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Theorem pcidlem 13245
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 444 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  Prime )
2 prmnn 13082 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
31, 2syl 16 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  NN )
4 simpr 448 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  e.  NN0 )
53, 4nnexpcld 11544 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  e.  NN )
61, 5pccld 13224 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e. 
NN0 )
76nn0red 10275 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e.  RR )
87leidd 9593 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  <_ 
( P  pCnt  ( P ^ A ) ) )
95nnzd 10374 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  e.  ZZ )
10 pcdvdsb 13242 . . . . . 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 1184 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  (
( P  pCnt  ( P ^ A ) )  <_  ( P  pCnt  ( P ^ A ) )  <->  ( P ^
( P  pCnt  ( P ^ A ) ) )  ||  ( P ^ A ) ) )
128, 11mpbid 202 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  ||  ( P ^ A ) )
133, 6nnexpcld 11544 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  e.  NN )
1413nnzd 10374 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  e.  ZZ )
15 dvdsle 12895 . . . . 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 643 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  (
( P ^ ( P  pCnt  ( P ^ A ) ) ) 
||  ( P ^ A )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  <_  ( P ^ A ) ) )
1712, 16mpd 15 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  <_  ( P ^ A ) )
183nnred 10015 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  RR )
196nn0zd 10373 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e.  ZZ )
20 nn0z 10304 . . . . 5  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2120adantl 453 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  e.  ZZ )
22 prmuz2 13097 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
231, 22syl 16 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  ( ZZ>= `  2 )
)
24 eluz2b1 10547 . . . . . 6  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  ZZ  /\  1  < 
P ) )
2524simprbi 451 . . . . 5  |-  ( P  e.  ( ZZ>= `  2
)  ->  1  <  P )
2623, 25syl 16 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  1  <  P )
2718, 19, 21, 26leexp2d 11553 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  (
( P  pCnt  ( P ^ A ) )  <_  A  <->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  <_  ( P ^ A ) ) )
2817, 27mpbird 224 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  <_  A )
29 iddvds 12863 . . . 4  |-  ( ( P ^ A )  e.  ZZ  ->  ( P ^ A )  ||  ( P ^ A ) )
309, 29syl 16 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  ||  ( P ^ A ) )
31 pcdvdsb 13242 . . . 4  |-  ( ( P  e.  Prime  /\  ( P ^ A )  e.  ZZ  /\  A  e. 
NN0 )  ->  ( A  <_  ( P  pCnt  ( P ^ A ) )  <->  ( P ^ A )  ||  ( P ^ A ) ) )
321, 9, 4, 31syl3anc 1184 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( A  <_  ( P  pCnt  ( P ^ A ) )  <->  ( P ^ A )  ||  ( P ^ A ) ) )
3330, 32mpbird 224 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  <_  ( P  pCnt  ( P ^ A ) ) )
34 nn0re 10230 . . . 4  |-  ( A  e.  NN0  ->  A  e.  RR )
3534adantl 453 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  e.  RR )
367, 35letri3d 9215 . 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 889 1  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   RRcr 8989   1c1 8991    < clt 9120    <_ cle 9121   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ^cexp 11382    || cdivides 12852   Primecprime 13079    pCnt cpc 13210
This theorem is referenced by:  pcid  13246  pcmpt  13261  dvdsppwf1o  20971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080  df-pc 13211
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