Theorem sgmmul 15726

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 1029	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> M e. NN )
2		simpr2 1030	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> N e. NN )
3		simpr3 1031	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( M gcd N ) = 1 )
4		eqid 2231	. . 3 |- { x e. NN | x || M } = { x e. NN | x || M }
5		eqid 2231	. . 3 |- { x e. NN | x || N } = { x e. NN | x || N }
6		eqid 2231	. . 3 |- { x e. NN | x || ( M x. N ) } = { x e. NN | x || ( M x. N ) }
7		ssrab2 3312	. . . . . 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 3225	. . . . 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 9929	. . . 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 15657	. . 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 3312	. . . . . 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 3225	. . . . 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 9929	. . . 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 15657	. . 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 9929	. . . . 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 9929	. . . . 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 15639	. . . . 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 1273	. . . 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 2237	. . 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 6025	. . 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 15721	. 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 15713	. . . 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 15713	. . . 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 6036	. 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 9192	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( M x. N ) e. NN )
35		sgmval 15713	. . 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 2275	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 1004   = wceq 1397   e. wcel 2202   {crab 2514   class class class wbr 4088  (class class class)co 6018   CCcc 8030   1c1 8033   x. cmul 8037   NNcn 9143   RR+crp 9888   sum_csu 11918   || cdvds 12353   gcd cgcd 12529   ^c ccxp 15587   sigma csgm 15711

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152  ax-pre-suploc 8153  ax-addf 8154  ax-mulf 8155

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-disj 4065  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-of 6235  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-oadd 6586  df-er 6702  df-map 6819  df-pm 6820  df-en 6910  df-dom 6911  df-fin 6912  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-xneg 10007  df-xadd 10008  df-ioo 10127  df-ico 10129  df-icc 10130  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-fac 10989  df-bc 11011  df-ihash 11039  df-shft 11380  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564  df-clim 11844  df-sumdc 11919  df-ef 12214  df-e 12215  df-dvds 12354  df-gcd 12530  df-rest 13329  df-topgen 13348  df-psmet 14563  df-xmet 14564  df-met 14565  df-bl 14566  df-mopn 14567  df-top 14728  df-topon 14741  df-bases 14773  df-ntr 14826  df-cn 14918  df-cnp 14919  df-tx 14983  df-cncf 15301  df-limced 15386  df-dvap 15387  df-relog 15588  df-rpcxp 15589  df-sgm 15712

This theorem is referenced by:  perfect1  15728  perfectlem1  15729  perfectlem2  15730