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Theorem pcidlem 12924
Description: The prime count of a prime power. (Contributed by Mario Carneiro, 12-Mar-2014.)
Assertion
Ref Expression
pcidlem  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  =  A )

Proof of Theorem pcidlem
StepHypRef Expression
1 simpl 443 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  Prime )
2 prmnn 12761 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
31, 2syl 15 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  NN )
4 simpr 447 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  e.  NN0 )
53, 4nnexpcld 11266 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  e.  NN )
61, 5pccld 12903 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e. 
NN0 )
76nn0red 10019 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e.  RR )
87leidd 9339 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  <_ 
( P  pCnt  ( P ^ A ) ) )
95nnzd 10116 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  e.  ZZ )
10 pcdvdsb 12921 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( P ^ A )  e.  ZZ  /\  ( P 
pCnt  ( P ^ A ) )  e. 
NN0 )  ->  (
( P  pCnt  ( P ^ A ) )  <_  ( P  pCnt  ( P ^ A ) )  <->  ( P ^
( P  pCnt  ( P ^ A ) ) )  ||  ( P ^ A ) ) )
111, 9, 6, 10syl3anc 1182 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  (
( P  pCnt  ( P ^ A ) )  <_  ( P  pCnt  ( P ^ A ) )  <->  ( P ^
( P  pCnt  ( P ^ A ) ) )  ||  ( P ^ A ) ) )
128, 11mpbid 201 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  ||  ( P ^ A ) )
133, 6nnexpcld 11266 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  e.  NN )
1413nnzd 10116 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  e.  ZZ )
15 dvdsle 12574 . . . . 5  |-  ( ( ( P ^ ( P  pCnt  ( P ^ A ) ) )  e.  ZZ  /\  ( P ^ A )  e.  NN )  ->  (
( P ^ ( P  pCnt  ( P ^ A ) ) ) 
||  ( P ^ A )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  <_  ( P ^ A ) ) )
1614, 5, 15syl2anc 642 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  (
( P ^ ( P  pCnt  ( P ^ A ) ) ) 
||  ( P ^ A )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  <_  ( P ^ A ) ) )
1712, 16mpd 14 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  <_  ( P ^ A ) )
183nnred 9761 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  RR )
196nn0zd 10115 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e.  ZZ )
20 nn0z 10046 . . . . 5  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2120adantl 452 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  e.  ZZ )
22 prmuz2 12776 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
231, 22syl 15 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  ( ZZ>= `  2 )
)
24 eluz2b1 10289 . . . . . 6  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  ZZ  /\  1  < 
P ) )
2524simprbi 450 . . . . 5  |-  ( P  e.  ( ZZ>= `  2
)  ->  1  <  P )
2623, 25syl 15 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  1  <  P )
2718, 19, 21, 26leexp2d 11275 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  (
( P  pCnt  ( P ^ A ) )  <_  A  <->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  <_  ( P ^ A ) ) )
2817, 27mpbird 223 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  <_  A )
29 iddvds 12542 . . . 4  |-  ( ( P ^ A )  e.  ZZ  ->  ( P ^ A )  ||  ( P ^ A ) )
309, 29syl 15 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  ||  ( P ^ A ) )
31 pcdvdsb 12921 . . . 4  |-  ( ( P  e.  Prime  /\  ( P ^ A )  e.  ZZ  /\  A  e. 
NN0 )  ->  ( A  <_  ( P  pCnt  ( P ^ A ) )  <->  ( P ^ A )  ||  ( P ^ A ) ) )
321, 9, 4, 31syl3anc 1182 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( A  <_  ( P  pCnt  ( P ^ A ) )  <->  ( P ^ A )  ||  ( P ^ A ) ) )
3330, 32mpbird 223 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  <_  ( P  pCnt  ( P ^ A ) ) )
34 nn0re 9974 . . . 4  |-  ( A  e.  NN0  ->  A  e.  RR )
3534adantl 452 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  e.  RR )
367, 35letri3d 8961 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  (
( P  pCnt  ( P ^ A ) )  =  A  <->  ( ( P  pCnt  ( P ^ A ) )  <_  A  /\  A  <_  ( P  pCnt  ( P ^ A ) ) ) ) )
3728, 33, 36mpbir2and 888 1  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   1c1 8738    < clt 8867    <_ cle 8868   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ^cexp 11104    || cdivides 12531   Primecprime 12758    pCnt cpc 12889
This theorem is referenced by:  pcid  12925  pcmpt  12940  dvdsppwf1o  20426
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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