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Theorem pcid 14480
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 10874 . 2  |-  ( A  e.  ZZ  <->  ( A  e.  NN0  \/  ( A  e.  RR  /\  -u A  e.  NN ) ) )
2 pcidlem 14479 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
3 prmnn 14304 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
43adantr 463 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  NN )
54nncnd 10547 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  CC )
6 simprl 754 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  A  e.  RR )
76recnd 9611 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  A  e.  CC )
8 nnnn0 10798 . . . . . . 7  |-  ( -u A  e.  NN  ->  -u A  e.  NN0 )
98ad2antll 726 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  -u A  e.  NN0 )
10 expneg2 12157 . . . . . 6  |-  ( ( P  e.  CC  /\  A  e.  CC  /\  -u A  e.  NN0 )  ->  ( P ^ A )  =  ( 1  /  ( P ^ -u A ) ) )
115, 7, 9, 10syl3anc 1226 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P ^ A
)  =  ( 1  /  ( P ^ -u A ) ) )
1211oveq2d 6286 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ A ) )  =  ( P  pCnt  ( 1  /  ( P ^ -u A ) ) ) )
13 simpl 455 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  P  e.  Prime )
14 1zzd 10891 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
1  e.  ZZ )
15 ax-1ne0 9550 . . . . . . 7  |-  1  =/=  0
1615a1i 11 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
1  =/=  0 )
174, 9nnexpcld 12313 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P ^ -u A
)  e.  NN )
18 pcdiv 14460 . . . . . 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 1231 . . . . 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 14463 . . . . . . . 8  |-  ( P  e.  Prime  ->  ( P 
pCnt  1 )  =  0 )
2120adantr 463 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  1
)  =  0 )
22 pcidlem 14479 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  -u A  e.  NN0 )  ->  ( P  pCnt  ( P ^ -u A ) )  = 
-u A )
239, 22syldan 468 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  ( P ^ -u A ) )  =  -u A
)
2421, 23oveq12d 6288 . . . . . 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 9799 . . . . . . 7  |-  -u -u A  =  ( 0  - 
-u A )
267negnegd 9913 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  ->  -u -u A  =  A
)
2725, 26syl5eqr 2509 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( 0  -  -u A
)  =  A )
2824, 27eqtrd 2495 . . . . 5  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( ( P  pCnt  1 )  -  ( P  pCnt  ( P ^ -u A ) ) )  =  A )
2919, 28eqtrd 2495 . . . 4  |-  ( ( P  e.  Prime  /\  ( A  e.  RR  /\  -u A  e.  NN ) )  -> 
( P  pCnt  (
1  /  ( P ^ -u A ) ) )  =  A )
3012, 29eqtrd 2495 . . 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 473 1  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    - cmin 9796   -ucneg 9797    / cdiv 10202   NNcn 10531   NN0cn0 10791   ZZcz 10860   ^cexp 12148   Primecprime 14301    pCnt cpc 14444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-dvds 14071  df-gcd 14229  df-prm 14302  df-pc 14445
This theorem is referenced by:  pcprmpw2  14489  pcaddlem  14491  expnprm  14505  sylow1lem1  16817  pgpfi  16824  ablfaclem3  17333  isppw2  23587  dvdsppwf1o  23660  lgsval2lem  23779  dchrisum0flblem1  23891  ostth3  24021
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