Theorem fsumdvdsmul 24921

Description: Product of two divisor sums. (This is also the main part of the proof that " sum_ k || N F ( k ) is a multiplicative function if F is".) (Contributed by Mario Carneiro, 2-Jul-2015.)

Hypotheses

Ref	Expression
dvdsmulf1o.1	|- ( ph -> M e. NN )
dvdsmulf1o.2	|- ( ph -> N e. NN )
dvdsmulf1o.3	|- ( ph -> ( M gcd N ) = 1 )
dvdsmulf1o.x	|- X = { x e. NN | x || M }
dvdsmulf1o.y	|- Y = { x e. NN | x || N }
dvdsmulf1o.z	|- Z = { x e. NN | x || ( M x. N ) }
fsumdvdsmul.4	|- ( ( ph /\ j e. X ) -> A e. CC )
fsumdvdsmul.5	|- ( ( ph /\ k e. Y ) -> B e. CC )
fsumdvdsmul.6	|- ( ( ph /\ ( j e. X /\ k e. Y ) ) -> ( A x. B ) = D )
fsumdvdsmul.7	|- ( i = ( j x. k ) -> C = D )

Assertion

Ref	Expression
fsumdvdsmul	|- ( ph -> ( sum_ j e. X A x. sum_ k e. Y B ) = sum_ i e. Z C )

Distinct variable groups:   A, k   D, i   x, M   x, N   i, j, k, X   B, j   C, j, k   i, Y, j, k   i, Z, j   x, i, j, k   ph, i, j, k

Allowed substitution hints:   ph( x)   A( x, i, j)   B( x, i, k)   C( x, i)   D( x, j, k)   M( i, j, k)   N( i, j, k)   X( x)   Y( x)   Z( x, k)

