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Theorem fsumdvdsmul 23987
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 12183 . . . 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 23896 . . . . . 6  |-  ( M  e.  NN  ->  { x  e.  NN  |  x  ||  M }  C_  ( 1 ... M ) )
53, 4syl 17 . . . . 5  |-  ( ph  ->  { x  e.  NN  |  x  ||  M }  C_  ( 1 ... M
) )
62, 5syl5eqss 3514 . . . 4  |-  ( ph  ->  X  C_  ( 1 ... M ) )
7 ssfi 7798 . . . 4  |-  ( ( ( 1 ... M
)  e.  Fin  /\  X  C_  ( 1 ... M ) )  ->  X  e.  Fin )
81, 6, 7syl2anc 665 . . 3  |-  ( ph  ->  X  e.  Fin )
9 fzfid 12183 . . . . 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 23896 . . . . . . 7  |-  ( N  e.  NN  ->  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N ) )
1311, 12syl 17 . . . . . 6  |-  ( ph  ->  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N
) )
1410, 13syl5eqss 3514 . . . . 5  |-  ( ph  ->  Y  C_  ( 1 ... N ) )
15 ssfi 7798 . . . . 5  |-  ( ( ( 1 ... N
)  e.  Fin  /\  Y  C_  ( 1 ... N ) )  ->  Y  e.  Fin )
169, 14, 15syl2anc 665 . . . 4  |-  ( ph  ->  Y  e.  Fin )
17 fsumdvdsmul.5 . . . 4  |-  ( (
ph  /\  k  e.  Y )  ->  B  e.  CC )
1816, 17fsumcl 13777 . . 3  |-  ( ph  -> 
sum_ k  e.  Y  B  e.  CC )
19 fsumdvdsmul.4 . . 3  |-  ( (
ph  /\  j  e.  X )  ->  A  e.  CC )
208, 18, 19fsummulc1 13824 . 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 466 . . . . 5  |-  ( (
ph  /\  j  e.  X )  ->  Y  e.  Fin )
2217adantlr 719 . . . . 5  |-  ( ( ( ph  /\  j  e.  X )  /\  k  e.  Y )  ->  B  e.  CC )
2321, 19, 22fsummulc2 13823 . . . 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 652 . . . . 5  |-  ( ( ( ph  /\  j  e.  X )  /\  k  e.  Y )  ->  ( A  x.  B )  =  D )
2625sumeq2dv 13747 . . . 4  |-  ( (
ph  /\  j  e.  X )  ->  sum_ k  e.  Y  ( A  x.  B )  =  sum_ k  e.  Y  D
)
2723, 26eqtrd 2470 . . 3  |-  ( (
ph  /\  j  e.  X )  ->  ( A  x.  sum_ k  e.  Y  B )  = 
sum_ k  e.  Y  D )
2827sumeq2dv 13747 . 2  |-  ( ph  -> 
sum_ j  e.  X  ( A  x.  sum_ k  e.  Y  B )  =  sum_ j  e.  X  sum_ k  e.  Y  D
)
29 fveq2 5881 . . . . . . 7  |-  ( z  =  <. j ,  k
>.  ->  (  x.  `  z )  =  (  x.  `  <. j ,  k >. )
)
30 df-ov 6308 . . . . . . 7  |-  ( j  x.  k )  =  (  x.  `  <. j ,  k >. )
3129, 30syl6eqr 2488 . . . . . 6  |-  ( z  =  <. j ,  k
>.  ->  (  x.  `  z )  =  ( j  x.  k ) )
3231csbeq1d 3408 . . . . 5  |-  ( z  =  <. j ,  k
>.  ->  [_ (  x.  `  z )  /  i ]_ C  =  [_ (
j  x.  k )  /  i ]_ C
)
33 ovex 6333 . . . . . 6  |-  ( j  x.  k )  e. 
