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Theorem pcid 12925
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 10037 . 2  |-  ( A  e.  ZZ  <->  ( A  e.  NN0  \/  ( A  e.  RR  /\  -u A  e.  NN ) ) )
2 pcidlem 12924 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
3 prmnn 12761 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
43adantr 451 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  NN )
54nncnd 9762 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  CC )
6 simprl 732 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  A  e.  RR )
76recnd 8861 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  A  e.  CC )
8 nnnn0 9972 . . . . . . 7  |-  ( -u A  e.  NN  ->  -u A  e.  NN0 )
98ad2antll 709 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  -u A  e.  NN0 )
10 expneg2 11112 . . . . . 6  |-  ( ( P  e.  CC  /\  A  e.  CC  /\  -u A  e.  NN0 )  ->  ( P ^ A )  =  ( 1  /  ( P ^ -u A ) ) )
115, 7, 9, 10syl3anc 1182 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P ^ A
)  =  ( 1  /  ( P ^ -u A ) ) )
1211oveq2d 5874 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ A ) )  =  ( P  pCnt  ( 1  /  ( P ^ -u A ) ) ) )
13 simpl 443 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  Prime )
14 1z 10053 . . . . . . 7  |-  1  e.  ZZ
1514a1i 10 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
1  e.  ZZ )
16 ax-1ne0 8806 . . . . . . 7  |-  1  =/=  0
1716a1i 10 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
1  =/=  0 )
184, 9nnexpcld 11266 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P ^ -u A
)  e.  NN )
19 pcdiv 12905 . . . . . 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 1187 . . . . 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 12908 . . . . . . . 8  |-  ( P  e.  Prime  ->  ( P 
pCnt  1 )  =  0 )
2221adantr 451 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  1
)  =  0 )
23 pcidlem 12924 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  -u A  e.  NN0 )  ->  ( P  pCnt  ( P ^ -u A ) )  = 
-u A )
249, 23syldan 456 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ -u A ) )  =  -u A
)
2522, 24oveq12d 5876 . . . . . 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 9040 . . . . . . 7  |-  -u -u A  =  ( 0  - 
-u A )
277negnegd 9148 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  -u -u A  =  A
)
2826, 27syl5eqr 2329 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( 0  -  -u A
)  =  A )
2925, 28eqtrd 2315 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( ( P  pCnt  1 )  -  ( P  pCnt  ( P ^ -u A ) ) )  =  A )
3020, 29eqtrd 2315 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  (
1  /  ( P ^ -u A ) ) )  =  A )
3112, 30eqtrd 2315 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ A ) )  =  A )
322, 31jaodan 760 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  NN0  \/  ( A  e.  RR  /\  -u A  e.  NN ) ) )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
331, 32sylan2b 461 1  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   ^cexp 11104   Primecprime 12758    pCnt cpc 12889
This theorem is referenced by:  pcprmpw2  12934  pcaddlem  12936  expnprm  12950  sylow1lem1  14909  pgpfi  14916  ablfaclem3  15322  isppw2  20353  dvdsppwf1o  20426  lgsval2lem  20545  dchrisum0flblem1  20657  ostth3  20787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890
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