Theorem pcid 12341

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 9285	. 2 |- ( A e. ZZ <-> ( A e. NN0 \/ ( A e. RR /\ -u A e. NN ) ) )
2		pcidlem 12340	. . 3 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) = A )
3		prmnn 12128	. . . . . . . 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 8951	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> P e. CC )
6	4	nnap0d 8983	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> P # 0 )
7		simprl 529	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> A e. RR )
8	7	recnd 8004	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> A e. CC )
9		nnnn0 9201	. . . . . . 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 10547	. . . . . 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 1250	. . . . 5 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P ^ A ) = ( 1 / ( P ^ -u A ) ) )
13	12	oveq2d 5907	. . . 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 9298	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> 1 e. ZZ )
16		1ne0 9005	. . . . . . 7 |- 1 =/= 0
17	16	a1i 9	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> 1 =/= 0 )
18	4, 10	nnexpcld 10694	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P ^ -u A ) e. NN )
19		pcdiv 12320	. . . . . 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 1254	. . . . 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 12323	. . . . . . . 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 12340	. . . . . . . 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 5909	. . . . . 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 8149	. . . . . . 7 |- -u -u A = ( 0 - -u A )
27	8	negnegd 8277	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> -u -u A = A )
28	26, 27	eqtr3id 2236	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( 0 - -u A ) = A )
29	25, 28	eqtrd 2222	. . . . 5 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( ( P pCnt 1 ) - ( P pCnt ( P ^ -u A ) ) ) = A )
30	20, 29	eqtrd 2222	. . . 4 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P pCnt ( 1 / ( P ^ -u A ) ) ) = A )
31	13, 30	eqtrd 2222	. . 3 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P pCnt ( P ^ A ) ) = A )
32	2, 31	jaodan 798	. 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 709   = wceq 1364   e. wcel 2160   =/= wne 2360   class class class wbr 4018  (class class class)co 5891   CCcc 7827   RRcr 7828   0cc0 7829   1c1 7830   - cmin 8146   -ucneg 8147   # cap 8556   / cdiv 8647   NNcn 8937   NN0cn0 9194   ZZcz 9271   ^cexp 10537   Primecprime 12125   pCnt cpc 12302

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947  ax-arch 7948  ax-caucvg 7949

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-isom 5240  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-1o 6435  df-2o 6436  df-er 6553  df-en 6759  df-sup 7001  df-inf 7002  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648  df-inn 8938  df-2 8996  df-3 8997  df-4 8998  df-n0 9195  df-z 9272  df-uz 9547  df-q 9638  df-rp 9672  df-fz 10027  df-fzo 10161  df-fl 10288  df-mod 10341  df-seqfrec 10464  df-exp 10538  df-cj 10869  df-re 10870  df-im 10871  df-rsqrt 11025  df-abs 11026  df-dvds 11813  df-gcd 11962  df-prm 12126  df-pc 12303

This theorem is referenced by:  pcprmpw2  12350  pcaddlem  12356  expnprm  12369  lgsval2lem  14808