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Theorem pcneg 13168
Description: The prime count of a negative number. (Contributed by Mario Carneiro, 13-Mar-2014.)
Assertion
Ref Expression
pcneg  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  A
) )

Proof of Theorem pcneg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elq 10502 . . 3  |-  ( A  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  A  =  ( x  /  y ) )
2 zcn 10213 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
32ad2antrl 709 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  x  e.  CC )
4 nncn 9934 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  CC )
54ad2antll 710 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  y  e.  CC )
6 nnne0 9958 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  =/=  0 )
76ad2antll 710 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  y  =/=  0 )
83, 5, 7divnegd 9729 . . . . . . 7  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  -u ( x  /  y )  =  ( -u x  / 
y ) )
98oveq2d 6030 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  -u ( x  / 
y ) )  =  ( P  pCnt  ( -u x  /  y ) ) )
10 neg0 9273 . . . . . . . . . 10  |-  -u 0  =  0
11 simpr 448 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  x  =  0 )
1211negeqd 9226 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  -u x  =  -u 0 )
1310, 12, 113eqtr4a 2439 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  -u x  =  x )
1413oveq1d 6029 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  ( -u x  /  y )  =  ( x  / 
y ) )
1514oveq2d 6030 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  ( P  pCnt  ( -u x  /  y ) )  =  ( P  pCnt  ( x  /  y ) ) )
16 simpll 731 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  P  e.  Prime )
17 simplrl 737 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  x  e.  ZZ )
1817znegcld 10303 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  -u x  e.  ZZ )
19 simpr 448 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  x  =/=  0 )
202negne0bd 9330 . . . . . . . . . . 11  |-  ( x  e.  ZZ  ->  (
x  =/=  0  <->  -u x  =/=  0 ) )
2117, 20syl 16 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  (
x  =/=  0  <->  -u x  =/=  0 ) )
2219, 21mpbid 202 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  -u x  =/=  0 )
23 simplrr 738 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  y  e.  NN )
24 pcdiv 13147 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( -u x  e.  ZZ  /\  -u x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  (
-u x  /  y
) )  =  ( ( P  pCnt  -u x
)  -  ( P 
pCnt  y ) ) )
2516, 18, 22, 23, 24syl121anc 1189 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  ( -u x  /  y ) )  =  ( ( P 
pCnt  -u x )  -  ( P  pCnt  y ) ) )
26 pcdiv 13147 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 )  /\  y  e.  NN )  ->  ( P  pCnt  (
x  /  y ) )  =  ( ( P  pCnt  x )  -  ( P  pCnt  y ) ) )
2716, 17, 19, 23, 26syl121anc 1189 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  ( x  / 
y ) )  =  ( ( P  pCnt  x )  -  ( P 
pCnt  y ) ) )
28 eqid 2381 . . . . . . . . . . . . 13  |-  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  -u x } ,  RR ,  <  )  =  sup ( { y  e.  NN0  |  ( P ^ y )  ||  -u x } ,  RR ,  <  )
2928pczpre 13142 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  ( -u x  e.  ZZ  /\  -u x  =/=  0 ) )  ->  ( P  pCnt  -u x )  =  sup ( { y  e.  NN0  |  ( P ^ y )  ||  -u x } ,  RR ,  <  ) )
3016, 18, 22, 29syl12anc 1182 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  -u x )  =  sup ( { y  e.  NN0  |  ( P ^ y )  ||  -u x } ,  RR ,  <  ) )
31 eqid 2381 . . . . . . . . . . . . . 14  |-  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  x } ,  RR ,  <  )  =  sup ( { y  e.  NN0  |  ( P ^ y )  ||  x } ,  RR ,  <  )
3231pczpre 13142 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  =  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  x } ,  RR ,  <  )
)
33 prmz 13004 . . . . . . . . . . . . . . . . . 18  |-  ( P  e.  Prime  ->  P  e.  ZZ )
34 zexpcl 11317 . . . . . . . . . . . . . . . . . 18  |-  ( ( P  e.  ZZ  /\  y  e.  NN0 )  -> 
( P ^ y
)  e.  ZZ )
3533, 34sylan 458 . . . . . . . . . . . . . . . . 17  |-  ( ( P  e.  Prime  /\  y  e.  NN0 )  ->  ( P ^ y )  e.  ZZ )
36 simpl 444 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  ZZ  /\  x  =/=  0 )  ->  x  e.  ZZ )
37 dvdsnegb 12788 . . . . . . . . . . . . . . . . 17  |-  ( ( ( P ^ y
)  e.  ZZ  /\  x  e.  ZZ )  ->  ( ( P ^
y )  ||  x  <->  ( P ^ y ) 
||  -u x ) )
3835, 36, 37syl2an 464 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  y  e.  NN0 )  /\  ( x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( ( P ^ y )  ||  x 
<->  ( P ^ y
)  ||  -u x ) )
3938an32s 780 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  x  =/=  0 ) )  /\  y  e. 
