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Theorem pcid 13246
Description: The prime count of a prime power. (Contributed by Mario Carneiro, 9-Sep-2014.)
Assertion
Ref Expression
pcid  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( P ^ A ) )  =  A )

Proof of Theorem pcid
StepHypRef Expression
1 elznn0nn 10295 . 2  |-  ( A  e.  ZZ  <->  ( A  e.  NN0  \/  ( A  e.  RR  /\  -u A  e.  NN ) ) )
2 pcidlem 13245 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
3 prmnn 13082 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
43adantr 452 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  NN )
54nncnd 10016 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  CC )
6 simprl 733 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  A  e.  RR )
76recnd 9114 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  A  e.  CC )
8 nnnn0 10228 . . . . . . 7  |-  ( -u A  e.  NN  ->  -u A  e.  NN0 )
98ad2antll 710 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  -u A  e.  NN0 )
10 expneg2 11390 . . . . . 6  |-  ( ( P  e.  CC  /\  A  e.  CC  /\  -u A  e.  NN0 )  ->  ( P ^ A )  =  ( 1  /  ( P ^ -u A ) ) )
115, 7, 9, 10syl3anc 1184 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P ^ A
)  =  ( 1  /  ( P ^ -u A ) ) )
1211oveq2d 6097 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ A ) )  =  ( P  pCnt  ( 1  /  ( P ^ -u A ) ) ) )
13 simpl 444 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  Prime )
14 1z 10311 . . . . . . 7  |-  1  e.  ZZ
1514a1i 11 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
1  e.  ZZ )
16 ax-1ne0 9059 . . . . . . 7  |-  1  =/=  0
1716a1i 11 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
1  =/=  0 )
184, 9nnexpcld 11544 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P ^ -u A
)  e.  NN )
19 pcdiv 13226 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
1  e.  ZZ  /\  1  =/=  0 )  /\  ( P ^ -u A
)  e.  NN )  ->  ( P  pCnt  ( 1  /  ( P ^ -u A ) ) )  =  ( ( P  pCnt  1
)  -  ( P 
pCnt  ( P ^ -u A ) ) ) )
2013, 15, 17, 18, 19syl121anc 1189 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  (
1  /  ( P ^ -u A ) ) )  =  ( ( P  pCnt  1
)  -  ( P 
pCnt  ( P ^ -u A ) ) ) )
21 pc1 13229 . . . . . . . 8  |-  ( P  e.  Prime  ->  ( P 
pCnt  1 )  =  0 )
2221adantr 452 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  1
)  =  0 )
23 pcidlem 13245 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  -u A  e.  NN0 )  ->  ( P  pCnt  ( P ^ -u A ) )  = 
-u A )
249, 23syldan 457 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ -u A ) )  =  -u A
)
2522, 24oveq12d 6099 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( ( P  pCnt  1 )  -  ( P  pCnt  ( P ^ -u A ) ) )  =  ( 0  - 
-u A ) )
26 df-neg 9294 . . . . . . 7  |-  -u -u A  =  ( 0  - 
-u A )
277negnegd 9402 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  -u -u A  =  A
)
2826, 27syl5eqr 2482 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( 0  -  -u A
)  =  A )
2925, 28eqtrd 2468 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( ( P  pCnt  1 )  -  ( P  pCnt  ( P ^ -u A ) ) )  =  A )
3020, 29eqtrd 2468 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  (
1  /  ( P ^ -u A ) ) )  =  A )
3112, 30eqtrd 2468 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ A ) )  =  A )
322, 31jaodan 761 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  NN0  \/  ( A  e.  RR  /\  -u A  e.  NN ) ) )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
331, 32sylan2b 462 1  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    - cmin 9291   -ucneg 9292    / cdiv 9677   NNcn 10000   NN0cn0 10221   ZZcz 10282   ^cexp 11382   Primecprime 13079    pCnt cpc 13210
This theorem is referenced by:  pcprmpw2  13255  pcaddlem  13257  expnprm  13271  sylow1lem1  15232  pgpfi  15239  ablfaclem3  15645  isppw2  20898  dvdsppwf1o  20971  lgsval2lem  21090  dchrisum0flblem1  21202  ostth3  21332
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080  df-pc 13211
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