Theorem sgmmul 16323

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 10112	. . . 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 16195	. . 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 10112	. . . 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 16195	. . 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 10112	. . . . 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 10112	. . . . 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 16175	. . . . 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 6095	. . 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 16318	. 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 16285	. . . 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 16285	. . . 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 6106	. 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 9363	. . 3 |- ( ( A e. CC /\ ( M e. NN /\ N e. NN /\ ( M gcd N ) = 1 ) ) -> ( M x. N ) e. NN )
35		sgmval 16285	. . 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   {crab 2532   class class class wbr 4131  (class class class)co 6088   CCcc 8184   1c1 8187   x. cmul 8191   NNcn 9314   RR+crp 10071   sum_csu 12144   || cdvds 12579   gcd cgcd 12755   ^c ccxp 16119   sigma csgm 16268

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306  ax-pre-suploc 8307  ax-addf 8308  ax-mulf 8309

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-1st 6378  df-2nd 6379  df-recs 6581  df-irdg 6646  df-frec 6667  df-1o 6692  df-oadd 6696  df-er 6812  df-map 6929  df-pm 6930  df-en 7029  df-dom 7030  df-fin 7031  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9655  df-uz 9938  df-q 10036  df-rp 10072  df-xneg 10191  df-xadd 10192  df-ioo 10311  df-ico 10313  df-icc 10314  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-fac 11186  df-bc 11208  df-ihash 11237  df-shft 11602  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-sumdc 12145  df-ef 12440  df-e 12441  df-dvds 12580  df-gcd 12756  df-rest 13654  df-topgen 13673  df-psmet 14989  df-xmet 14990  df-met 14991  df-bl 14992  df-mopn 14993  df-top 15257  df-topon 15270  df-bases 15302  df-ntr 15355  df-cn 15447  df-cnp 15448  df-tx 15512  df-cncf 15830  df-limced 15915  df-dvap 15916  df-relog 16120  df-rpcxp 16121  df-sgm 16271

This theorem is used by:  perfect1  16331  perfectlem1  16332  perfectlem2  16333