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Theorem pcidlem 13025
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 12811 . . . . . . . . . 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 11061 . . . . . . . 8 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ A ) e. NN )
61, 5pccld 13002 . . . . . . 7 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) e. NN0 )
76nn0red 9556 . . . . . 6 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) e. RR )
87leidd 8790 . . . . 5 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) <_ ( P pCnt ( P ^ A ) ) )
95nnzd 9702 . . . . . 6 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ A ) e. ZZ )
10 pcdvdsb 13022 . . . . . 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 1274 . . . . 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 11061 . . . . . 6 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ ( P pCnt ( P ^ A ) ) ) e. NN )
1413nnzd 9702 . . . . 5 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P ^ ( P pCnt ( P ^ A ) ) ) e. ZZ )
15 dvdsle 12534 . . . . 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 9252 . . . 4 |- ( ( P e. Prime /\ A e. NN0 ) -> P e. RR )
19 prmuz2 12832 . . . . 5 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
20 eluz2gt1 9937 . . . . 5 |- ( P e. ( ZZ>= ` 2 ) -> 1 < P )
211, 19, 203syl 17 . . . 4 |- ( ( P e. Prime /\ A e. NN0 ) -> 1 < P )
22 nn0leexp2 11076 . . . 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 1277 . . 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 12494 . . . 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 13022 . . . 4 |- ( ( P e. Prime /\ ( P ^ A ) e. ZZ /\ A e. NN0 ) -> ( A <_ ( P pCnt ( P ^ A ) ) <-> ( P ^ A ) || ( P ^ A ) ) )
281, 9, 4, 27syl3anc 1274 . . 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 9507 . . . 4 |- ( A e. NN0 -> A e. RR )
3130adantl 277 . . 3 |- ( ( P e. Prime /\ A e. NN0 ) -> A e. RR )
327, 31letri3d 8391 . 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 953 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 1398   e. wcel 2205   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   RRcr 8128   1c1 8130   < clt 8310   <_ cle 8311   NNcn 9239   2c2 9290   NN0cn0 9498   ZZcz 9579   ZZ>=cuz 9856   ^cexp 10904   || cdvds 12477   Primecprime 12808   pCnt cpc 12986
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248  ax-caucvg 8249
This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-2o 6650  df-er 6769  df-en 6978  df-sup 7277  df-inf 7278  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-fz 10346  df-fzo 10481  df-fl 10634  df-mod 10689  df-seqfrec 10814  df-exp 10905  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-dvds 12478  df-gcd 12654  df-prm 12809  df-pc 12987
This theorem is referenced by:  pcid  13026  pcmpt  13045  dvdsppwf1o  15874
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