Theorem pcid 13022

Description: The prime count of a prime power. (Contributed by Mario Carneiro, 9-Sep-2014.)

Assertion

Ref	Expression
pcid	|- ( ( P e. Prime /\ A e. ZZ ) -> ( P pCnt ( P ^ A ) ) = A )

Proof of Theorem pcid

Step	Hyp	Ref	Expression
1		elznn0nn 9591	. 2 |- ( A e. ZZ <-> ( A e. NN0 \/ ( A e. RR /\ -u A e. NN ) ) )
2		pcidlem 13021	. . 3 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) = A )
3		prmnn 12807	. . . . . . . 8 |- ( P e. Prime -> P e. NN )
4	3	adantr 276	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> P e. NN )
5	4	nncnd 9251	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> P e. CC )
6	4	nnap0d 9283	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> P # 0 )
7		simprl 531	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> A e. RR )
8	7	recnd 8302	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> A e. CC )
9		nnnn0 9503	. . . . . . 7 |- ( -u A e. NN -> -u A e. NN0 )
10	9	ad2antll 491	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> -u A e. NN0 )
11		expineg2 10910	. . . . . 6 |- ( ( ( P e. CC /\ P # 0 ) /\ ( A e. CC /\ -u A e. NN0 ) ) -> ( P ^ A ) = ( 1 / ( P ^ -u A ) ) )
12	5, 6, 8, 10, 11	syl22anc 1275	. . . . 5 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P ^ A ) = ( 1 / ( P ^ -u A ) ) )
13	12	oveq2d 6066	. . . 4 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P pCnt ( P ^ A ) ) = ( P pCnt ( 1 / ( P ^ -u A ) ) ) )
14		simpl 109	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> P e. Prime )
15		1zzd 9604	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> 1 e. ZZ )
16		1ne0 9305	. . . . . . 7 |- 1 =/= 0
17	16	a1i 9	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> 1 =/= 0 )
18	4, 10	nnexpcld 11057	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P ^ -u A ) e. NN )
19		pcdiv 13000	. . . . . 6 |- ( ( P e. Prime /\ ( 1 e. ZZ /\ 1 =/= 0 ) /\ ( P ^ -u A ) e. NN ) -> ( P pCnt ( 1 / ( P ^ -u A ) ) ) = ( ( P pCnt 1 ) - ( P pCnt ( P ^ -u A ) ) ) )
20	14, 15, 17, 18, 19	syl121anc 1279	. . . . 5 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P pCnt ( 1 / ( P ^ -u A ) ) ) = ( ( P pCnt 1 ) - ( P pCnt ( P ^ -u A ) ) ) )
21		pc1 13003	. . . . . . . 8 |- ( P e. Prime -> ( P pCnt 1 ) = 0 )
22	21	adantr 276	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P pCnt 1 ) = 0 )
23		pcidlem 13021	. . . . . . . 8 |- ( ( P e. Prime /\ -u A e. NN0 ) -> ( P pCnt ( P ^ -u A ) ) = -u A )
24	10, 23	syldan 282	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P pCnt ( P ^ -u A ) ) = -u A )
25	22, 24	oveq12d 6068	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( ( P pCnt 1 ) - ( P pCnt ( P ^ -u A ) ) ) = ( 0 - -u A ) )
26		df-neg 8447	. . . . . . 7 |- -u -u A = ( 0 - -u A )
27	8	negnegd 8575	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> -u -u A = A )
28	26, 27	eqtr3id 2279	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( 0 - -u A ) = A )
29	25, 28	eqtrd 2265	. . . . 5 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( ( P pCnt 1 ) - ( P pCnt ( P ^ -u A ) ) ) = A )
30	20, 29	eqtrd 2265	. . . 4 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P pCnt ( 1 / ( P ^ -u A ) ) ) = A )
31	13, 30	eqtrd 2265	. . 3 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P pCnt ( P ^ A ) ) = A )
32	2, 31	jaodan 805	. 2 |- ( ( P e. Prime /\ ( A e. NN0 \/ ( A e. RR /\ -u A e. NN ) ) ) -> ( P pCnt ( P ^ A ) ) = A )
33	1, 32	sylan2b 287	1 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P pCnt ( P ^ A ) ) = A )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2203   =/= wne 2412   class class class wbr 4109  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   1c1 8128   - cmin 8444   -ucneg 8445   # cap 8855   / cdiv 8946   NNcn 9237   NN0cn0 9496   ZZcz 9577   ^cexp 10900   Primecprime 12804   pCnt cpc 12982

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-2o 6648  df-er 6767  df-en 6976  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-dvds 12474  df-gcd 12650  df-prm 12805  df-pc 12983

This theorem is referenced by:  pcprmpw2  13031  pcaddlem  13037  expnprm  13051  dvdsppwf1o  15857  lgsval2lem  15883