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Theorem pcqcl 14239
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 11184 . . 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 10544 . . . . . . . . . . . 12  |-  ( y  e.  NN  ->  y  e.  CC )
5 nnne0 10568 . . . . . . . . . . . 12  |-  ( y  e.  NN  ->  y  =/=  0 )
64, 5div0d 10319 . . . . . . . . . . 11  |-  ( y  e.  NN  ->  (
0  /  y )  =  0 )
76ad2antll 728 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( 0  /  y )  =  0 )
8 oveq1 6291 . . . . . . . . . . 11  |-  ( x  =  0  ->  (
x  /  y )  =  ( 0  / 
y ) )
98eqeq1d 2469 . . . . . . . . . 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 2691 . . . . . . . 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 14235 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  (
x  /  y ) )  =  ( ( P  pCnt  x )  -  ( P  pCnt  y ) ) )
14 pczcl 14231 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  e.  NN0 )
1514nn0zd 10964 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  e.  ZZ )
16153adant3 1016 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  x
)  e.  ZZ )
17 nnz 10886 . . . . . . . . . . . . . . . 16  |-  ( y  e.  NN  ->  y  e.  ZZ )
1817, 5jca 532 . . . . . . . . . . . . . . 15  |-  ( y  e.  NN  ->  (
y  e.  ZZ  /\  y  =/=  0 ) )
19 pczcl 14231 . . . . . . . . . . . . . . . 16  |-  ( ( P  e.  Prime  /\  (
y  e.  ZZ  /\  y  =/=  0 ) )  ->  ( P  pCnt  y )  e.  NN0 )
2019nn0zd 10964 . . . . . . . . . . . . . . 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 1015 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  y
)  e.  ZZ )
2316, 22zsubcld 10971 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( ( P  pCnt  x )  -  ( P 
pCnt  y ) )  e.  ZZ )
2413, 23eqeltrd 2555 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  (
x  /  y ) )  e.  ZZ )
25243expb 1197 . . . . . . . . . 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 2748 . . . . . . . 8  |-  ( N  =  ( x  / 
y )  ->  ( N  =/=  0  <->  ( x  /  y )  =/=  0 ) )
30 oveq2 6292 . . . . . . . . 9  |-  ( N  =  ( x  / 
y )  ->  ( P  pCnt  N )  =  ( P  pCnt  (
x  /  y ) ) )
3130eleq1d 2536 . . . . . . . 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 2961 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815  (class class class)co 6284   0cc0 9492    - cmin 9805    / cdiv 10206   NNcn 10536   ZZcz 10864   QQcq 11182   Primecprime 14076    pCnt cpc 14219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-q 11183  df-rp 11221  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-dvds 13848  df-gcd 14004  df-prm 14077  df-pc 14220
This theorem is referenced by:  pcqdiv  14240  pcexp  14242  pcxcl  14243  pcadd  14267  qexpz  14279  expnprm  14280  padicabv  23571  padicabvf  23572  padicabvcxp  23573
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