Proof of Theorem fsumdvdsmul

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 12772	. . . 4 |- ( ph -> ( 1 ... M ) e. Fin )
2		dvdsmulf1o.x	. . . . 5 |- X = { x e. NN | x || M }
3		dvdsmulf1o.1	. . . . . 6 |- ( ph -> M e. NN )
4		dvdsssfz1 15040	. . . . . 6 |- ( M e. NN -> { x e. NN | x || M } C_ ( 1 ... M ) )
5	3, 4	syl 17	. . . . 5 |- ( ph -> { x e. NN | x || M } C_ ( 1 ... M ) )
6	2, 5	syl5eqss 3649	. . . 4 |- ( ph -> X C_ ( 1 ... M ) )
7		ssfi 8180	. . . 4 |- ( ( ( 1 ... M ) e. Fin /\ X C_ ( 1 ... M ) ) -> X e. Fin )
8	1, 6, 7	syl2anc 693	. . 3 |- ( ph -> X e. Fin )
9		fzfid 12772	. . . . 5 |- ( ph -> ( 1 ... N ) e. Fin )
10		dvdsmulf1o.y	. . . . . 6 |- Y = { x e. NN | x || N }
11		dvdsmulf1o.2	. . . . . . 7 |- ( ph -> N e. NN )
12		dvdsssfz1 15040	. . . . . . 7 |- ( N e. NN -> { x e. NN | x || N } C_ ( 1 ... N ) )
13	11, 12	syl 17	. . . . . 6 |- ( ph -> { x e. NN | x || N } C_ ( 1 ... N ) )
14	10, 13	syl5eqss 3649	. . . . 5 |- ( ph -> Y C_ ( 1 ... N ) )
15		ssfi 8180	. . . . 5 |- ( ( ( 1 ... N ) e. Fin /\ Y C_ ( 1 ... N ) ) -> Y e. Fin )
16	9, 14, 15	syl2anc 693	. . . 4 |- ( ph -> Y e. Fin )
17		fsumdvdsmul.5	. . . 4 |- ( ( ph /\ k e. Y ) -> B e. CC )
18	16, 17	fsumcl 14464	. . 3 |- ( ph -> sum_ k e. Y B e. CC )
19		fsumdvdsmul.4	. . 3 |- ( ( ph /\ j e. X ) -> A e. CC )
20	8, 18, 19	fsummulc1 14517	. 2 |- ( ph -> ( sum_ j e. X A x. sum_ k e. Y B ) = sum_ j e. X ( A x. sum_ k e. Y B ) )
21	16	adantr 481	. . . . 5 |- ( ( ph /\ j e. X ) -> Y e. Fin )
22	17	adantlr 751	. . . . 5 |- ( ( ( ph /\ j e. X ) /\ k e. Y ) -> B e. CC )
23	21, 19, 22	fsummulc2 14516	. . . 4 |- ( ( ph /\ j e. X ) -> ( A x. sum_ k e. Y B ) = sum_ k e. Y ( A x. B ) )
24		fsumdvdsmul.6	. . . . . 6 |- ( ( ph /\ ( j e. X /\ k e. Y ) ) -> ( A x. B ) = D )
25	24	anassrs 680	. . . . 5 |- ( ( ( ph /\ j e. X ) /\ k e. Y ) -> ( A x. B ) = D )
26	25	sumeq2dv 14433	. . . 4 |- ( ( ph /\ j e. X ) -> sum_ k e. Y ( A x. B ) = sum_ k e. Y D )
27	23, 26	eqtrd 2656	. . 3 |- ( ( ph /\ j e. X ) -> ( A x. sum_ k e. Y B ) = sum_ k e. Y D )
28	27	sumeq2dv 14433	. 2 |- ( ph -> sum_ j e. X ( A x. sum_ k e. Y B ) = sum_ j e. X sum_ k e. Y D )
29		fveq2 6191	. . . . . . 7 |- ( z = <. j , k >. -> ( x. ` z ) = ( x. ` <. j , k >. ) )
30		df-ov 6653	. . . . . . 7 |- ( j x. k ) = ( x. ` <. j , k >. )
31	29, 30	syl6eqr 2674	. . . . . 6 |- ( z = <. j , k >. -> ( x. ` z ) = ( j x. k ) )
32	31	csbeq1d 3540	. . . . 5 |- ( z = <. j , k >. -> [_ ( x. ` z ) / i ]_ C = [_ ( j x. k ) / i ]_ C )
33		ovex 6678	. . . . . 6 |- ( j x. k ) e. _V
34		fsumdvdsmul.7	. . . . . 6 |- ( i = ( j x. k ) -> C = D )
35	33, 34	csbie 3559	. . . . 5 |- [_ ( j x. k ) / i ]_ C = D
36	32, 35	syl6eq 2672	. . . 4 |- ( z = <. j , k >. -> [_ ( x. ` z ) / i ]_ C = D )
37	19	adantrr 753	. . . . . 6 |- ( ( ph /\ ( j e. X /\ k e. Y ) ) -> A e. CC )
38	17	adantrl 752	. . . . . 6 |- ( ( ph /\ ( j e. X /\ k e. Y ) ) -> B e. CC )
39	37, 38	mulcld 10060	. . . . 5 |- ( ( ph /\ ( j e. X /\ k e. Y ) ) -> ( A x. B ) e. CC )
40	24, 39	eqeltrrd 2702	. . . 4 |- ( ( ph /\ ( j e. X /\ k e. Y ) ) -> D e. CC )
41	36, 8, 16, 40	fsumxp 14503	. . 3 |- ( ph -> sum_ j e. X sum_ k e. Y D = sum_ z e. ( X X. Y ) [_ ( x. ` z ) / i ]_ C )
42		nfcv 2764	. . . . 5 |- F/_ w C
43		nfcsb1v 3549	. . . . 5 |- F/_ i [_ w / i ]_ C
44		csbeq1a 3542	. . . . 5 |- ( i = w -> C = [_ w / i ]_ C )
45	42, 43, 44	cbvsumi 14427	. . . 4 |- sum_ i e. Z C = sum_ w e. Z [_ w / i ]_ C
46		csbeq1 3536	. . . . 5 |- ( w = ( x. ` z ) -> [_ w / i ]_ C = [_ ( x. ` z ) / i ]_ C )
47		xpfi 8231	. . . . . 6 |- ( ( X e. Fin /\ Y e. Fin ) -> ( X X. Y ) e. Fin )
48	8, 16, 47	syl2anc 693	. . . . 5 |- ( ph -> ( X X. Y ) e. Fin )
49		dvdsmulf1o.3	. . . . . 6 |- ( ph -> ( M gcd N ) = 1 )
50		dvdsmulf1o.z	. . . . . 6 |- Z = { x e. NN | x || ( M x. N ) }