_V
34 fsumdvdsmul.7 . . . . . 6  |-  ( i  =  ( j  x.  k )  ->  C  =  D )
3533, 34csbie 3427 . . . . 5  |-  [_ (
j  x.  k )  /  i ]_ C  =  D
3632, 35syl6eq 2486 . . . 4  |-  ( z  =  <. j ,  k
>.  ->  [_ (  x.  `  z )  /  i ]_ C  =  D
)
3719adantrr 721 . . . . . 6  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  A  e.  CC )
3817adantrl 720 . . . . . 6  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  B  e.  CC )
3937, 38mulcld 9662 . . . . 5  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  -> 
( A  x.  B
)  e.  CC )
4024, 39eqeltrrd 2518 . . . 4  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  D  e.  CC )
4136, 8, 16, 40fsumxp 13811 . . 3  |-  ( ph  -> 
sum_ j  e.  X  sum_ k  e.  Y  D  =  sum_ z  e.  ( X  X.  Y )
[_ (  x.  `  z )  /  i ]_ C )
42 nfcv 2591 . . . . 5  |-  F/_ w C
43 nfcsb1v 3417 . . . . 5  |-  F/_ i [_ w  /  i ]_ C
44 csbeq1a 3410 . . . . 5  |-  ( i  =  w  ->  C  =  [_ w  /  i ]_ C )
4542, 43, 44cbvsumi 13741 . . . 4  |-  sum_ i  e.  Z  C  =  sum_ w  e.  Z  [_ w  /  i ]_ C
46 csbeq1 3404 . . . . 5  |-  ( w  =  (  x.  `  z )  ->  [_ w  /  i ]_ C  =  [_ (  x.  `  z )  /  i ]_ C )
47 xpfi 7848 . . . . . 6  |-  ( ( X  e.  Fin  /\  Y  e.  Fin )  ->  ( X  X.  Y
)  e.  Fin )
488, 16, 47syl2anc 665 . . . . 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 ) }
513, 11, 49, 2, 10, 50dvdsmulf1o 23986 . . . . 5  |-  ( ph  ->  (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -1-1-onto-> Z )
52 fvres 5895 . . . . . 6  |-  ( z  e.  ( X  X.  Y )  ->  (
(  x.  |`  ( X  X.  Y ) ) `
 z )  =  (  x.  `  z
) )
5352adantl 467 . . . . 5  |-  ( (
ph  /\  z  e.  ( X  X.  Y
) )  ->  (
(  x.  |`  ( X  X.  Y ) ) `
 z )  =  (  x.  `  z
) )
5440ralrimivva 2853 . . . . . . . 8  |-  ( ph  ->  A. j  e.  X  A. k  e.  Y  D  e.  CC )
5536eleq1d 2498 . . . . . . . . 9  |-  ( z  =  <. j ,  k
>.  ->  ( [_ (  x.  `  z )  / 
i ]_ C  e.  CC  <->  D  e.  CC ) )
5655ralxp 4996 . . . . . . . 8  |-  ( A. z  e.  ( X  X.  Y ) [_ (  x.  `  z )  / 
i ]_ C  e.  CC  <->  A. j  e.  X  A. k  e.  Y  D  e.  CC )
5754, 56sylibr 215 . . . . . . 7  |-  ( ph  ->  A. z  e.  ( X  X.  Y )
[_ (  x.  `  z )  /  i ]_ C  e.  CC )
58 ax-mulf 9618 . . . . . . . . . 10  |-  x.  :
( CC  X.  CC )
--> CC
59 ffn 5746 . . . . . . . . . 10  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
6058, 59ax-mp 5 . . . . . . . . 9  |-  x.  Fn  ( CC  X.  CC )
61 ssrab2 3552 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  M }  C_  NN
622, 61eqsstri 3500 . . . . . . . . . . 11  |-  X  C_  NN
63 nnsscn 10614 . . . . . . . . . . 11  |-  NN  C_  CC
6462, 63sstri 3479 . . . . . . . . . 10  |-  X  C_  CC
65 ssrab2 3552 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  N }  C_  NN
