Theorem pcid 12762

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 9421	. 2 |- ( A e. ZZ <-> ( A e. NN0 \/ ( A e. RR /\ -u A e. NN ) ) )
2		pcidlem 12761	. . 3 |- ( ( P e. Prime /\ A e. NN0 ) -> ( P pCnt ( P ^ A ) ) = A )
3		prmnn 12547	. . . . . . . 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 9085	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> P e. CC )
6	4	nnap0d 9117	. . . . . 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 8136	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> A e. CC )
9		nnnn0 9337	. . . . . . 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 10730	. . . . . 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 1251	. . . . 5 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P ^ A ) = ( 1 / ( P ^ -u A ) ) )
13	12	oveq2d 5983	. . . 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 9434	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> 1 e. ZZ )
16		1ne0 9139	. . . . . . 7 |- 1 =/= 0
17	16	a1i 9	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> 1 =/= 0 )
18	4, 10	nnexpcld 10877	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P ^ -u A ) e. NN )
19		pcdiv 12740	. . . . . 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 1255	. . . . 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 12743	. . . . . . . 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 12761	. . . . . . . 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 5985	. . . . . 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 8281	. . . . . . 7 |- -u -u A = ( 0 - -u A )
27	8	negnegd 8409	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> -u -u A = A )
28	26, 27	eqtr3id 2254	. . . . . 6 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( 0 - -u A ) = A )
29	25, 28	eqtrd 2240	. . . . 5 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( ( P pCnt 1 ) - ( P pCnt ( P ^ -u A ) ) ) = A )
30	20, 29	eqtrd 2240	. . . 4 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P pCnt ( 1 / ( P ^ -u A ) ) ) = A )
31	13, 30	eqtrd 2240	. . 3 |- ( ( P e. Prime /\ ( A e. RR /\ -u A e. NN ) ) -> ( P pCnt ( P ^ A ) ) = A )
32	2, 31	jaodan 799	. 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 710   = wceq 1373   e. wcel 2178   =/= wne 2378   class class class wbr 4059  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   1c1 7961   - cmin 8278   -ucneg 8279   # cap 8689   / cdiv 8780   NNcn 9071   NN0cn0 9330   ZZcz 9407   ^cexp 10720   Primecprime 12544   pCnt cpc 12722

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-2o 6526  df-er 6643  df-en 6851  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-gcd 12390  df-prm 12545  df-pc 12723

This theorem is referenced by:  pcprmpw2  12771  pcaddlem  12777  expnprm  12791  dvdsppwf1o  15576  lgsval2lem  15602