Theorem sgmppw 15852

Description: The value of the divisor function at a prime power. (Contributed by Mario Carneiro, 17-May-2016.)

Assertion

Ref	Expression
sgmppw	|- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> ( A sigma ( P ^ N ) ) = sum_ k e. ( 0 ... N ) ( ( P ^c A ) ^ k ) )

Distinct variable groups:   A, k   k, N   P, k

Proof of Theorem sgmppw

Dummy variables i n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1024	. . 3 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> A e. CC )
2		simp2 1025	. . . . 5 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> P e. Prime )
3		prmnn 12803	. . . . 5 |- ( P e. Prime -> P e. NN )
4	2, 3	syl 14	. . . 4 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> P e. NN )
5		simp3 1026	. . . 4 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> N e. NN0 )
6	4, 5	nnexpcld 11056	. . 3 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> ( P ^ N ) e. NN )
7		sgmval 15843	. . 3 |- ( ( A e. CC /\ ( P ^ N ) e. NN ) -> ( A sigma ( P ^ N ) ) = sum_ n e. { x e. NN | x || ( P ^ N ) } ( n ^c A ) )
8	1, 6, 7	syl2anc 411	. 2 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> ( A sigma ( P ^ N ) ) = sum_ n e. { x e. NN | x || ( P ^ N ) } ( n ^c A ) )
9		oveq1 6056	. . 3 |- ( n = ( P ^ k ) -> ( n ^c A ) = ( ( P ^ k ) ^c A ) )
10		0zd 9588	. . . 4 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> 0 e. ZZ )
11	5	nn0zd 9697	. . . 4 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> N e. ZZ )
12	10, 11	fzfigd 10792	. . 3 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> ( 0 ... N ) e. Fin )
13		eqid 2232	. . . . 5 |- ( i e. ( 0 ... N ) |-> ( P ^ i ) ) = ( i e. ( 0 ... N ) |-> ( P ^ i ) )
14	13	dvdsppwf1o 15849	. . . 4 |- ( ( P e. Prime /\ N e. NN0 ) -> ( i e. ( 0 ... N ) |-> ( P ^ i ) ) : ( 0 ... N ) -1-1-onto-> { x e. NN | x || ( P ^ N ) } )
15	2, 5, 14	syl2anc 411	. . 3 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> ( i e. ( 0 ... N ) |-> ( P ^ i ) ) : ( 0 ... N ) -1-1-onto-> { x e. NN | x || ( P ^ N ) } )
16		oveq2 6057	. . . 4 |- ( i = k -> ( P ^ i ) = ( P ^ k ) )
17		simpr 110	. . . 4 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> k e. ( 0 ... N ) )
18	4	adantr 276	. . . . 5 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> P e. NN )
19		elfznn0 10447	. . . . . 6 |- ( k e. ( 0 ... N ) -> k e. NN0 )
20	19	adantl 277	. . . . 5 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> k e. NN0 )
21	18, 20	nnexpcld 11056	. . . 4 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( P ^ k ) e. NN )
22	13, 16, 17, 21	fvmptd3 5770	. . 3 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( ( i e. ( 0 ... N ) |-> ( P ^ i ) ) ` k ) = ( P ^ k ) )
23		elrabi 2969	. . . . . 6 |- ( n e. { x e. NN | x || ( P ^ N ) } -> n e. NN )
24	23	adantl 277	. . . . 5 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ n e. { x e. NN | x || ( P ^ N ) } ) -> n e. NN )
25	24	nnrpd 10026	. . . 4 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ n e. { x e. NN | x || ( P ^ N ) } ) -> n e. RR+ )
26	1	adantr 276	. . . 4 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ n e. { x e. NN | x || ( P ^ N ) } ) -> A e. CC )
27	25, 26	rpcncxpcld 15784	. . 3 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ n e. { x e. NN | x || ( P ^ N ) } ) -> ( n ^c A ) e. CC )
28	9, 12, 15, 22, 27	fsumf1o 12072	. 2 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> sum_ n e. { x e. NN | x || ( P ^ N ) } ( n ^c A ) = sum_ k e. ( 0 ... N ) ( ( P ^ k ) ^c A ) )
29	20	nn0cnd 9554	. . . . . 6 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> k e. CC )
30	1	adantr 276	. . . . . 6 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> A e. CC )
31	29, 30	mulcomd 8294	. . . . 5 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( k x. A ) = ( A x. k ) )
32	31	oveq2d 6065	. . . 4 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( P ^c ( k x. A ) ) = ( P ^c ( A x. k ) ) )
33	18	nnrpd 10026	. . . . . 6 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> P e. RR+ )
34	20	nn0red 9553	. . . . . 6 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> k e. RR )
35	33, 34, 30	cxpmuld 15794	. . . . 5 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( P ^c ( k x. A ) ) = ( ( P ^c k ) ^c A ) )
36	20	nn0zd 9697	. . . . . . 7 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> k e. ZZ )
37		cxpexpnn 15753	. . . . . . 7 |- ( ( P e. NN /\ k e. ZZ ) -> ( P ^c k ) = ( P ^ k ) )
38	18, 36, 37	syl2anc 411	. . . . . 6 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( P ^c k ) = ( P ^ k ) )
39	38	oveq1d 6064	. . . . 5 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( ( P ^c k ) ^c A ) = ( ( P ^ k ) ^c A ) )
40	35, 39	eqtrd 2265	. . . 4 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( P ^c ( k x. A ) ) = ( ( P ^ k ) ^c A ) )
41	33, 30, 20	rpcxpmul2d 15789	. . . 4 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( P ^c ( A x. k ) ) = ( ( P ^c A ) ^ k ) )
42	32, 40, 41	3eqtr3d 2273	. . 3 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( ( P ^ k ) ^c A ) = ( ( P ^c A ) ^ k ) )
43	42	sumeq2dv 12049	. 2 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> sum_ k e. ( 0 ... N ) ( ( P ^ k ) ^c A ) = sum_ k e. ( 0 ... N ) ( ( P ^c A ) ^ k ) )
44	8, 28, 43	3eqtrd 2269	1 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> ( A sigma ( P ^ N ) ) = sum_ k e. ( 0 ... N ) ( ( P ^c A ) ^ k ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   {crab 2524   class class class wbr 4108   |-> cmpt 4170   -1-1-onto->wf1o 5350  (class class class)co 6049   CCcc 8124   0cc0 8126   x. cmul 8131   NNcn 9236   NN0cn0 9495   ZZcz 9576   ...cfz 10341   ^cexp 10899   sum_csu 12034   || cdvds 12469   Primecprime 12800   ^c ccxp 15714   sigma csgm 15841