NN0 )  ->  (
( P ^ y
)  ||  x  <->  ( P ^ y )  ||  -u x ) )
4039rabbidva 2884 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  { y  e. 
NN0  |  ( P ^ y )  ||  x }  =  {
y  e.  NN0  | 
( P ^ y
)  ||  -u x }
)
4140supeq1d 7380 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  sup ( { y  e.  NN0  |  ( P ^ y )  ||  x } ,  RR ,  <  )  =  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  -u x } ,  RR ,  <  )
)
4232, 41eqtrd 2413 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  =  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  -u x } ,  RR ,  <  )
)
4316, 17, 19, 42syl12anc 1182 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  x )  =  sup ( { y  e.  NN0  |  ( P ^ y )  ||  -u x } ,  RR ,  <  ) )
4430, 43eqtr4d 2416 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  -u x )  =  ( P  pCnt  x
) )
4544oveq1d 6029 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  (
( P  pCnt  -u x
)  -  ( P 
pCnt  y ) )  =  ( ( P 
pCnt  x )  -  ( P  pCnt  y ) ) )
4627, 45eqtr4d 2416 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  ( x  / 
y ) )  =  ( ( P  pCnt  -u x )  -  ( P  pCnt  y ) ) )
4725, 46eqtr4d 2416 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  ( -u x  /  y ) )  =  ( P  pCnt  ( x  /  y ) ) )
4815, 47pm2.61dane 2622 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  ( -u x  / 
y ) )  =  ( P  pCnt  (
x  /  y ) ) )
499, 48eqtrd 2413 . . . . 5  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  -u ( x  / 
y ) )  =  ( P  pCnt  (
x  /  y ) ) )
50 negeq 9224 . . . . . . 7  |-  ( A  =  ( x  / 
y )  ->  -u A  =  -u ( x  / 
y ) )
5150oveq2d 6030 . . . . . 6  |-  ( A  =  ( x  / 
y )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  -u (
x  /  y ) ) )
52 oveq2 6022 . . . . . 6  |-  ( A  =  ( x  / 
y )  ->  ( P  pCnt  A )  =  ( P  pCnt  (
x  /  y ) ) )
5351, 52eqeq12d 2395 . . . . 5  |-  ( A  =  ( x  / 
y )  ->  (
( P  pCnt  -u A
)  =  ( P 
pCnt  A )  <->  ( P  pCnt  -u ( x  / 
y ) )  =  ( P  pCnt  (
x  /  y ) ) ) )
5449, 53syl5ibrcom 214 . . . 4  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( A  =  ( x  / 
y )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  A
) ) )
5554rexlimdvva 2774 . . 3  |-  ( P  e.  Prime  ->  ( E. x  e.  ZZ  E. y  e.  NN  A  =  ( x  / 
y )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  A
) ) )
561, 55syl5bi 209 . 2  |-  ( P  e.  Prime  ->  ( A  e.  QQ  ->  ( P  pCnt  -u A )  =  ( P  pCnt  A
) ) )
5756imp 419 1  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2544   E.wrex 2644   {crab 2647   class class class wbr 4147  (class class class)co 6014   supcsup 7374   CCcc 8915   RRcr 8916   0cc0 8917    < clt 9047    - cmin 9217   -ucneg 9218    / cdiv 9603   NNcn 9926   NN0cn0 10147   ZZcz 10208   QQcq 10500   ^cexp 11303    || cdivides 12773   Primecprime 13000    pCnt cpc 13131
This theorem is referenced by:  pcabs  13169  pcadd2  13180  lgsneg  20964
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 2362  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635  ax-cnex 8973  ax-resscn 8974  ax-1cn 8975  ax-icn 8976  ax-addcl 8977  ax-addrcl 8978  ax-mulcl 8979  ax-mulrcl 8980  ax-mulcom 8981  ax-addass 8982  ax-mulass 8983  ax-distr 8984  ax-i2m1 8985  ax-1ne0 8986  ax-1rid 8987  ax-rnegex 8988  ax-rrecex 8989  ax-cnre 8990  ax-pre-lttri 8991  ax-pre-lttrn 8992  ax-pre-ltadd 8993  ax-pre-mulgt0 8994  ax-pre-sup 8995
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-nel 2547  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-int 3987  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-riota 6479  df-recs 6563  df-rdg 6598  df-1o 6654  df-2o 6655  df-oadd 6658  df-er 6835  df-en 7040  df-dom 7041  df-sdom 7042  df-fin 7043  df-sup 7375  df-pnf 9049  df-mnf 9050  df-xr 9051  df-ltxr 9052  df-le 9053  df-sub 9219  df-neg 9220  df-div 9604  df-nn 9927  df-2 9984  df-3 9985  df-n0 10148  df-z 10209  df-uz 10415  df-q 10501  df-rp 10539  df-fl 11123  df-mod 11172  df-seq 11245  df-exp 11304  df-cj 11825  df-re 11826  df-im 11827  df-sqr 11961  df-abs 11962  df-dvds 12774  df-gcd 12928  df-prm 13001  df-pc 13132
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