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Theorem fsumdvdsmul 23337
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.
StepHypRef Expression
1 fzfid 12063 . . . 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 sgmss 23246 . . . . . 6  |-  ( M  e.  NN  ->  { x  e.  NN  |  x  ||  M }  C_  ( 1 ... M ) )
53, 4syl 16 . . . . 5  |-  ( ph  ->  { x  e.  NN  |  x  ||  M }  C_  ( 1 ... M
) )
62, 5syl5eqss 3553 . . . 4  |-  ( ph  ->  X  C_  ( 1 ... M ) )
7 ssfi 7752 . . . 4  |-  ( ( ( 1 ... M
)  e.  Fin  /\  X  C_  ( 1 ... M ) )  ->  X  e.  Fin )
81, 6, 7syl2anc 661 . . 3  |-  ( ph  ->  X  e.  Fin )
9 fzfid 12063 . . . . 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 sgmss 23246 . . . . . . 7  |-  ( N  e.  NN  ->  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N ) )
1311, 12syl 16 . . . . . 6  |-  ( ph  ->  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N
) )
1410, 13syl5eqss 3553 . . . . 5  |-  ( ph  ->  Y  C_  ( 1 ... N ) )
15 ssfi 7752 . . . . 5  |-  ( ( ( 1 ... N
)  e.  Fin  /\  Y  C_  ( 1 ... N ) )  ->  Y  e.  Fin )
169, 14, 15syl2anc 661 . . . 4  |-  ( ph  ->  Y  e.  Fin )
17 fsumdvdsmul.5 . . . 4  |-  ( (
ph  /\  k  e.  Y )  ->  B  e.  CC )
1816, 17fsumcl 13535 . . 3  |-  ( ph  -> 
sum_ k  e.  Y  B  e.  CC )
19 fsumdvdsmul.4 . . 3  |-  ( (
ph  /\  j  e.  X )  ->  A  e.  CC )
208, 18, 19fsummulc1 13580 . 2  |-  ( ph  ->  ( sum_ j  e.  X  A  x.  sum_ k  e.  Y  B )  = 
sum_ j  e.  X  ( A  x.  sum_ k  e.  Y  B )
)
2116adantr 465 . . . . 5  |-  ( (
ph  /\  j  e.  X )  ->  Y  e.  Fin )
2217adantlr 714 . . . . 5  |-  ( ( ( ph  /\  j  e.  X )  /\  k  e.  Y )  ->  B  e.  CC )
2321, 19, 22fsummulc2 13579 . . . 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 )
2524anassrs 648 . . . . 5  |-  ( ( ( ph  /\  j  e.  X )  /\  k  e.  Y )  ->  ( A  x.  B )  =  D )
2625sumeq2dv 13505 . . . 4  |-  ( (
ph  /\  j  e.  X )  ->  sum_ k  e.  Y  ( A  x.  B )  =  sum_ k  e.  Y  D
)
2723, 26eqtrd 2508 . . 3  |-  ( (
ph  /\  j  e.  X )  ->  ( A  x.  sum_ k  e.  Y  B )  = 
sum_ k  e.  Y  D )
2827sumeq2dv 13505 . 2  |-  ( ph  -> 
sum_ j  e.  X  ( A  x.  sum_ k  e.  Y  B )  =  sum_ j  e.  X  sum_ k  e.  Y  D
)
29 fveq2 5872 . . . . . . 7  |-  ( z  =  <. j ,  k
>.  ->  (  x.  `  z )  =  (  x.  `  <. j ,  k >. )
)
30 df-ov 6298 . . . . . . 7  |-  ( j  x.  k )  =  (  x.  `  <. j ,  k >. )
3129, 30syl6eqr 2526 . . . . . 6  |-  ( z  =  <. j ,  k
>.  ->  (  x.  `  z )  =  ( j  x.  k ) )
3231csbeq1d 3447 . . . . 5  |-  ( z  =  <. j ,  k
>.  ->  [_ (  x.  `  z )  /  i ]_ C  =  [_ (
j  x.  k )  /  i ]_ C
)
33 ovex 6320 . . . . . 6  |-  ( j  x.  k )  e. 
