Theorem sgmmul 16093

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 1034	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> M e. NN )
2		simpr2 1035	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> N e. NN )
3		simpr3 1036	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( M gcd N ) = 1 )
4		eqid 2238	. . 3 |- { x e. NN | x || M } = { x e. NN | x || M }
5		eqid 2238	. . 3 |- { x e. NN | x || N } = { x e. NN | x || N }
6		eqid 2238	. . 3 |- { x e. NN | x || ( M x. N ) } = { x e. NN | x || ( M x. N ) }
7		ssrab2 3333	. . . . . 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 3246	. . . . 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 10078	. . . 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 531	. . . 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 16012	. . 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 3333	. . . . . 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 3246	. . . . 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 10078	. . . 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 531	. . . 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 16012	. . 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 483	. . . . . 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 10078	. . . . 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 482	. . . . . 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 10078	. . . . 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 531	. . . . 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 15994	. . . . 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 1278	. . . 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 2244	. . 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 6086	. . 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 16088	. 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 16080	. . . 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 16080	. . . 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 6097	. 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 9336	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( M x. N ) e. NN )
35		sgmval 16080	. . 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 2282	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 1009   = wceq 1402   e. wcel 2209   {crab 2532   class class class wbr 4128  (class class class)co 6079   CCcc 8171   1c1 8174   x. cmul 8178   NNcn 9287   RR+crp 10037   sum_csu 12102   || cdvds 12537   gcd cgcd 12713   ^c ccxp 15941   sigma csgm 16078

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293  ax-pre-suploc 8294  ax-addf 8295  ax-mulf 8296

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-disj 4105  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-of 6296  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-oadd 6685  df-er 6801  df-map 6918  df-pm 6919  df-en 7017  df-dom 7018  df-fin 7019  df-sup 7318  df-inf 7319  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-xneg 10157  df-xadd 10158  df-ioo 10277  df-ico 10279  df-icc 10280  df-fz 10395  df-fzo 10533  df-fl 10688  df-mod 10743  df-seqfrec 10868  df-exp 10959  df-fac 11147  df-bc 11169  df-ihash 11198  df-shft 11563  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-clim 12028  df-sumdc 12103  df-ef 12398  df-e 12399  df-dvds 12538  df-gcd 12714  df-rest 13578  df-topgen 13597  df-psmet 14863  df-xmet 14864  df-met 14865  df-bl 14866  df-mopn 14867  df-top 15082  df-topon 15095  df-bases 15127  df-ntr 15180  df-cn 15272  df-cnp 15273  df-tx 15337  df-cncf 15655  df-limced 15740  df-dvap 15741  df-relog 15942  df-rpcxp 15943  df-sgm 16079

This theorem is referenced by:  perfect1  16095  perfectlem1  16096  perfectlem2  16097