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Theorem pcidlem 12862
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 109 . . . . . . . 8 |- ( ( P e. Prime /\ A e. NN0 ) -> P e. Prime )
2 prmnn 12648 . . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
31, 2syl 14 . . . . . . . . 9 |- ( ( P e. Prime /\ A e. NN0 ) -> P e. NN )
4 simpr 110 . . . . . . . . 9 |- ( ( P e. Prime /\ A e. NN0 ) -> A e. NN0 )
53, 4nnexpcld 10929 . . . . . . . 8 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ A ) e. NN )
61, 5pccld 12839 . . . . . . 7 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) e. NN0 )
76nn0red 9434 . . . . . 6 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) e. RR )
87leidd 8672 . . . . 5 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) <_ ( P pCnt ( P ^ A ) ) )
95nnzd 9579 . . . . . 6 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ A ) e. ZZ )
10 pcdvdsb 12859 . . . . . 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 1271 . . . . 5 |- ( ( P e. Prime /\ A e. NN0 ) -> ( ( P pCnt ( P ^ A ) ) <_ ( P pCnt ( P ^ A ) ) <-> ( P ^ ( P pCnt ( P ^ A ) ) ) || ( P ^ A ) ) )
128, 11mpbid 147 . . . 4 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ ( P pCnt ( P ^ A ) ) ) || ( P ^ A ) )
133, 6nnexpcld 10929 . . . . . 6 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ ( P pCnt ( P ^ A ) ) ) e. NN )
1413nnzd 9579 . . . . 5 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ ( P pCnt ( P ^ A ) ) ) e. ZZ )
15 dvdsle 12371 . . . . 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 411 . . . 4 |- ( ( P e. Prime /\ A e. NN0 ) -> ( ( P ^ ( P pCnt ( P ^ A ) ) ) || ( P ^ A ) -> ( P ^ ( P pCnt ( P ^ A ) ) ) <_ ( P ^ A ) ) )
1712, 16mpd 13 . . 3 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ ( P pCnt ( P ^ A ) ) ) <_ ( P ^ A ) )
183nnred 9134 . . . 4 |- ( ( P e. Prime /\ A e. NN0 ) -> P e. RR )
19 prmuz2 12669 . . . . 5 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
20 eluz2gt1 9809 . . . . 5 |- ( P e. ( ZZ>= ` 2 ) -> 1 < P )
211, 19, 203syl 17 . . . 4 |- ( ( P e. Prime /\ A e. NN0 ) -> 1 < P )
22 nn0leexp2 10944 . . . 4 |- ( ( ( P e. RR /\ ( P pCnt ( P ^ A ) ) e. NN0 /\ A e. NN0 ) /\ 1 < P ) -> ( ( P pCnt ( P ^ A ) ) <_ A <-> ( P ^ ( P pCnt ( P ^ A ) ) ) <_ ( P ^ A ) ) )
2318, 6, 4, 21, 22syl31anc 1274 . . 3 |- ( ( P e. Prime /\ A e. NN0 ) -> ( ( P pCnt ( P ^ A ) ) <_ A <-> ( P ^ ( P pCnt ( P ^ A ) ) ) <_ ( P ^ A ) ) )
2417, 23mpbird 167 . 2 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) <_ A )
25 iddvds 12331 . . . 4 |- ( ( P ^ A ) e. ZZ -> ( P ^ A ) || ( P ^ A ) )
269, 25syl 14 . . 3 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ A ) || ( P ^ A ) )
27 pcdvdsb 12859 . . . 4 |- ( ( P e. Prime /\ ( P ^ A ) e. ZZ /\ A e. NN0 ) -> ( A <_ ( P pCnt ( P ^ A ) ) <-> ( P ^ A ) || ( P ^ A ) ) )
281, 9, 4, 27syl3anc 1271 . . 3 |- ( ( P e. Prime /\ A e. NN0 ) -> ( A <_ ( P pCnt ( P ^ A ) ) <-> ( P ^ A ) || ( P ^ A ) ) )
2926, 28mpbird 167 . 2 |- ( ( P e. Prime /\ A e. NN0 ) -> A <_ ( P pCnt ( P ^ A ) ) )
30 nn0re 9389 . . . 4 |- ( A e. NN0 -> A e. RR )
3130adantl 277 . . 3 |- ( ( P e. Prime /\ A e. NN0 ) -> A e. RR )
327, 31letri3d 8273 . 2 |- ( ( P e. Prime /\ A e. NN0 ) -> ( ( P pCnt ( P ^ A ) ) = A <-> ( ( P pCnt ( P ^ A ) ) <_ A /\ A <_ ( P pCnt ( P ^ A ) ) ) ) )
3324, 29, 32mpbir2and 950 1 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) = A )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   RRcr 8009   1c1 8011   < clt 8192   <_ cle 8193   NNcn 9121   2c2 9172   NN0cn0 9380   ZZcz 9457   ZZ>=cuz 9733   ^cexp 10772   || cdvds 12314   Primecprime 12645   pCnt cpc 12823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130
This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-cj 11369  df-re 11370  df-im 11371  df-rsqrt 11525  df-abs 11526  df-dvds 12315  df-gcd 12491  df-prm 12646  df-pc 12824
This theorem is referenced by:  pcid  12863  pcmpt  12882  dvdsppwf1o  15679
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