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Theorem pcidlem 14049
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 13877 . . . . . . . . . 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 12139 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  e.  NN )
61, 5pccld 14028 . . . . . . 7  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e. 
NN0 )
76nn0red 10741 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e.  RR )
87leidd 10010 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  <_ 
( P  pCnt  ( P ^ A ) ) )
95nnzd 10850 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ A )  e.  ZZ )
10 pcdvdsb 14046 . . . . . 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 1219 . . . . 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 12139 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  e.  NN )
1413nnzd 10850 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P ^ ( P  pCnt  ( P ^ A ) ) )  e.  ZZ )
15 dvdsle 13689 . . . . 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 10441 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  RR )
196nn0zd 10849 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  ( P  pCnt  ( P ^ A ) )  e.  ZZ )
20 nn0z 10773 . . . . 5  |-  ( A  e.  NN0  ->  A  e.  ZZ )
2120adantl 466 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  e.  ZZ )
22 prmuz2 13892 . . . . . 6  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
231, 22syl 16 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  P  e.  ( ZZ>= `  2 )
)
24 eluz2b1 11030 . . . . . 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 12148 . . 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 13657 . . . 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 14046 . . . 4  |-  ( ( P  e.  Prime  /\  ( P ^ A )  e.  ZZ  /\  A  e. 
NN0 )  ->  ( A  <_  ( P  pCnt  ( P ^ A ) )  <->  ( P ^ A )  ||  ( P ^ A ) ) )
321, 9, 4, 31syl3anc 1219 . . 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 10692 . . . 4  |-  ( A  e.  NN0  ->  A  e.  RR )
3534adantl 466 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  NN0 )  ->  A  e.  RR )
367, 35letri3d 9620 . 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 913 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 1370    e. wcel 1758   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   RRcr 9385   1c1 9387    < clt 9522    <_ cle 9523   NNcn 10426   2c2 10475   NN0cn0 10683   ZZcz 10750   ZZ>=cuz 10965   ^cexp 11975    || cdivides 13646   Primecprime 13874    pCnt cpc 14014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-q 11058  df-rp 11096  df-fl 11752  df-mod 11819  df-seq 11917  df-exp 11976  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-dvds 13647  df-gcd 13802  df-prm 13875  df-pc 14015
This theorem is referenced by:  pcid  14050  pcmpt  14065  dvdsppwf1o  22652
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