Theorem sgmmul 15318

Description: The divisor function for fixed parameter A is a multiplicative function. (Contributed by Mario Carneiro, 2-Jul-2015.)

Assertion

Ref	Expression
sgmmul	|- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( A sigma ( M x. N ) ) = ( ( A sigma M ) x. ( A sigma N ) ) )

Proof of Theorem sgmmul

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

Step	Hyp	Ref	Expression
1		simpr1 1005	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> M e. NN )
2		simpr2 1006	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> N e. NN )
3		simpr3 1007	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( M gcd N ) = 1 )
4		eqid 2196	. . 3 |- { x e. NN | x || M } = { x e. NN | x || M }
5		eqid 2196	. . 3 |- { x e. NN | x || N } = { x e. NN | x || N }
6		eqid 2196	. . 3 |- { x e. NN | x || ( M x. N ) } = { x e. NN | x || ( M x. N ) }
7		ssrab2 3269	. . . . . 6 |- { x e. NN | x || M } C_ NN
8		simpr 110	. . . . . 6 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ j e. { x e. NN | x || M } ) -> j e. { x e. NN | x || M } )
9	7, 8	sselid 3182	. . . . 5 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ j e. { x e. NN | x || M } ) -> j e. NN )
10	9	nnrpd 9788	. . . 4 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ j e. { x e. NN | x || M } ) -> j e. RR+ )
11		simpll 527	. . . 4 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ j e. { x e. NN | x || M } ) -> A e. CC )
12	10, 11	rpcncxpcld 15249	. . 3 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ j e. { x e. NN | x || M } ) -> ( j ^c A ) e. CC )
13		ssrab2 3269	. . . . . 6 |- { x e. NN | x || N } C_ NN
14		simpr 110	. . . . . 6 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ k e. { x e. NN | x || N } ) -> k e. { x e. NN | x || N } )
15	13, 14	sselid 3182	. . . . 5 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ k e. { x e. NN | x || N } ) -> k e. NN )
16	15	nnrpd 9788	. . . 4 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ k e. { x e. NN | x || N } ) -> k e. RR+ )
17		simpll 527	. . . 4 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ k e. { x e. NN | x || N } ) -> A e. CC )
18	16, 17	rpcncxpcld 15249	. . 3 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ k e. { x e. NN | x || N } ) -> ( k ^c A ) e. CC )
19	9	adantrr 479	. . . . . 6 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ ( j e. { x e. NN | x || M } /\ k e. { x e. NN | x || N } ) ) -> j e. NN )
20	19	nnrpd 9788	. . . . 5 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ ( j e. { x e. NN | x || M } /\ k e. { x e. NN | x || N } ) ) -> j e. RR+ )
21	15	adantrl 478	. . . . . 6 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ ( j e. { x e. NN | x || M } /\ k e. { x e. NN | x || N } ) ) -> k e. NN )
22	21	nnrpd 9788	. . . . 5 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ ( j e. { x e. NN | x || M } /\ k e. { x e. NN | x || N } ) ) -> k e. RR+ )
23		simpll 527	. . . . 5 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ ( j e. { x e. NN | x || M } /\ k e. { x e. NN | x || N } ) ) -> A e. CC )
24		rpmulcxp 15231	. . . . 5 |- ( ( j e. RR+ /\ k e. RR+ /\ A e. CC ) -> ( ( j x. k ) ^c A ) = ( ( j ^c A ) x. ( k ^c A ) ) )
25	20, 22, 23, 24	syl3anc 1249	. . . 4 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ ( j e. { x e. NN | x || M } /\ k e. { x e. NN | x || N } ) ) -> ( ( j x. k ) ^c A ) = ( ( j ^c A ) x. ( k ^c A ) ) )
26	25	eqcomd 2202	. . 3 |- ( ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) /\ ( j e. { x e. NN | x || M } /\ k e. { x e. NN | x || N } ) ) -> ( ( j ^c A ) x. ( k ^c A ) ) = ( ( j x. k ) ^c A ) )
27		oveq1 5932	. . 3 |- ( i = ( j x. k ) -> ( i ^c A ) = ( ( j x. k ) ^c A ) )
28	1, 2, 3, 4, 5, 6, 12, 18, 26, 27	fsumdvdsmul 15313	. 2 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( sum_ j e. { x e. NN | x || M } ( j ^c A ) x. sum_ k e. { x e. NN | x || N } ( k ^c A ) ) = sum_ i e. { x e. NN | x || ( M x. N ) } ( i ^c A ) )
29		sgmval 15305	. . . 4 |- ( ( A e. CC /\ M e. NN ) -> ( A sigma M ) = sum_ j e. { x e. NN | x || M } ( j ^c A ) )
30	1, 29	syldan 282	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( A sigma M ) = sum_ j e. { x e. NN | x || M } ( j ^c A ) )
31		sgmval 15305	. . . 4 |- ( ( A e. CC /\ N e. NN ) -> ( A sigma N ) = sum_ k e. { x e. NN | x || N } ( k ^c A ) )
32	2, 31	syldan 282	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( A sigma N ) = sum_ k e. { x e. NN | x || N } ( k ^c A ) )
33	30, 32	oveq12d 5943	. 2 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( ( A sigma M ) x. ( A sigma N ) ) = ( sum_ j e. { x e. NN | x || M } ( j ^c A ) x. sum_ k e. { x e. NN | x || N } ( k ^c A ) ) )
34	1, 2	nnmulcld 9058	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( M x. N ) e. NN )
35		sgmval 15305	. . 3 |- ( ( A e. CC /\ ( M x. N ) e. NN ) -> ( A sigma ( M x. N ) ) = sum_ i e. { x e. NN | x || ( M x. N ) } ( i ^c A ) )
36	34, 35	syldan 282	. 2 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( A sigma ( M x. N ) ) = sum_ i e. { x e. NN | x || ( M x. N ) } ( i ^c A ) )
37	28, 33, 36	3eqtr4rd 2240	1 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( A sigma ( M x. N ) ) = ( ( A sigma M ) x. ( A sigma N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   {crab 2479   class class class wbr 4034  (class class class)co 5925   CCcc 7896   1c1 7899   x. cmul 7903   NNcn 9009   RR+crp 9747   sum_csu 11537   || cdvds 11971   gcd cgcd 12147   ^c ccxp 15179   sigma csgm 15303

