ->  (
( P  pCnt  N
)  e.  ZZ  <->  ( P  pCnt  ( x  /  y
) )  e.  ZZ ) )
3229, 31imbi12d 311 . . . . . . 7  |-  ( N  =  ( x  / 
y )  ->  (
( N  =/=  0  ->  ( P  pCnt  N
)  e.  ZZ )  <-> 
( ( x  / 
y )  =/=  0  ->  ( P  pCnt  (
x  /  y ) )  e.  ZZ ) ) )
3328, 32syl5ibrcom 213 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( N  =  ( x  / 
y )  ->  ( N  =/=  0  ->  ( P  pCnt  N )  e.  ZZ ) ) )
3433com23 72 . . . . 5  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( N  =/=  0  ->  ( N  =  ( x  / 
y )  ->  ( P  pCnt  N )  e.  ZZ ) ) )
3534impancom 427 . . . 4  |-  ( ( P  e.  Prime  /\  N  =/=  0 )  ->  (
( x  e.  ZZ  /\  y  e.  NN )  ->  ( N  =  ( x  /  y
)  ->  ( P  pCnt  N )  e.  ZZ ) ) )
3635adantrl 696 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  -> 
( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( N  =  ( x  /  y )  -> 
( P  pCnt  N
)  e.  ZZ ) ) )
3736rexlimdvv 2686 . 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 14 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557  (class class class)co 5874   0cc0 8753    - cmin 9053    / cdiv 9439   NNcn 9762   ZZcz 10040   QQcq 10332   Primecprime 12774    pCnt cpc 12905
This theorem is referenced by:  pcqdiv  12926  pcexp  12928  pcxcl  12929  pcadd  12953  qexpz  12965  expnprm  12966  padicabv  20795  padicabvf  20796  padicabvcxp  20797
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  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906
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