Theorem sgmppw 15238

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 999	. . 3 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> A e. CC )
2		simp2 1000	. . . . 5 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> P e. Prime )
3		prmnn 12288	. . . . 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 1001	. . . 4 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> N e. NN0 )
6	4, 5	nnexpcld 10789	. . 3 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> ( P ^ N ) e. NN )
7		sgmval 15229	. . 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 5930	. . 3 |- ( n = ( P ^ k ) -> ( n ^c A ) = ( ( P ^ k ) ^c A ) )
10		0zd 9340	. . . 4 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> 0 e. ZZ )
11	5	nn0zd 9448	. . . 4 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> N e. ZZ )
12	10, 11	fzfigd 10525	. . 3 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> ( 0 ... N ) e. Fin )
13		eqid 2196	. . . . 5 |- ( i e. ( 0 ... N ) |-> ( P ^ i ) ) = ( i e. ( 0 ... N ) |-> ( P ^ i ) )
14	13	dvdsppwf1o 15235	. . . 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 5931	. . . 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 10191	. . . . . 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 10789	. . . 4 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( P ^ k ) e. NN )
22	13, 16, 17, 21	fvmptd3 5656	. . 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 2917	. . . . . 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 9771	. . . 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 15173	. . 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 11557	. 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 9306	. . . . . 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 8050	. . . . 5 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( k x. A ) = ( A x. k ) )
32	31	oveq2d 5939	. . . 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 9771	. . . . . 6 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> P e. RR+ )
34	20	nn0red 9305	. . . . . 6 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> k e. RR )
35	33, 34, 30	cxpmuld 15183	. . . . 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 9448	. . . . . . 7 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> k e. ZZ )
37		cxpexpnn 15142	. . . . . . 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 5938	. . . . 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 2229	. . . 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 15178	. . . 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 2237	. . 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 11535	. 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 2233	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 980   = wceq 1364   e. wcel 2167   {crab 2479   class class class wbr 4034   |-> cmpt 4095   -1-1-onto->wf1o 5258  (class class class)co 5923   CCcc 7879   0cc0 7881   x. cmul 7886   NNcn 8992   NN0cn0 9251   ZZcz 9328   ...cfz 10085   ^cexp 10632   sum_csu 11520   || cdvds 11954   Primecprime 12285   ^c ccxp 15103   sigma csgm 15227

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-mulrcl 7980  ax-addcom 7981  ax-mulcom 7982  ax-addass 7983  ax-mulass 7984  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-1rid 7988  ax-0id 7989  ax-rnegex 7990  ax-precex 7991  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-apti 7996  ax-pre-ltadd 7997  ax-pre-mulgt0 7998  ax-pre-mulext 7999  ax-arch 8000  ax-caucvg 8001  ax-pre-suploc 8002  ax-addf 8003  ax-mulf 8004

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-disj 4012  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-of 6136  df-1st 6199  df-2nd 6200  df-recs 6364  df-irdg 6429  df-frec 6450  df-1o 6475  df-2o 6476  df-oadd 6479  df-er 6593  df-map 6710  df-pm 6711  df-en 6801  df-dom 6802  df-fin 6803  df-sup 7051  df-inf 7052  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-reap 8604  df-ap 8611  df-div 8702  df-inn 8993  df-2 9051  df-3 9052  df-4 9053  df-n0 9252  df-xnn0 9315  df-z 9329  df-uz 9604  df-q 9696  df-rp 9731  df-xneg 9849  df-xadd 9850  df-ioo 9969  df-ico 9971  df-icc 9972  df-fz 10086  df-fzo 10220  df-fl 10362  df-mod 10417  df-seqfrec 10542  df-exp 10633  df-fac 10820  df-bc 10842  df-ihash 10870  df-shft 10982  df-cj 11009  df-re 11010  df-im 11011  df-rsqrt 11165  df-abs 11166  df-clim 11446  df-sumdc 11521  df-ef 11815  df-e 11816  df-dvds 11955  df-gcd 12131  df-prm 12286  df-pc 12464  df-rest 12922  df-topgen 12941  df-psmet 14109  df-xmet 14110  df-met 14111  df-bl 14112  df-mopn 14113  df-top 14244  df-topon 14257  df-bases 14289  df-ntr 14342  df-cn 14434  df-cnp 14435  df-tx 14499  df-cncf 14817  df-limced 14902  df-dvap 14903  df-relog 15104  df-rpcxp 15105  df-sgm 15228

This theorem is referenced by:  1sgmprm  15240  1sgm2ppw  15241