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 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016  ax-arch 8017  ax-caucvg 8018  ax-pre-suploc 8019  ax-addf 8020  ax-mulf 8021

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 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-of 6139  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-map 6718  df-pm 6719  df-en 6809  df-dom 6810  df-fin 6811  df-sup 7059  df-inf 7060  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628  df-div 8719  df-inn 9010  df-2 9068  df-3 9069  df-4 9070  df-n0 9269  df-z 9346  df-uz 9621  df-q 9713  df-rp 9748  df-xneg 9866  df-xadd 9867  df-ioo 9986  df-ico 9988  df-icc 9989  df-fz 10103  df-fzo 10237  df-fl 10379  df-mod 10434  df-seqfrec 10559  df-exp 10650  df-fac 10837  df-bc 10859  df-ihash 10887  df-shft 10999  df-cj 11026  df-re 11027  df-im 11028  df-rsqrt 11182  df-abs 11183  df-clim 11463  df-sumdc 11538  df-ef 11832  df-e 11833  df-dvds 11972  df-gcd 12148  df-rest 12945  df-topgen 12964  df-psmet 14177  df-xmet 14178  df-met 14179  df-bl 14180  df-mopn 14181  df-top 14320  df-topon 14333  df-bases 14365  df-ntr 14418  df-cn 14510  df-cnp 14511  df-tx 14575  df-cncf 14893  df-limced 14978  df-dvap 14979  df-relog 15180  df-rpcxp 15181  df-sgm 15304

This theorem is referenced by:  perfect1  15320  perfectlem1  15321  perfectlem2  15322