6610, 65eqsstri 3500 . . . . . . . . . . 11  |-  Y  C_  NN
6766, 63sstri 3479 . . . . . . . . . 10  |-  Y  C_  CC
68 xpss12 4960 . . . . . . . . . 10  |-  ( ( X  C_  CC  /\  Y  C_  CC )  ->  ( X  X.  Y )  C_  ( CC  X.  CC ) )
6964, 67, 68mp2an 676 . . . . . . . . 9  |-  ( X  X.  Y )  C_  ( CC  X.  CC )
7046eleq1d 2498 . . . . . . . . . 10  |-  ( w  =  (  x.  `  z )  ->  ( [_ w  /  i ]_ C  e.  CC  <->  [_ (  x.  `  z
)  /  i ]_ C  e.  CC )
)
7170ralima 6160 . . . . . . . . 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 ) )
7260, 69, 71mp2an 676 . . . . . . . 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 4867 . . . . . . . . . 10  |-  (  x.  " ( X  X.  Y ) )  =  ran  (  x.  |`  ( X  X.  Y ) )
74 f1ofo 5838 . . . . . . . . . . 11  |-  ( (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y
)
-1-1-onto-> Z  ->  (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -onto-> Z )
75 forn 5813 . . . . . . . . . . 11  |-  ( (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y
) -onto-> Z  ->  ran  (  x.  |`  ( X  X.  Y ) )  =  Z )
7651, 74, 753syl 18 . . . . . . . . . 10  |-  ( ph  ->  ran  (  x.  |`  ( X  X.  Y ) )  =  Z )
7773, 76syl5eq 2482 . . . . . . . . 9  |-  ( ph  ->  (  x.  " ( X  X.  Y ) )  =  Z )
7877raleqdv 3038 . . . . . . . 8  |-  ( ph  ->  ( A. w  e.  (  x.  " ( X  X.  Y ) )
[_ w  /  i ]_ C  e.  CC  <->  A. w  e.  Z  [_ w  /  i ]_ C  e.  CC ) )
7972, 78syl5bbr 262 . . . . . . 7  |-  ( ph  ->  ( A. z  e.  ( X  X.  Y
) [_ (  x.  `  z )  /  i ]_ C  e.  CC  <->  A. w  e.  Z  [_ w  /  i ]_ C  e.  CC ) )
8057, 79mpbid 213 . . . . . 6  |-  ( ph  ->  A. w  e.  Z  [_ w  /  i ]_ C  e.  CC )
8180r19.21bi 2801 . . . . 5  |-  ( (
ph  /\  w  e.  Z )  ->  [_ w  /  i ]_ C  e.  CC )
8246, 48, 51, 53, 81fsumf1o 13767 . . . 4  |-  ( ph  -> 
sum_ w  e.  Z  [_ w  /  i ]_ C  =  sum_ z  e.  ( X  X.  Y
) [_ (  x.  `  z )  /  i ]_ C )
8345, 82syl5eq 2482 . . 3  |-  ( ph  -> 
sum_ i  e.  Z  C  =  sum_ z  e.  ( X  X.  Y
) [_ (  x.  `  z )  /  i ]_ C )
8441, 83eqtr4d 2473 . 2  |-  ( ph  -> 
sum_ j  e.  X  sum_ k  e.  Y  D  =  sum_ i  e.  Z  C )
8520, 28, 843eqtrd 2474 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   {crab 2786   [_csb 3401    C_ wss 3442   <.cop 4008   class class class wbr 4426    X. cxp 4852   ran crn 4855    |` cres 4856   "cima 4857    Fn wfn 5596   -->wf 5597   -onto->wfo 5599   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305   Fincfn 7577   CCcc 9536   1c1 9539    x. cmul 9543   NNcn 10609   ...cfz 11782   sum_csu 13730    || cdvds 14283    gcd cgcd 14442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-dvds 14284  df-gcd 14443
This theorem is referenced by:  sgmmul  23992  dchrisum0fmul  24207
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