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Theorem pcqcl 13923
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 755 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  N  e.  QQ )
2 elq 10955 . . 3  |-  ( N  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  /  y ) )
31, 2sylib 196 . 2  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  / 
y ) )
4 nncn 10330 . . . . . . . . . . . 12  |-  ( y  e.  NN  ->  y  e.  CC )
5 nnne0 10354 . . . . . . . . . . . 12  |-  ( y  e.  NN  ->  y  =/=  0 )
64, 5div0d 10106 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  (
0  /  y )  =  0 )
76ad2antll 728 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( 0  /  y )  =  0 )
8 oveq1 6098 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
x  /  y )  =  ( 0  / 
y ) )
98eqeq1d 2451 . . . . . . . . . 10  |-  ( x  =  0  ->  (
( x  /  y
)  =  0  <->  (
0  /  y )  =  0 ) )
107, 9syl5ibrcom 222 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( x  =  0  ->  (
x  /  y )  =  0 ) )
1110necon3d 2646 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( (
x  /  y )  =/=  0  ->  x  =/=  0 ) )
12 an32 796 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  x  =/=  0
)  <->  ( ( x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN ) )
13 pcdiv 13919 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  (
x  /  y ) )  =  ( ( P  pCnt  x )  -  ( P  pCnt  y ) ) )
14 pczcl 13915 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  e.  NN0 )
1514nn0zd 10745 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  e.  ZZ )
16153adant3 1008 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  x
)  e.  ZZ )
17 nnz 10668 . . . . . . . . . . . . . . . 16  |-  ( y  e.  NN  ->  y  e.  ZZ )
1817, 5jca 532 . . . . . . . . . . . . . . 15  |-  ( y  e.  NN  ->  (
y  e.  ZZ  /\  y  =/=  0 ) )
19 pczcl 13915 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  (
y  e.  ZZ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  NN0 )
2019nn0zd 10745 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  (
y  e.  ZZ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  ZZ )
2118, 20sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  y  e.  NN )  ->  ( P  pCnt  y )  e.  ZZ )
22213adant2 1007 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  y
)  e.  ZZ )
2316, 22zsubcld 10752 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( ( P  pCnt  x )  -  ( P 
pCnt  y ) )  e.  ZZ )
2413, 23eqeltrd 2517 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  (
x  /  y ) )  e.  ZZ )
25243expb 1188 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  (
( x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN ) )  ->  ( P  pCnt  ( x  / 
y ) )  e.  ZZ )
2612, 25sylan2b 475 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  (
( x  e.  ZZ  /\  y  e.  NN )  /\  x  =/=  0
) )  ->  ( P  pCnt  ( x  / 
y ) )  e.  ZZ )
2726expr 615 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( x  =/=  0  ->  ( P 
pCnt  ( x  / 
y ) )  e.  ZZ ) )
2811, 27syld 44 . . . . . . 7  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( (
x  /  y )  =/=  0  ->  ( P  pCnt  ( x  / 
y ) )  e.  ZZ ) )
29 neeq1 2616 . . . . . . . 8  |-  ( N  =  ( x  / 
y )  ->  ( N  =/=  0  <->  ( x  /  y )  =/=  0 ) )
30 oveq2 6099 . . . . . . . . 9  |-  ( N  =  ( x  / 
y )  ->  ( P  pCnt  N )  =  ( P  pCnt  (
x  /  y ) ) )
3130eleq1d 2509 . . . . . . . 8  |-  ( N  =  ( x  / 
y )  ->  (
( P  pCnt  N
)  e.  ZZ  <->  ( P  pCnt  ( x  /  y
) )  e.  ZZ ) )
3229, 31imbi12d 320 . . . . . . 7  |-  ( N  =  ( x  / 
y )  ->  (
( N  =/=  0  ->  ( P  pCnt  N
)  e.  ZZ )  <-> 
( ( x  / 
y )  =/=  0  ->  ( P  pCnt  (
x  /  y ) )  e.  ZZ ) ) )
3328, 32syl5ibrcom 222 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( N  =  ( x  / 
y )  ->  ( N  =/=  0  ->  ( P  pCnt  N )  e.  ZZ ) ) )
3433com23 78 . . . . 5  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( N  =/=  0  ->  ( N  =  ( x  / 
y )  ->  ( P  pCnt  N )  e.  ZZ ) ) )
3534impancom 440 . . . 4  |-  ( ( P  e.  Prime  /\  N  =/=  0 )  ->  (
( x  e.  ZZ  /\  y  e.  NN )  ->  ( N  =  ( x  /  y
)  ->  ( P  pCnt  N )  e.  ZZ ) ) )
3635adantrl 715 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  -> 
( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( N  =  ( x  /  y )  -> 
( P  pCnt  N
)  e.  ZZ ) ) )
3736rexlimdvv 2847 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716  (class class class)co 6091   0cc0 9282    - cmin 9595    / cdiv 9993   NNcn 10322   ZZcz 10646   QQcq 10953   Primecprime 13763    pCnt cpc 13903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-dvds 13536  df-gcd 13691  df-prm 13764  df-pc 13904
This theorem is referenced by:  pcqdiv  13924  pcexp  13926  pcxcl  13927  pcadd  13951  qexpz  13963  expnprm  13964  padicabv  22879  padicabvf  22880  padicabvcxp  22881
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