_V
34 nfcv 2629 . . . . . 6  |-  F/_ i D
35 fsumdvdsmul.7 . . . . . 6  |-  ( i  =  ( j  x.  k )  ->  C  =  D )
3633, 34, 35csbief 3465 . . . . 5  |-  [_ (
j  x.  k )  /  i ]_ C  =  D
3732, 36syl6eq 2524 . . . 4  |-  ( z  =  <. j ,  k
>.  ->  [_ (  x.  `  z )  /  i ]_ C  =  D
)
3819adantrr 716 . . . . . 6  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  A  e.  CC )
3917adantrl 715 . . . . . 6  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  B  e.  CC )
4038, 39mulcld 9628 . . . . 5  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  -> 
( A  x.  B
)  e.  CC )
4124, 40eqeltrrd 2556 . . . 4  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  D  e.  CC )
4237, 8, 16, 41fsumxp 13567 . . 3  |-  ( ph  -> 
sum_ j  e.  X  sum_ k  e.  Y  D  =  sum_ z  e.  ( X  X.  Y )
[_ (  x.  `  z )  /  i ]_ C )
43 nfcv 2629 . . . . 5  |-  F/_ w C
44 nfcsb1v 3456 . . . . 5  |-  F/_ i [_ w  /  i ]_ C
45 csbeq1a 3449 . . . . 5  |-  ( i  =  w  ->  C  =  [_ w  /  i ]_ C )
4643, 44, 45cbvsumi 13499 . . . 4  |-  sum_ i  e.  Z  C  =  sum_ w  e.  Z  [_ w  /  i ]_ C
47 csbeq1 3443 . . . . 5  |-  ( w  =  (  x.  `  z )  ->  [_ w  /  i ]_ C  =  [_ (  x.  `  z )  /  i ]_ C )
48 xpfi 7803 . . . . . 6  |-  ( ( X  e.  Fin  /\  Y  e.  Fin )  ->  ( X  X.  Y
)  e.  Fin )
498, 16, 48syl2anc 661 . . . . 5  |-  ( ph  ->  ( X  X.  Y
)  e.  Fin )
50 dvdsmulf1o.3 . . . . . 6  |-  ( ph  ->  ( M  gcd  N
)  =  1 )
51 dvdsmulf1o.z . . . . . 6  |-  Z  =  { x  e.  NN  |  x  ||  ( M  x.  N ) }
523, 11, 50, 2, 10, 51dvdsmulf1o 23336 . . . . 5  |-  ( ph  ->  (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -1-1-onto-> Z )
53 fvres 5886 . . . . . 6  |-  ( z  e.  ( X  X.  Y )  ->  (
(  x.  |`  ( X  X.  Y ) ) `
 z )  =  (  x.  `  z
) )
5453adantl 466 . . . . 5  |-  ( (
ph  /\  z  e.  ( X  X.  Y
) )  ->  (
(  x.  |`  ( X  X.  Y ) ) `
 z )  =  (  x.  `  z
) )
5541ralrimivva 2888 . . . . . . . 8  |-  ( ph  ->  A. j  e.  X  A. k  e.  Y  D  e.  CC )
5637eleq1d 2536 . . . . . . . . 9  |-  ( z  =  <. j ,  k
>.  ->  ( [_ (  x.  `  z )  / 
i ]_ C  e.  CC  <->  D  e.  CC ) )
5756ralxp 5150 . . . . . . . 8  |-  ( A. z  e.  ( X  X.  Y ) [_ (  x.  `  z )  / 
i ]_ C  e.  CC  <->  A. j  e.  X  A. k  e.  Y  D  e.  CC )
5855, 57sylibr 212 . . . . . . 7  |-  ( ph  ->  A. z  e.  ( X  X.  Y )
[_ (  x.  `  z )  /  i ]_ C  e.  CC )
59 ax-mulf 9584 . . . . . . . . . 10  |-  x.  :
( CC  X.  CC )
--> CC
60 ffn 5737 . . . . . . . . . 10  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
6159, 60ax-mp 5 . . . . . . . . 9  |-  x.  Fn  ( CC  X.  CC )
62 ssrab2 3590 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  M }  C_  NN
632, 62eqsstri 3539 . . . . . . . . . . 11  |-  X  C_  NN
64 nnsscn 10553 . . . . . . . . . . 11  |-  NN  C_  CC
6563, 64sstri 3518 . . . . . . . . . 10  |-  X  C_  CC
