Theorem sgmppw 15674

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 1021	. . 3 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> A e. CC )
2		simp2 1022	. . . . 5 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> P e. Prime )
3		prmnn 12640	. . . . 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 1023	. . . 4 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> N e. NN0 )
6	4, 5	nnexpcld 10925	. . 3 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> ( P ^ N ) e. NN )
7		sgmval 15665	. . 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 6014	. . 3 |- ( n = ( P ^ k ) -> ( n ^c A ) = ( ( P ^ k ) ^c A ) )
10		0zd 9466	. . . 4 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> 0 e. ZZ )
11	5	nn0zd 9575	. . . 4 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> N e. ZZ )
12	10, 11	fzfigd 10661	. . 3 |- ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) -> ( 0 ... N ) e. Fin )
13		eqid 2229	. . . . 5 |- ( i e. ( 0 ... N ) |-> ( P ^ i ) ) = ( i e. ( 0 ... N ) |-> ( P ^ i ) )
14	13	dvdsppwf1o 15671	. . . 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 6015	. . . 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 10318	. . . . . 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 10925	. . . 4 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( P ^ k ) e. NN )
22	13, 16, 17, 21	fvmptd3 5730	. . 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 2956	. . . . . 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 9898	. . . 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 15609	. . 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 11909	. 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 9432	. . . . . 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 8176	. . . . 5 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> ( k x. A ) = ( A x. k ) )
32	31	oveq2d 6023	. . . 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 9898	. . . . . 6 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> P e. RR+ )
34	20	nn0red 9431	. . . . . 6 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> k e. RR )
35	33, 34, 30	cxpmuld 15619	. . . . 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 9575	. . . . . . 7 |- ( ( ( A e. CC /\ P e. Prime /\ N e. NN0 ) /\ k e. ( 0 ... N ) ) -> k e. ZZ )
37		cxpexpnn 15578	. . . . . . 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 6022	. . . . 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 2262	. . . 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 15614	. . . 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 2270	. . 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 11887	. 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 2266	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 1002   = wceq 1395   e. wcel 2200   {crab 2512   class class class wbr 4083   |-> cmpt 4145   -1-1-onto->wf1o 5317  (class class class)co 6007   CCcc 8005   0cc0 8007   x. cmul 8012   NNcn 9118   NN0cn0 9377   ZZcz 9454   ...cfz 10212   ^cexp 10768   sum_csu 11872   || cdvds 12306   Primecprime 12637   ^c ccxp 15539   sigma csgm 15663

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126  ax-caucvg 8127  ax-pre-suploc 8128  ax-addf 8129  ax-mulf 8130

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-disj 4060  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-of 6224  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-2o 6569  df-oadd 6572  df-er 6688  df-map 6805  df-pm 6806  df-en 6896  df-dom 6897  df-fin 6898  df-sup 7159  df-inf 7160  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-n0 9378  df-xnn0 9441  df-z 9455  df-uz 9731  df-q 9823  df-rp 9858  df-xneg 9976  df-xadd 9977  df-ioo 10096  df-ico 10098  df-icc 10099  df-fz 10213  df-fzo 10347  df-fl 10498  df-mod 10553  df-seqfrec 10678  df-exp 10769  df-fac 10956  df-bc 10978  df-ihash 11006  df-shft 11334  df-cj 11361  df-re 11362  df-im 11363  df-rsqrt 11517  df-abs 11518  df-clim 11798  df-sumdc 11873  df-ef 12167  df-e 12168  df-dvds 12307  df-gcd 12483  df-prm 12638  df-pc 12816  df-rest 13282  df-topgen 13301  df-psmet 14515  df-xmet 14516  df-met 14517  df-bl 14518  df-mopn 14519  df-top 14680  df-topon 14693  df-bases 14725  df-ntr 14778  df-cn 14870  df-cnp 14871  df-tx 14935  df-cncf 15253  df-limced 15338  df-dvap 15339  df-relog 15540  df-rpcxp 15541  df-sgm 15664

This theorem is referenced by:  1sgmprm  15676  1sgm2ppw  15677