51	3, 11, 49, 2, 10, 50	dvdsmulf1o 24920	. . . . 5 |- ( ph -> ( x. |` ( X X. Y ) ) : ( X X. Y ) -1-1-onto-> Z )
52		fvres 6207	. . . . . 6 |- ( z e. ( X X. Y ) -> ( ( x. |` ( X X. Y ) ) ` z ) = ( x. ` z ) )
53	52	adantl 482	. . . . 5 |- ( ( ph /\ z e. ( X X. Y ) ) -> ( ( x. |` ( X X. Y ) ) ` z ) = ( x. ` z ) )
54	40	ralrimivva 2971	. . . . . . . 8 |- ( ph -> A. j e. X A. k e. Y D e. CC )
55	36	eleq1d 2686	. . . . . . . . 9 |- ( z = <. j , k >. -> ( [_ ( x. ` z ) / i ]_ C e. CC <-> D e. CC ) )
56	55	ralxp 5263	. . . . . . . 8 |- ( A. z e. ( X X. Y ) [_ ( x. ` z ) / i ]_ C e. CC <-> A. j e. X A. k e. Y D e. CC )
57	54, 56	sylibr 224	. . . . . . 7 |- ( ph -> A. z e. ( X X. Y ) [_ ( x. ` z ) / i ]_ C e. CC )
58		ax-mulf 10016	. . . . . . . . . 10 |- x. : ( CC X. CC ) --> CC
59		ffn 6045	. . . . . . . . . 10 |- ( x. : ( CC X. CC ) --> CC -> x. Fn ( CC X. CC ) )
60	58, 59	ax-mp 5	. . . . . . . . 9 |- x. Fn ( CC X. CC )
61		ssrab2 3687	. . . . . . . . . . . 12 |- { x e. NN | x || M } C_ NN
62	2, 61	eqsstri 3635	. . . . . . . . . . 11 |- X C_ NN
63		nnsscn 11025	. . . . . . . . . . 11 |- NN C_ CC
64	62, 63	sstri 3612	. . . . . . . . . 10 |- X C_ CC
65		ssrab2 3687	. . . . . . . . . . . 12 |- { x e. NN | x || N } C_ NN
66	10, 65	eqsstri 3635	. . . . . . . . . . 11 |- Y C_ NN
67	66, 63	sstri 3612	. . . . . . . . . 10 |- Y C_ CC
68		xpss12 5225	. . . . . . . . . 10 |- ( ( X C_ CC /\ Y C_ CC ) -> ( X X. Y ) C_ ( CC X. CC ) )
69	64, 67, 68	mp2an 708	. . . . . . . . 9 |- ( X X. Y ) C_ ( CC X. CC )
70	46	eleq1d 2686	. . . . . . . . . 10 |- ( w = ( x. ` z ) -> ( [_ w / i ]_ C e. CC <-> [_ ( x. ` z ) / i ]_ C e. CC ) )
71	70	ralima 6498	. . . . . . . . 9 |- ( ( x. Fn ( CC X. CC ) /\ ( X X. Y ) C_ ( CC X. CC ) ) -> ( A. w e. ( x. " ( X X. Y ) ) [_ w / i ]_ C e. CC <-> A. z e. ( X X. Y ) [_ ( x. ` z ) / i ]_ C e. CC ) )
72	60, 69, 71	mp2an 708	. . . . . . . 8 |- ( A. w e. ( x. " ( X X. Y ) ) [_ w / i ]_ C e. CC <-> A. z e. ( X X. Y ) [_ ( x. ` z ) / i ]_ C e. CC )
73		df-ima 5127	. . . . . . . . . 10 |- ( x. " ( X X. Y ) ) = ran ( x. |` ( X X. Y ) )
74		f1ofo 6144	. . . . . . . . . . 11 |- ( ( x. |` ( X X. Y ) ) : ( X X. Y ) -1-1-onto-> Z -> ( x. |` ( X X. Y ) ) : ( X X. Y ) -onto-> Z )
75		forn 6118	. . . . . . . . . . 11 |- ( ( x. |` ( X X. Y ) ) : ( X X. Y ) -onto-> Z -> ran ( x. |` ( X X. Y ) ) = Z )
76	51, 74, 75	3syl 18	. . . . . . . . . 10 |- ( ph -> ran ( x. |` ( X X. Y ) ) = Z )
77	73, 76	syl5eq 2668	. . . . . . . . 9 |- ( ph -> ( x. " ( X X. Y ) ) = Z )
78	77	raleqdv 3144	. . . . . . . 8 |- ( ph -> ( A. w e. ( x. " ( X X. Y ) ) [_ w / i ]_ C e. CC <-> A. w e. Z [_ w / i ]_ C e. CC ) )
79	72, 78	syl5bbr 274	. . . . . . 7 |- ( ph -> ( A. z e. ( X X. Y ) [_ ( x. ` z ) / i ]_ C e. CC <-> A. w e. Z [_ w / i ]_ C e. CC ) )
80	57, 79	mpbid 222	. . . . . 6 |- ( ph -> A. w e. Z [_ w / i ]_ C e. CC )
81	80	r19.21bi 2932	. . . . 5 |- ( ( ph /\ w e. Z ) -> [_ w / i ]_ C e. CC )
82	46, 48, 51, 53, 81	fsumf1o 14454	. . . 4 |- ( ph -> sum_ w e. Z [_ w / i ]_ C = sum_ z e. ( X X. Y ) [_ ( x. ` z ) / i ]_ C )
83	45, 82	syl5eq 2668	. . 3 |- ( ph -> sum_ i e. Z C = sum_ z e. ( X X. Y ) [_ ( x. ` z ) / i ]_ C )
84	41, 83	eqtr4d 2659	. 2 |- ( ph -> sum_ j e. X sum_ k e. Y D = sum_ i e. Z C )
85	20, 28, 84	3eqtrd 2660	1 |- ( ph -> ( sum_ j e. X A x. sum_ k e. Y B ) = sum_ i e. Z C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   [_csb 3533   C_ wss 3574   <.cop 4183   class class class wbr 4653   X. cxp 5112   ran crn 5115   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   1c1 9937   x. cmul 9941   NNcn 11020   ...cfz 12326   sum_csu 14416   || cdvds 14983   gcd cgcd 15216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217

This theorem is referenced by:  sgmmul  24926  dchrisum0fmul  25195