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Theorem pcneg 14273
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 11196 . . 3  |-  ( A  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  A  =  ( x  /  y ) )
2 zcn 10881 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
32ad2antrl 727 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  x  e.  CC )
4 nncn 10556 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  CC )
54ad2antll 728 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  y  e.  CC )
6 nnne0 10580 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  =/=  0 )
76ad2antll 728 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  y  =/=  0 )
83, 5, 7divnegd 10345 . . . . . . 7  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  -u ( x  /  y )  =  ( -u x  / 
y ) )
98oveq2d 6311 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  -u ( x  / 
y ) )  =  ( P  pCnt  ( -u x  /  y ) ) )
10 neg0 9877 . . . . . . . . . 10  |-  -u 0  =  0
11 simpr 461 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  x  =  0 )
1211negeqd 9826 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  -u x  =  -u 0 )
1310, 12, 113eqtr4a 2534 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  -u x  =  x )
1413oveq1d 6310 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  ( -u x  /  y )  =  ( x  / 
y ) )
1514oveq2d 6311 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  ( P  pCnt  ( -u x  /  y ) )  =  ( P  pCnt  ( x  /  y ) ) )
16 simpll 753 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  P  e.  Prime )
17 simplrl 759 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  x  e.  ZZ )
1817znegcld 10980 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  -u x  e.  ZZ )
19 simpr 461 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  x  =/=  0 )
202negne0bd 9935 . . . . . . . . . . 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 210 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  -u x  =/=  0 )
23 simplrr 760 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  y  e.  NN )
24 pcdiv 14252 . . . . . . . . 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 1233 . . . . . . . 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 14252 . . . . . . . . . 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 1233 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  ( x  / 
y ) )  =  ( ( P  pCnt  x )  -  ( P 
pCnt  y ) ) )
28 eqid 2467 . . . . . . . . . . . . 13  |-  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  -u x } ,  RR ,  <  )  =  sup ( { y  e.  NN0  |  ( P ^ y )  ||  -u x } ,  RR ,  <  )
2928pczpre 14247 . . . . . . . . . . . 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 1226 . . . . . . . . . . 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 2467 . . . . . . . . . . . . . 14  |-  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  x } ,  RR ,  <  )  =  sup ( { y  e.  NN0  |  ( P ^ y )  ||  x } ,  RR ,  <  )
3231pczpre 14247 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  =  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  x } ,  RR ,  <  )
)
33 prmz 14097 . . . . . . . . . . . . . . . . . 18  |-  ( P  e.  Prime  ->  P  e.  ZZ )
34 zexpcl 12161 . . . . . . . . . . . . . . . . . 18  |-  ( ( P  e.  ZZ  /\  y  e.  NN0 )  -> 
( P ^ y
)  e.  ZZ )
3533, 34sylan 471 . . . . . . . . . . . . . . . . 17  |-  ( ( P  e.  Prime  /\  y  e.  NN0 )  ->  ( P ^ y )  e.  ZZ )
36 simpl 457 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  ZZ  /\  x  =/=  0 )  ->  x  e.  ZZ )
37 dvdsnegb 13879 . . . . . . . . . . . . . . . . 17  |-  ( ( ( P ^ y
)  e.  ZZ  /\  x  e.  ZZ )  ->  ( ( P ^
y )  ||  x  <->  ( P ^ y ) 
||  -u x ) )
3835, 36, 37syl2an 477 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  y  e.  NN0 )  /\  ( x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( ( P ^ y )  ||  x 
<->  ( P ^ y
)  ||  -u x ) )
3938an32s 802 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  x  =/=  0 ) )  /\  y  e. 
NN0 )  ->  (
( P ^ y
)  ||  x  <->  ( P ^ y )  ||  -u x ) )
4039rabbidva 3109 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  { y  e. 
NN0  |  ( P ^ y )  ||  x }  =  {
y  e.  NN0  | 
( P ^ y
)  ||  -u x }
)
4140supeq1d 7918 . . . . . . . . . . . . 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 2508 . . . . . . . . . . . 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 1226 . . . . . . . . . . 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 2511 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  -u x )  =  ( P  pCnt  x
) )
4544oveq1d 6310 . . . . . . . . 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 2511 . . . . . . . 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 2511 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  ( -u x  /  y ) )  =  ( P  pCnt  ( x  /  y ) ) )
4815, 47pm2.61dane 2785 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  ( -u x  / 
y ) )  =  ( P  pCnt  (
x  /  y ) ) )
499, 48eqtrd 2508 . . . . 5  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  -u ( x  / 
y ) )  =  ( P  pCnt  (
x  /  y ) ) )
50 negeq 9824 . . . . . . 7  |-  ( A  =  ( x  / 
y )  ->  -u A  =  -u ( x  / 
y ) )
5150oveq2d 6311 . . . . . 6  |-  ( A  =  ( x  / 
y )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  -u (
x  /  y ) ) )
52 oveq2 6303 . . . . . 6  |-  ( A  =  ( x  / 
y )  ->  ( P  pCnt  A )  =  ( P  pCnt  (
x  /  y ) ) )
5351, 52eqeq12d 2489 . . . . 5  |-  ( A  =  ( x  / 
y )  ->  (
( P  pCnt  -u A
)  =  ( P 
pCnt  A )  <->  ( P  pCnt  -u ( x  / 
y ) )  =  ( P  pCnt  (
x  /  y ) ) ) )
5449, 53syl5ibrcom 222 . . . 4  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( A  =  ( x  / 
y )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  A
) ) )
5554rexlimdvva 2966 . . 3  |-  ( P  e.  Prime  ->  ( E. x  e.  ZZ  E. y  e.  NN  A  =  ( x  / 
y )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  A
) ) )
561, 55syl5bi 217 . 2  |-  ( P  e.  Prime  ->  ( A  e.  QQ  ->  ( P  pCnt  -u A )  =  ( P  pCnt  A
) ) )
5756imp 429 1  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   {crab 2821   class class class wbr 4453  (class class class)co 6295   supcsup 7912   CCcc 9502   RRcr 9503   0cc0 9504    < clt 9640    - cmin 9817   -ucneg 9818    / cdiv 10218   NNcn 10548   NN0cn0 10807   ZZcz 10876   QQcq 11194   ^cexp 12146    || cdivides 13864   Primecprime 14093    pCnt cpc 14236
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-prm 14094  df-pc 14237
This theorem is referenced by:  pcabs  14274  pcadd2  14285  lgsneg  23460
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