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Theorem pcidlem 12940
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 12777 . . . . . . . . . 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 11282 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  e.  NN )
61, 5pccld 12919 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e. 
NN0 )
76nn0red 10035 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e.  RR )
87leidd 9355 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  <_ 
( P  pCnt  ( P ^ A ) ) )
95nnzd 10132 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  e.  ZZ )
10 pcdvdsb 12937 . . . . . 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 11282 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  e.  NN )
1413nnzd 10132 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  e.  ZZ )
15 dvdsle 12590 . . . . 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 9777 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  RR )
196nn0zd 10131 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e.  ZZ )
20 nn0z 10062 . . . . 5  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2120adantl 452 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  e.  ZZ )
22 prmuz2 12792 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
231, 22syl 15 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  ( ZZ>= `  2 )
)
24 eluz2b1 10305 . . . . . 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 11291 . . 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 12558 . . . 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 12937 . . . 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 9990 . . . 4  |-  ( A  e.  NN0  ->  A  e.  RR )
3534adantl 452 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  e.  RR )
367, 35letri3d 8977 . 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   1c1 8754    < clt 8883    <_ cle 8884   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ^cexp 11120    || cdivides 12547   Primecprime 12774    pCnt cpc 12905
This theorem is referenced by:  pcid  12941  pcmpt  12956  dvdsppwf1o  20442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906
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