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 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246  ax-pre-suploc 8247  ax-addf 8248  ax-mulf 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 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 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-disj 4085  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-isom 5360  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-of 6265  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-2o 6647  df-oadd 6650  df-er 6766  df-map 6883  df-pm 6884  df-en 6975  df-dom 6976  df-fin 6977  df-sup 7274  df-inf 7275  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-xnn0 9563  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-xneg 10104  df-xadd 10105  df-ioo 10224  df-ico 10226  df-icc 10227  df-fz 10342  df-fzo 10476  df-fl 10629  df-mod 10684  df-seqfrec 10809  df-exp 10900  df-fac 11087  df-bc 11109  df-ihash 11137  df-shft 11496  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680  df-clim 11960  df-sumdc 12035  df-ef 12330  df-e 12331  df-dvds 12470  df-gcd 12646  df-prm 12801  df-pc 12979  df-rest 13446  df-topgen 13465  df-psmet 14683  df-xmet 14684  df-met 14685  df-bl 14686  df-mopn 14687  df-top 14855  df-topon 14868  df-bases 14900  df-ntr 14953  df-cn 15045  df-cnp 15046  df-tx 15110  df-cncf 15428  df-limced 15513  df-dvap 15514  df-relog 15715  df-rpcxp 15716  df-sgm 15842

This theorem is referenced by:  1sgmprm  15854  1sgm2ppw  15855