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Theorem pcqcl 14769
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 762 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  N  e.  QQ )
2 elq 11266 . . 3  |-  ( N  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  /  y ) )
31, 2sylib 199 . 2  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  ->  E. x  e.  ZZ  E. y  e.  NN  N  =  ( x  / 
y ) )
4 nncn 10617 . . . . . . . . . . . 12  |-  ( y  e.  NN  ->  y  e.  CC )
5 nnne0 10642 . . . . . . . . . . . 12  |-  ( y  e.  NN  ->  y  =/=  0 )
64, 5div0d 10381 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  (
0  /  y )  =  0 )
76ad2antll 733 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( 0  /  y )  =  0 )
8 oveq1 6312 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
x  /  y )  =  ( 0  / 
y ) )
98eqeq1d 2431 . . . . . . . . . 10  |-  ( x  =  0  ->  (
( x  /  y
)  =  0  <->  (
0  /  y )  =  0 ) )
107, 9syl5ibrcom 225 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( x  =  0  ->  (
x  /  y )  =  0 ) )
1110necon3d 2655 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( (
x  /  y )  =/=  0  ->  x  =/=  0 ) )
12 an32 805 . . . . . . . . . 10  |-  ( ( ( x  e.  ZZ  /\  y  e.  NN )  /\  x  =/=  0
)  <->  ( ( x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN ) )
13 pcdiv 14765 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  (
x  /  y ) )  =  ( ( P  pCnt  x )  -  ( P  pCnt  y ) ) )
14 pczcl 14761 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  e.  NN0 )
1514nn0zd 11038 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  e.  ZZ )
16153adant3 1025 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  x
)  e.  ZZ )
17 nnz 10959 . . . . . . . . . . . . . . . 16  |-  ( y  e.  NN  ->  y  e.  ZZ )
1817, 5jca 534 . . . . . . . . . . . . . . 15  |-  ( y  e.  NN  ->  (
y  e.  ZZ  /\  y  =/=  0 ) )
19 pczcl 14761 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  (
y  e.  ZZ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  NN0 )
2019nn0zd 11038 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  (
y  e.  ZZ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  ZZ )
2118, 20sylan2 476 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  y  e.  NN )  ->  ( P  pCnt  y )  e.  ZZ )
22213adant2 1024 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  y
)  e.  ZZ )
2316, 22zsubcld 11045 . . . . . . . . . . . 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 1206 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  (
( x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN ) )  ->  ( P  pCnt  ( x  / 
y ) )  e.  ZZ )
2612, 25sylan2b 477 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  (
( x  e.  ZZ  /\  y  e.  NN )  /\  x  =/=  0
) )  ->  ( P  pCnt  ( x  / 
y ) )  e.  ZZ )
2726expr 618 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( x  =/=  0  ->  ( P 
pCnt  ( x  / 
y ) )  e.  ZZ ) )
2811, 27syld 45 . . . . . . 7  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( (
x  /  y )  =/=  0  ->  ( P  pCnt  ( x  / 
y ) )  e.  ZZ ) )
29 neeq1 2712 . . . . . . . 8  |-  ( N  =  ( x  / 
y )  ->  ( N  =/=  0  <->  ( x  /  y )  =/=  0 ) )
30 oveq2 6313 . . . . . . . . 9  |-  ( N  =  ( x  / 
y )  ->  ( P  pCnt  N )  =  ( P  pCnt  (
x  /  y ) ) )
3130eleq1d 2498 . . . . . . . 8  |-  ( N  =  ( x  / 
y )  ->  (
( P  pCnt  N
)  e.  ZZ  <->  ( P  pCnt  ( x  /  y
) )  e.  ZZ ) )
3229, 31imbi12d 321 . . . . . . 7  |-  ( N  =  ( x  / 
y )  ->  (
( N  =/=  0  ->  ( P  pCnt  N
)  e.  ZZ )  <-> 
( ( x  / 
y )  =/=  0  ->  ( P  pCnt  (
x  /  y ) )  e.  ZZ ) ) )
3328, 32syl5ibrcom 225 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( N  =  ( x  / 
y )  ->  ( N  =/=  0  ->  ( P  pCnt  N )  e.  ZZ ) ) )
3433com23 81 . . . . 5  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( N  =/=  0  ->  ( N  =  ( x  / 
y )  ->  ( P  pCnt  N )  e.  ZZ ) ) )
3534impancom 441 . . . 4  |-  ( ( P  e.  Prime  /\  N  =/=  0 )  ->  (
( x  e.  ZZ  /\  y  e.  NN )  ->  ( N  =  ( x  /  y
)  ->  ( P  pCnt  N )  e.  ZZ ) ) )
3635adantrl 720 . . 3  |-  ( ( P  e.  Prime  /\  ( N  e.  QQ  /\  N  =/=  0 ) )  -> 
( ( x  e.  ZZ  /\  y  e.  NN )  ->  ( N  =  ( x  /  y )  -> 
( P  pCnt  N
)  e.  ZZ ) ) )
3736rexlimdvv 2930 . 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783  (class class class)co 6305   0cc0 9538    - cmin 9859    / cdiv 10268   NNcn 10609   ZZcz 10937   QQcq 11264   Primecprime 14593    pCnt cpc 14749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-dvds 14284  df-gcd 14443  df-prm 14594  df-pc 14750
This theorem is referenced by:  pcqdiv  14770  pcexp  14772  pcxcl  14773  pcadd  14797  qexpz  14809  expnprm  14810  padicabv  24331  padicabvf  24332  padicabvcxp  24333
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