66 ssrab2 3590 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  N }  C_  NN
6710, 66eqsstri 3539 . . . . . . . . . . 11  |-  Y  C_  NN
6867, 64sstri 3518 . . . . . . . . . 10  |-  Y  C_  CC
69 xpss12 5114 . . . . . . . . . 10  |-  ( ( X  C_  CC  /\  Y  C_  CC )  ->  ( X  X.  Y )  C_  ( CC  X.  CC ) )
7065, 68, 69mp2an 672 . . . . . . . . 9  |-  ( X  X.  Y )  C_  ( CC  X.  CC )
7147eleq1d 2536 . . . . . . . . . 10  |-  ( w  =  (  x.  `  z )  ->  ( [_ w  /  i ]_ C  e.  CC  <->  [_ (  x.  `  z
)  /  i ]_ C  e.  CC )
)
7271ralima 6151 . . . . . . . . 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 ) )
7361, 70, 72mp2an 672 . . . . . . . 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 )
74 df-ima 5018 . . . . . . . . . 10  |-  (  x.  " ( X  X.  Y ) )  =  ran  (  x.  |`  ( X  X.  Y ) )
75 f1ofo 5829 . . . . . . . . . . 11  |-  ( (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y
)
-1-1-onto-> Z  ->  (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -onto-> Z )
76 forn 5804 . . . . . . . . . . 11  |-  ( (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y
) -onto-> Z  ->  ran  (  x.  |`  ( X  X.  Y ) )  =  Z )
7752, 75, 763syl 20 . . . . . . . . . 10  |-  ( ph  ->  ran  (  x.  |`  ( X  X.  Y ) )  =  Z )
7874, 77syl5eq 2520 . . . . . . . . 9  |-  ( ph  ->  (  x.  " ( X  X.  Y ) )  =  Z )
7978raleqdv 3069 . . . . . . . 8  |-  ( ph  ->  ( A. w  e.  (  x.  " ( X  X.  Y ) )
[_ w  /  i ]_ C  e.  CC  <->  A. w  e.  Z  [_ w  /  i ]_ C  e.  CC ) )
8073, 79syl5bbr 259 . . . . . . 7  |-  ( ph  ->  ( A. z  e.  ( X  X.  Y
) [_ (  x.  `  z )  /  i ]_ C  e.  CC  <->  A. w  e.  Z  [_ w  /  i ]_ C  e.  CC ) )
8158, 80mpbid 210 . . . . . 6  |-  ( ph  ->  A. w  e.  Z  [_ w  /  i ]_ C  e.  CC )
8281r19.21bi 2836 . . . . 5  |-  ( (
ph  /\  w  e.  Z )  ->  [_ w  /  i ]_ C  e.  CC )
8347, 49, 52, 54, 82fsumf1o 13525 . . . 4  |-  ( ph  -> 
sum_ w  e.  Z  [_ w  /  i ]_ C  =  sum_ z  e.  ( X  X.  Y
) [_ (  x.  `  z )  /  i ]_ C )
8446, 83syl5eq 2520 . . 3  |-  ( ph  -> 
sum_ i  e.  Z  C  =  sum_ z  e.  ( X  X.  Y
) [_ (  x.  `  z )  /  i ]_ C )
8542, 84eqtr4d 2511 . 2  |-  ( ph  -> 
sum_ j  e.  X  sum_ k  e.  Y  D  =  sum_ i  e.  Z  C )
8620, 28, 853eqtrd 2512 1  |-  ( ph  ->  ( sum_ j  e.  X  A  x.  sum_ k  e.  Y  B )  = 
sum_ i  e.  Z  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821   [_csb 3440    C_ wss 3481   <.cop 4039   class class class wbr 4453    X. cxp 5003   ran crn 5006    |` cres 5007   "cima 5008    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   Fincfn 7528   CCcc 9502   1c1 9505    x. cmul 9509   NNcn 10548   ...cfz 11684   sum_csu 13488    || cdivides 13864    gcd cgcd 14020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-sum 13489  df-dvds 13865  df-gcd 14021
This theorem is referenced by:  sgmmul  23342  dchrisum0fmul  23557
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