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Theorem pcid 12921
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 10034 . 2  |-  ( A  e.  ZZ  <->  ( A  e.  NN0  \/  ( A  e.  RR  /\  -u A  e.  NN ) ) )
2 pcidlem 12920 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
3 prmnn 12757 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
43adantr 453 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  NN )
54nncnd 9759 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  CC )
6 simprl 734 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  A  e.  RR )
76recnd 8858 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  A  e.  CC )
8 nnnn0 9969 . . . . . . 7  |-  ( -u A  e.  NN  ->  -u A  e.  NN0 )
98ad2antll 711 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  -u A  e.  NN0 )
10 expneg2 11108 . . . . . 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 5837 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ A ) )  =  ( P  pCnt  ( 1  /  ( P ^ -u A ) ) ) )
13 simpl 445 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  Prime )
14 1z 10050 . . . . . . 7  |-  1  e.  ZZ
1514a1i 12 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
1  e.  ZZ )
16 ax-1ne0 8803 . . . . . . 7  |-  1  =/=  0
1716a1i 12 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
1  =/=  0 )
184, 9nnexpcld 11262 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P ^ -u A
)  e.  NN )
19 pcdiv 12901 . . . . . 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 12904 . . . . . . . 8  |-  ( P  e.  Prime  ->  ( P 
pCnt  1 )  =  0 )
2221adantr 453 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  1
)  =  0 )
23 pcidlem 12920 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  -u A  e.  NN0 )  ->  ( P  pCnt  ( P ^ -u A ) )  = 
-u A )
249, 23syldan 458 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ -u A ) )  =  -u A
)
2522, 24oveq12d 5839 . . . . . 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 9037 . . . . . . 7  |-  -u -u A  =  ( 0  - 
-u A )
277negnegd 9145 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  -u -u A  =  A
)
2826, 27syl5eqr 2332 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( 0  -  -u A
)  =  A )
2925, 28eqtrd 2318 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( ( P  pCnt  1 )  -  ( P  pCnt  ( P ^ -u A ) ) )  =  A )
3020, 29eqtrd 2318 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  (
1  /  ( P ^ -u A ) ) )  =  A )
3112, 30eqtrd 2318 . . 3  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ A ) )  =  A )
322, 31jaodan 762 . 2  |-  ( ( P  e.  Prime  /\  ( A  e.  NN0  \/  ( A  e.  RR  /\  -u A  e.  NN ) ) )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
331, 32sylan2b 463 1  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    = wceq 1625    e. wcel 1687    =/= wne 2449  (class class class)co 5821   CCcc 8732   RRcr 8733   0cc0 8734   1c1 8735    - cmin 9034   -ucneg 9035    / cdiv 9420   NNcn 9743   NN0cn0 9962   ZZcz 10021   ^cexp 11100   Primecprime 12754    pCnt cpc 12885
This theorem is referenced by:  pcprmpw2  12930  pcaddlem  12932  expnprm  12946  sylow1lem1  14905  pgpfi  14912  ablfaclem3  15318  isppw2  20349  dvdsppwf1o  20422  lgsval2lem  20541  dchrisum0flblem1  20653  ostth3  20783
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-cnex 8790  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-mulcom 8798  ax-addass 8799  ax-mulass 8800  ax-distr 8801  ax-i2m1 8802  ax-1ne0 8803  ax-1rid 8804  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-lttri 8808  ax-pre-lttrn 8809  ax-pre-ltadd 8810  ax-pre-mulgt0 8811  ax-pre-sup 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-reu 2553  df-rmo 2554  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-1st 6085  df-2nd 6086  df-iota 6254  df-riota 6301  df-recs 6385  df-rdg 6420  df-1o 6476  df-2o 6477  df-oadd 6480  df-er 6657  df-en 6861  df-dom 6862  df-sdom 6863  df-fin 6864  df-sup 7191  df-pnf 8866  df-mnf 8867  df-xr 8868  df-ltxr 8869  df-le 8870  df-sub 9036  df-neg 9037  df-div 9421  df-nn 9744  df-2 9801  df-3 9802  df-n0 9963  df-z 10022  df-uz 10228  df-q 10314  df-rp 10352  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-dvds 12528  df-gcd 12682  df-prm 12755  df-pc 12886
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