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Theorem pcid 13939
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 10660 . 2  |-  ( A  e.  ZZ  <->  ( A  e.  NN0  \/  ( A  e.  RR  /\  -u A  e.  NN ) ) )
2 pcidlem 13938 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
3 prmnn 13766 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
43adantr 465 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  NN )
54nncnd 10338 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  CC )
6 simprl 755 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  A  e.  RR )
76recnd 9412 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  A  e.  CC )
8 nnnn0 10586 . . . . . . 7  |-  ( -u A  e.  NN  ->  -u A  e.  NN0 )
98ad2antll 728 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  -u A  e.  NN0 )
10 expneg2 11874 . . . . . 6  |-  ( ( P  e.  CC  /\  A  e.  CC  /\  -u A  e.  NN0 )  ->  ( P ^ A )  =  ( 1  /  ( P ^ -u A ) ) )
115, 7, 9, 10syl3anc 1218 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P ^ A
)  =  ( 1  /  ( P ^ -u A ) ) )
1211oveq2d 6107 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ A ) )  =  ( P  pCnt  ( 1  /  ( P ^ -u A ) ) ) )
13 simpl 457 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  Prime )
14 1zzd 10677 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
1  e.  ZZ )
15 ax-1ne0 9351 . . . . . . 7  |-  1  =/=  0
1615a1i 11 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
1  =/=  0 )
174, 9nnexpcld 12029 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P ^ -u A
)  e.  NN )
18 pcdiv 13919 . . . . . 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 ) ) ) )
1913, 14, 16, 17, 18syl121anc 1223 . . . . 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 ) ) ) )
20 pc1 13922 . . . . . . . 8  |-  ( P  e.  Prime  ->  ( P 
pCnt  1 )  =  0 )
2120adantr 465 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  1
)  =  0 )
22 pcidlem 13938 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  -u A  e.  NN0 )  ->  ( P  pCnt  ( P ^ -u A ) )  = 
-u A )
239, 22syldan 470 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ -u A ) )  =  -u A
)
2421, 23oveq12d 6109 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( ( P  pCnt  1 )  -  ( P  pCnt  ( P ^ -u A ) ) )  =  ( 0  - 
-u A ) )
25 df-neg 9598 . . . . . . 7  |-  -u -u A  =  ( 0  - 
-u A )
267negnegd 9710 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  -u -u A  =  A
)
2725, 26syl5eqr 2489 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( 0  -  -u A
)  =  A )
2824, 27eqtrd 2475 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( ( P  pCnt  1 )  -  ( P  pCnt  ( P ^ -u A ) ) )  =  A )
2919, 28eqtrd 2475 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  (
1  /  ( P ^ -u A ) ) )  =  A )
3012, 29eqtrd 2475 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ A ) )  =  A )
312, 30jaodan 783 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  NN0  \/  ( A  e.  RR  /\  -u A  e.  NN ) ) )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
321, 31sylan2b 475 1  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    - cmin 9595   -ucneg 9596    / cdiv 9993   NNcn 10322   NN0cn0 10579   ZZcz 10646   ^cexp 11865   Primecprime 13763    pCnt cpc 13903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-dvds 13536  df-gcd 13691  df-prm 13764  df-pc 13904
This theorem is referenced by:  pcprmpw2  13948  pcaddlem  13950  expnprm  13964  sylow1lem1  16097  pgpfi  16104  ablfaclem3  16588  isppw2  22453  dvdsppwf1o  22526  lgsval2lem  22645  dchrisum0flblem1  22757  ostth3  22887
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