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Theorem pcneg 12942
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 10334 . . 3  |-  ( A  e.  QQ  <->  E. x  e.  ZZ  E. y  e.  NN  A  =  ( x  /  y ) )
2 zcn 10045 . . . . . . . . 9  |-  ( x  e.  ZZ  ->  x  e.  CC )
32ad2antrl 708 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  x  e.  CC )
4 nncn 9770 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  e.  CC )
54ad2antll 709 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  y  e.  CC )
6 nnne0 9794 . . . . . . . . 9  |-  ( y  e.  NN  ->  y  =/=  0 )
76ad2antll 709 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  y  =/=  0 )
83, 5, 7divnegd 9565 . . . . . . 7  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  -u ( x  /  y )  =  ( -u x  / 
y ) )
98oveq2d 5890 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  -u ( x  / 
y ) )  =  ( P  pCnt  ( -u x  /  y ) ) )
10 neg0 9109 . . . . . . . . . 10  |-  -u 0  =  0
11 simpr 447 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  x  =  0 )
1211negeqd 9062 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  -u x  =  -u 0 )
1310, 12, 113eqtr4a 2354 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  -u x  =  x )
1413oveq1d 5889 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  ( -u x  /  y )  =  ( x  / 
y ) )
1514oveq2d 5890 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =  0 )  ->  ( P  pCnt  ( -u x  /  y ) )  =  ( P  pCnt  ( x  /  y ) ) )
16 simpll 730 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  P  e.  Prime )
17 simplrl 736 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  x  e.  ZZ )
1817znegcld 10135 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  -u x  e.  ZZ )
19 simpr 447 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  x  =/=  0 )
202negne0bd 9166 . . . . . . . . . . 11  |-  ( x  e.  ZZ  ->  (
x  =/=  0  <->  -u x  =/=  0 ) )
2117, 20syl 15 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  (
x  =/=  0  <->  -u x  =/=  0 ) )
2219, 21mpbid 201 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  -u x  =/=  0 )
23 simplrr 737 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  y  e.  NN )
24 pcdiv 12921 . . . . . . . . 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 1187 . . . . . . . 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 12921 . . . . . . . . . 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 1187 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  ( x  / 
y ) )  =  ( ( P  pCnt  x )  -  ( P 
pCnt  y ) ) )
28 eqid 2296 . . . . . . . . . . . . 13  |-  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  -u x } ,  RR ,  <  )  =  sup ( { y  e.  NN0  |  ( P ^ y )  ||  -u x } ,  RR ,  <  )
2928pczpre 12916 . . . . . . . . . . . 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 1180 . . . . . . . . . . 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 2296 . . . . . . . . . . . . . 14  |-  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  x } ,  RR ,  <  )  =  sup ( { y  e.  NN0  |  ( P ^ y )  ||  x } ,  RR ,  <  )
3231pczpre 12916 . . . . . . . . . . . . 13  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( P  pCnt  x )  =  sup ( { y  e.  NN0  |  ( P ^ y
)  ||  x } ,  RR ,  <  )
)
33 prmz 12778 . . . . . . . . . . . . . . . . . 18  |-  ( P  e.  Prime  ->  P  e.  ZZ )
34 zexpcl 11134 . . . . . . . . . . . . . . . . . 18  |-  ( ( P  e.  ZZ  /\  y  e.  NN0 )  -> 
( P ^ y
)  e.  ZZ )
3533, 34sylan 457 . . . . . . . . . . . . . . . . 17  |-  ( ( P  e.  Prime  /\  y  e.  NN0 )  ->  ( P ^ y )  e.  ZZ )
36 simpl 443 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  ZZ  /\  x  =/=  0 )  ->  x  e.  ZZ )
37 dvdsnegb 12562 . . . . . . . . . . . . . . . . 17  |-  ( ( ( P ^ y
)  e.  ZZ  /\  x  e.  ZZ )  ->  ( ( P ^
y )  ||  x  <->  ( P ^ y ) 
||  -u x ) )
3835, 36, 37syl2an 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( P  e.  Prime  /\  y  e.  NN0 )  /\  ( x  e.  ZZ  /\  x  =/=  0 ) )  ->  ( ( P ^ y )  ||  x 
<->  ( P ^ y
)  ||  -u x ) )
3938an32s 779 . . . . . . . . . . . . . . 15  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  x  =/=  0 ) )  /\  y  e. 
NN0 )  ->  (
( P ^ y
)  ||  x  <->  ( P ^ y )  ||  -u x ) )
4039rabbidva 2792 . . . . . . . . . . . . . 14  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  x  =/=  0 ) )  ->  { y  e. 
NN0  |  ( P ^ y )  ||  x }  =  {
y  e.  NN0  | 
( P ^ y
)  ||  -u x }
)
4140supeq1d 7215 . . . . . . . . . . . . 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 2328 . . . . . . . . . . . 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 1180 . . . . . . . . . . 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 2331 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  -u x )  =  ( P  pCnt  x
) )
4544oveq1d 5889 . . . . . . . . 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 2331 . . . . . . . 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 2331 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  ( x  e.  ZZ  /\  y  e.  NN ) )  /\  x  =/=  0 )  ->  ( P  pCnt  ( -u x  /  y ) )  =  ( P  pCnt  ( x  /  y ) ) )
4815, 47pm2.61dane 2537 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  ( -u x  / 
y ) )  =  ( P  pCnt  (
x  /  y ) ) )
499, 48eqtrd 2328 . . . . 5  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( P  pCnt  -u ( x  / 
y ) )  =  ( P  pCnt  (
x  /  y ) ) )
50 negeq 9060 . . . . . . 7  |-  ( A  =  ( x  / 
y )  ->  -u A  =  -u ( x  / 
y ) )
5150oveq2d 5890 . . . . . 6  |-  ( A  =  ( x  / 
y )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  -u (
x  /  y ) ) )
52 oveq2 5882 . . . . . 6  |-  ( A  =  ( x  / 
y )  ->  ( P  pCnt  A )  =  ( P  pCnt  (
x  /  y ) ) )
5351, 52eqeq12d 2310 . . . . 5  |-  ( A  =  ( x  / 
y )  ->  (
( P  pCnt  -u A
)  =  ( P 
pCnt  A )  <->  ( P  pCnt  -u ( x  / 
y ) )  =  ( P  pCnt  (
x  /  y ) ) ) )
5449, 53syl5ibrcom 213 . . . 4  |-  ( ( P  e.  Prime  /\  (
x  e.  ZZ  /\  y  e.  NN )
)  ->  ( A  =  ( x  / 
y )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  A
) ) )
5554rexlimdvva 2687 . . 3  |-  ( P  e.  Prime  ->  ( E. x  e.  ZZ  E. y  e.  NN  A  =  ( x  / 
y )  ->  ( P  pCnt  -u A )  =  ( P  pCnt  A
) ) )
561, 55syl5bi 208 . 2  |-  ( P  e.  Prime  ->  ( A  e.  QQ  ->  ( P  pCnt  -u A )  =  ( P  pCnt  A
) ) )
5756imp 418 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560   class class class wbr 4039  (class class class)co 5874   supcsup 7209   CCcc 8751   RRcr 8752   0cc0 8753    < clt 8883    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   NN0cn0 9981   ZZcz 10040   QQcq 10332   ^cexp 11120    || cdivides 12547   Primecprime 12774    pCnt cpc 12905
This theorem is referenced by:  pcabs  12943  pcadd2  12954  lgsneg  20574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906
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