Theorem pcid 12977

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 9554	. 2 |- ( A e. ZZ <-> ( A e. NN0 \/ ( A e. RR /\ -u A e. NN ) ) )
2		pcidlem 12976	. . 3 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) = A )
3		prmnn 12762	. . . . . . . 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 9216	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> P e. CC )
6	4	nnap0d 9248	. . . . . 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 8267	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> A e. CC )
9		nnnn0 9468	. . . . . . 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 10873	. . . . . 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 6044	. . . 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 9567	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> 1 e. ZZ )
16		1ne0 9270	. . . . . . 7 |- 1 =/= 0
17	16	a1i 9	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> 1 =/= 0 )
18	4, 10	nnexpcld 11020	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P ^ -u A ) e. NN )
19		pcdiv 12955	. . . . . 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 12958	. . . . . . . 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 12976	. . . . . . . 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 6046	. . . . . 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 8412	. . . . . . 7 |- -u -u A = ( 0 - -u A )
27	8	negnegd 8540	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> -u -u A = A )
28	26, 27	eqtr3id 2278	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( 0 - -u A ) = A )
29	25, 28	eqtrd 2264	. . . . 5 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( ( P pCnt 1 ) - ( P pCnt ( P ^ -u A ) ) ) = A )
30	20, 29	eqtrd 2264	. . . 4 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P pCnt ( 1 / ( P ^ -u A ) ) ) = A )
31	13, 30	eqtrd 2264	. . 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 2202   =/= wne 2403   class class class wbr 4093  (class class class)co 6028   CCcc 8090   RRcr 8091   0cc0 8092   1c1 8093   - cmin 8409   -ucneg 8410   # cap 8820   / cdiv 8911   NNcn 9202   NN0cn0 9461   ZZcz 9540   ^cexp 10863   Primecprime 12759   pCnt cpc 12937

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-sup 7243  df-inf 7244  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-fz 10306  df-fzo 10440  df-fl 10593  df-mod 10648  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-dvds 12429  df-gcd 12605  df-prm 12760  df-pc 12938

This theorem is referenced by:  pcprmpw2  12986  pcaddlem  12992  expnprm  13006  dvdsppwf1o  15803  lgsval2lem  15829