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Theorem pcidlem 14243
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 457 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  Prime )
2 prmnn 14068 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
31, 2syl 16 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  NN )
4 simpr 461 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  e.  NN0 )
53, 4nnexpcld 12286 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  e.  NN )
61, 5pccld 14222 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e. 
NN0 )
76nn0red 10842 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e.  RR )
87leidd 10108 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  <_ 
( P  pCnt  ( P ^ A ) ) )
95nnzd 10954 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  e.  ZZ )
10 pcdvdsb 14240 . . . . . 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 1223 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  (
( P  pCnt  ( P ^ A ) )  <_  ( P  pCnt  ( P ^ A ) )  <->  ( P ^
( P  pCnt  ( P ^ A ) ) )  ||  ( P ^ A ) ) )
128, 11mpbid 210 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  ||  ( P ^ A ) )
133, 6nnexpcld 12286 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  e.  NN )
1413nnzd 10954 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  e.  ZZ )
15 dvdsle 13879 . . . . 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 661 . . . 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 10540 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  RR )
196nn0zd 10953 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e.  ZZ )
20 nn0z 10876 . . . . 5  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2120adantl 466 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  e.  ZZ )
22 prmuz2 14083 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
231, 22syl 16 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  ( ZZ>= `  2 )
)
24 eluz2b1 11142 . . . . . 6  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  ZZ  /\  1  < 
P ) )
2524simprbi 464 . . . . 5  |-  ( P  e.  ( ZZ>= `  2
)  ->  1  <  P )
2623, 25syl 16 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  1  <  P )
2718, 19, 21, 26leexp2d 12295 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  (
( P  pCnt  ( P ^ A ) )  <_  A  <->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  <_  ( P ^ A ) ) )
2817, 27mpbird 232 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  <_  A )
29 iddvds 13847 . . . 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 14240 . . . 4  |-  ( ( P  e.  Prime  /\  ( P ^ A )  e.  ZZ  /\  A  e. 
NN0 )  ->  ( A  <_  ( P  pCnt  ( P ^ A ) )  <->  ( P ^ A )  ||  ( P ^ A ) ) )
321, 9, 4, 31syl3anc 1223 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( A  <_  ( P  pCnt  ( P ^ A ) )  <->  ( P ^ A )  ||  ( P ^ A ) ) )
3330, 32mpbird 232 . 2  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  <_  ( P  pCnt  ( P ^ A ) ) )
34 nn0re 10793 . . . 4  |-  ( A  e.  NN0  ->  A  e.  RR )
3534adantl 466 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  e.  RR )
367, 35letri3d 9715 . 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 915 1  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   RRcr 9480   1c1 9482    < clt 9617    <_ cle 9618   NNcn 10525   2c2 10574   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   ^cexp 12122    || cdivides 13836   Primecprime 14065    pCnt cpc 14208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-q 11172  df-rp 11210  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-dvds 13837  df-gcd 13993  df-prm 14066  df-pc 14209
This theorem is referenced by:  pcid  14244  pcmpt  14259  dvdsppwf1o  23183
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