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Theorem pcneg 14061
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 11069 . . 3  |-  ( A  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  A  =  ( x  /  y ) )
2 zcn 10765 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
32ad2antrl 727 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  x  e.  CC )
4 nncn 10444 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  CC )
54ad2antll 728 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  y  e.  CC )
6 nnne0 10468 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  =/=  0 )
76ad2antll 728 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  y  =/=  0 )
83, 5, 7divnegd 10234 . . . . . . 7  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  -u ( x  /  y )  =  ( -u x  / 
y ) )
98oveq2d 6219 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  -u ( x  / 
y ) )  =  ( P  pCnt  ( -u x  /  y ) ) )
10 neg0 9769 . . . . . . . . . 10  |-  -u 0  =  0
11 simpr 461 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  x  =  0 )
1211negeqd 9718 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  -u x  =  -u 0 )
1310, 12, 113eqtr4a 2521 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  -u x  =  x )
1413oveq1d 6218 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  ( -u x  /  y )  =  ( x  / 
y ) )
1514oveq2d 6219 . . . . . . 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 10863 . . . . . . . . 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 9826 . . . . . . . . . . 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 14040 . . . . . . . . 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 1224 . . . . . . . 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 14040 . . . . . . . . . 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 1224 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  ( x  / 
y ) )  =  ( ( P  pCnt  x )  -  ( P 
pCnt  y ) ) )
28 eqid 2454 . . . . . . . . . . . . 13  |-  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  -u x } ,  RR ,  <  )  =  sup ( { y  e.  NN0  |  ( P ^ y )  ||  -u x } ,  RR ,  <  )
2928pczpre 14035 . . . . . . . . . . . 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 1217 . . . . . . . . . . 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 2454 . . . . . . . . . . . . . 14  |-  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  x } ,  RR ,  <  )  =  sup ( { y  e.  NN0  |  ( P ^ y )  ||  x } ,  RR ,  <  )
3231pczpre 14035 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  =  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  x } ,  RR ,  <  )
)
33 prmz 13888 . . . . . . . . . . . . . . . . . 18  |-  ( P  e.  Prime  ->  P  e.  ZZ )
34 zexpcl 12000 . . . . . . . . . . . . . . . . . 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 13671 . . . . . . . . . . . . . . . . 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 3069 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  { y  e. 
NN0  |  ( P ^ y )  ||  x }  =  {
y  e.  NN0  | 
( P ^ y
)  ||  -u x }
)
4140supeq1d 7810 . . . . . . . . . . . . 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 2495 . . . . . . . . . . . 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 1217 . . . . . . . . . . 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 2498 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  -u x )  =  ( P  pCnt  x
) )
4544oveq1d 6218 . . . . . . . . 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 2498 . . . . . . . 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 2498 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  ( -u x  /  y ) )  =  ( P  pCnt  ( x  /  y ) ) )
4815, 47pm2.61dane 2770 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  ( -u x  / 
y ) )  =  ( P  pCnt  (
x  /  y ) ) )
499, 48eqtrd 2495 . . . . 5  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  -u ( x  / 
y ) )  =  ( P  pCnt  (
x  /  y ) ) )
50 negeq 9716 . . . . . . 7  |-  ( A  =  ( x  / 
y )  ->  -u A  =  -u ( x  / 
y ) )
5150oveq2d 6219 . . . . . 6  |-  ( A  =  ( x  / 
y )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  -u (
x  /  y ) ) )
52 oveq2 6211 . . . . . 6  |-  ( A  =  ( x  / 
y )  ->  ( P  pCnt  A )  =  ( P  pCnt  (
x  /  y ) ) )
5351, 52eqeq12d 2476 . . . . 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 2954 . . 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 1370    e. wcel 1758    =/= wne 2648   E.wrex 2800   {crab 2803   class class class wbr 4403  (class class class)co 6203   supcsup 7804   CCcc 9394   RRcr 9395   0cc0 9396    < clt 9532    - cmin 9709   -ucneg 9710    / cdiv 10107   NNcn 10436   NN0cn0 10693   ZZcz 10760   QQcq 11067   ^cexp 11985    || cdivides 13656   Primecprime 13884    pCnt cpc 14024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-q 11068  df-rp 11106  df-fl 11762  df-mod 11829  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-dvds 13657  df-gcd 13812  df-prm 13885  df-pc 14025
This theorem is referenced by:  pcabs  14062  pcadd2  14073  lgsneg  22794
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