Theorem sgmmul 15538

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 1006	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> M e. NN )
2		simpr2 1007	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> N e. NN )
3		simpr3 1008	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( M gcd N ) = 1 )
4		eqid 2206	. . 3 |- { x e. NN | x || M } = { x e. NN | x || M }
5		eqid 2206	. . 3 |- { x e. NN | x || N } = { x e. NN | x || N }
6		eqid 2206	. . 3 |- { x e. NN | x || ( M x. N ) } = { x e. NN | x || ( M x. N ) }
7		ssrab2 3282	. . . . . 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 3195	. . . . 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 9831	. . . 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 15469	. . 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 3282	. . . . . 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 3195	. . . . 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 9831	. . . 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 15469	. . 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 9831	. . . . 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 9831	. . . . 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 15451	. . . . 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 1250	. . . 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 2212	. . 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 5963	. . 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 15533	. 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 15525	. . . 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 15525	. . . 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 5974	. 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 9100	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( M x. N ) e. NN )
35		sgmval 15525	. . 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 2250	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 981   = wceq 1373   e. wcel 2177   {crab 2489   class class class wbr 4050  (class class class)co 5956   CCcc 7938   1c1 7941   x. cmul 7945   NNcn 9051   RR+crp 9790   sum_csu 11734   || cdvds 12168   gcd cgcd 12344   ^c ccxp 15399   sigma csgm 15523

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643  ax-cnex 8031  ax-resscn 8032  ax-1cn 8033  ax-1re 8034  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-mulrcl 8039  ax-addcom 8040  ax-mulcom 8041  ax-addass 8042  ax-mulass 8043  ax-distr 8044  ax-i2m1 8045  ax-0lt1 8046  ax-1rid 8047  ax-0id 8048  ax-rnegex 8049  ax-precex 8050  ax-cnre 8051  ax-pre-ltirr 8052  ax-pre-ltwlin 8053  ax-pre-lttrn 8054  ax-pre-apti 8055  ax-pre-ltadd 8056  ax-pre-mulgt0 8057  ax-pre-mulext 8058  ax-arch 8059  ax-caucvg 8060  ax-pre-suploc 8061  ax-addf 8062  ax-mulf 8063

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-disj 4027  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-po 4350  df-iso 4351  df-iord 4420  df-on 4422  df-ilim 4423  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-isom 5288  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-of 6170  df-1st 6238  df-2nd 6239  df-recs 6403  df-irdg 6468  df-frec 6489  df-1o 6514  df-oadd 6518  df-er 6632  df-map 6749  df-pm 6750  df-en 6840  df-dom 6841  df-fin 6842  df-sup 7100  df-inf 7101  df-pnf 8124  df-mnf 8125  df-xr 8126  df-ltxr 8127  df-le 8128  df-sub 8260  df-neg 8261  df-reap 8663  df-ap 8670  df-div 8761  df-inn 9052  df-2 9110  df-3 9111  df-4 9112  df-n0 9311  df-z 9388  df-uz 9664  df-q 9756  df-rp 9791  df-xneg 9909  df-xadd 9910  df-ioo 10029  df-ico 10031  df-icc 10032  df-fz 10146  df-fzo 10280  df-fl 10430  df-mod 10485  df-seqfrec 10610  df-exp 10701  df-fac 10888  df-bc 10910  df-ihash 10938  df-shft 11196  df-cj 11223  df-re 11224  df-im 11225  df-rsqrt 11379  df-abs 11380  df-clim 11660  df-sumdc 11735  df-ef 12029  df-e 12030  df-dvds 12169  df-gcd 12345  df-rest 13143  df-topgen 13162  df-psmet 14375  df-xmet 14376  df-met 14377  df-bl 14378  df-mopn 14379  df-top 14540  df-topon 14553  df-bases 14585  df-ntr 14638  df-cn 14730  df-cnp 14731  df-tx 14795  df-cncf 15113  df-limced 15198  df-dvap 15199  df-relog 15400  df-rpcxp 15401  df-sgm 15524

This theorem is referenced by:  perfect1  15540  perfectlem1  15541  perfectlem2  15542