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Theorem pcneg 13239
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 10568 . . 3  |-  ( A  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  A  =  ( x  /  y ) )
2 zcn 10279 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
32ad2antrl 709 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  x  e.  CC )
4 nncn 10000 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  CC )
54ad2antll 710 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  y  e.  CC )
6 nnne0 10024 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  =/=  0 )
76ad2antll 710 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  y  =/=  0 )
83, 5, 7divnegd 9795 . . . . . . 7  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  -u ( x  /  y )  =  ( -u x  / 
y ) )
98oveq2d 6089 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  -u ( x  / 
y ) )  =  ( P  pCnt  ( -u x  /  y ) ) )
10 neg0 9339 . . . . . . . . . 10  |-  -u 0  =  0
11 simpr 448 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  x  =  0 )
1211negeqd 9292 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  -u x  =  -u 0 )
1310, 12, 113eqtr4a 2493 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  -u x  =  x )
1413oveq1d 6088 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  ( -u x  /  y )  =  ( x  / 
y ) )
1514oveq2d 6089 . . . . . . 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 10369 . . . . . . . . 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 9396 . . . . . . . . . . 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 13218 . . . . . . . . 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 13218 . . . . . . . . . 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 2435 . . . . . . . . . . . . 13  |-  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  -u x } ,  RR ,  <  )  =  sup ( { y  e.  NN0  |  ( P ^ y )  ||  -u x } ,  RR ,  <  )
2928pczpre 13213 . . . . . . . . . . . 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 2435 . . . . . . . . . . . . . 14  |-  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  x } ,  RR ,  <  )  =  sup ( { y  e.  NN0  |  ( P ^ y )  ||  x } ,  RR ,  <  )
3231pczpre 13213 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  =  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  x } ,  RR ,  <  )
)
33 prmz 13075 . . . . . . . . . . . . . . . . . 18  |-  ( P  e.  Prime  ->  P  e.  ZZ )
34 zexpcl 11388 . . . . . . . . . . . . . . . . . 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 12859 . . . . . . . . . . . . . . . . 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 2939 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  { y  e. 
NN0  |  ( P ^ y )  ||  x }  =  {
y  e.  NN0  | 
( P ^ y
)  ||  -u x }
)
4140supeq1d 7443 . . . . . . . . . . . . 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 2467 . . . . . . . . . . . 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 2470 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  -u x )  =  ( P  pCnt  x
) )
4544oveq1d 6088 . . . . . . . . 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 2470 . . . . . . . 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 2470 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  ( -u x  /  y ) )  =  ( P  pCnt  ( x  /  y ) ) )
4815, 47pm2.61dane 2676 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  ( -u x  / 
y ) )  =  ( P  pCnt  (
x  /  y ) ) )
499, 48eqtrd 2467 . . . . 5  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  -u ( x  / 
y ) )  =  ( P  pCnt  (
x  /  y ) ) )
50 negeq 9290 . . . . . . 7  |-  ( A  =  ( x  / 
y )  ->  -u A  =  -u ( x  / 
y ) )
5150oveq2d 6089 . . . . . 6  |-  ( A  =  ( x  / 
y )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  -u (
x  /  y ) ) )
52 oveq2 6081 . . . . . 6  |-  ( A  =  ( x  / 
y )  ->  ( P  pCnt  A )  =  ( P  pCnt  (
x  /  y ) ) )
5351, 52eqeq12d 2449 . . . . 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 2829 . . 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 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   {crab 2701   class class class wbr 4204  (class class class)co 6073   supcsup 7437   CCcc 8980   RRcr 8981   0cc0 8982    < clt 9112    - cmin 9283   -ucneg 9284    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   QQcq 10566   ^cexp 11374    || cdivides 12844   Primecprime 13071    pCnt cpc 13202
This theorem is referenced by:  pcabs  13240  pcadd2  13251  lgsneg  21095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203
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