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Theorem fsumdvdsmul 22420
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 11779 . . . 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 22329 . . . . . 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 3388 . . . 4  |-  ( ph  ->  X  C_  ( 1 ... M ) )
7 ssfi 7521 . . . 4  |-  ( ( ( 1 ... M
)  e.  Fin  /\  X  C_  ( 1 ... M ) )  ->  X  e.  Fin )
81, 6, 7syl2anc 654 . . 3  |-  ( ph  ->  X  e.  Fin )
9 fzfid 11779 . . . . 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 22329 . . . . . . 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 3388 . . . . 5  |-  ( ph  ->  Y  C_  ( 1 ... N ) )
15 ssfi 7521 . . . . 5  |-  ( ( ( 1 ... N
)  e.  Fin  /\  Y  C_  ( 1 ... N ) )  ->  Y  e.  Fin )
169, 14, 15syl2anc 654 . . . 4  |-  ( ph  ->  Y  e.  Fin )
17 fsumdvdsmul.5 . . . 4  |-  ( (
ph  /\  k  e.  Y )  ->  B  e.  CC )
1816, 17fsumcl 13194 . . 3  |-  ( ph  -> 
sum_ k  e.  Y  B  e.  CC )
19 fsumdvdsmul.4 . . 3  |-  ( (
ph  /\  j  e.  X )  ->  A  e.  CC )
208, 18, 19fsummulc1 13235 . 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 462 . . . . 5  |-  ( (
ph  /\  j  e.  X )  ->  Y  e.  Fin )
2217adantlr 707 . . . . 5  |-  ( ( ( ph  /\  j  e.  X )  /\  k  e.  Y )  ->  B  e.  CC )
2321, 19, 22fsummulc2 13234 . . . 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 641 . . . . 5  |-  ( ( ( ph  /\  j  e.  X )  /\  k  e.  Y )  ->  ( A  x.  B )  =  D )
2625sumeq2dv 13164 . . . 4  |-  ( (
ph  /\  j  e.  X )  ->  sum_ k  e.  Y  ( A  x.  B )  =  sum_ k  e.  Y  D
)
2723, 26eqtrd 2465 . . 3  |-  ( (
ph  /\  j  e.  X )  ->  ( A  x.  sum_ k  e.  Y  B )  = 
sum_ k  e.  Y  D )
2827sumeq2dv 13164 . 2  |-  ( ph  -> 
sum_ j  e.  X  ( A  x.  sum_ k  e.  Y  B )  =  sum_ j  e.  X  sum_ k  e.  Y  D
)
29 fveq2 5679 . . . . . . 7  |-  ( z  =  <. j ,  k
>.  ->  (  x.  `  z )  =  (  x.  `  <. j ,  k >. )
)
30 df-ov 6083 . . . . . . 7  |-  ( j  x.  k )  =  (  x.  `  <. j ,  k >. )
3129, 30syl6eqr 2483 . . . . . 6  |-  ( z  =  <. j ,  k
>.  ->  (  x.  `  z )  =  ( j  x.  k ) )
3231csbeq1d 3283 . . . . 5  |-  ( z  =  <. j ,  k
>.  ->  [_ (  x.  `  z )  /  i ]_ C  =  [_ (
j  x.  k )  /  i ]_ C
)
33 ovex 6105 . . . . . 6  |-  ( j  x.  k )  e. 
_V
34 nfcv 2569 . . . . . 6  |-  F/_ i D
35 fsumdvdsmul.7 . . . . . 6  |-  ( i  =  ( j  x.  k )  ->  C  =  D )
3633, 34, 35csbief 3301 . . . . 5  |-  [_ (
j  x.  k )  /  i ]_ C  =  D
3732, 36syl6eq 2481 . . . 4  |-  ( z  =  <. j ,  k
>.  ->  [_ (  x.  `  z )  /  i ]_ C  =  D
)
3819adantrr 709 . . . . . 6  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  A  e.  CC )
3917adantrl 708 . . . . . 6  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  B  e.  CC )
4038, 39mulcld 9394 . . . . 5  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  -> 
( A  x.  B
)  e.  CC )
4124, 40eqeltrrd 2508 . . . 4  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  D  e.  CC )
4237, 8, 16, 41fsumxp 13223 . . 3  |-  ( ph  -> 
sum_ j  e.  X  sum_ k  e.  Y  D  =  sum_ z  e.  ( X  X.  Y )
[_ (  x.  `  z )  /  i ]_ C )
43 nfcv 2569 . . . . 5  |-  F/_ w C
44 nfcsb1v 3292 . . . . 5  |-  F/_ i [_ w  /  i ]_ C
45 csbeq1a 3285 . . . . 5  |-  ( i  =  w  ->  C  =  [_ w  /  i ]_ C )
4643, 44, 45cbvsumi 13158 . . . 4  |-  sum_ i  e.  Z  C  =  sum_ w  e.  Z  [_ w  /  i ]_ C
47 csbeq1 3279 . . . . 5  |-  ( w  =  (  x.  `  z )  ->  [_ w  /  i ]_ C  =  [_ (  x.  `  z )  /  i ]_ C )
48 xpfi 7571 . . . . . 6  |-  ( ( X  e.  Fin  /\  Y  e.  Fin )  ->  ( X  X.  Y
)  e.  Fin )
498, 16, 48syl2anc 654 . . . . 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 22419 . . . . 5  |-  ( ph  ->  (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -1-1-onto-> Z )
53 fvres 5692 . . . . . 6  |-  ( z  e.  ( X  X.  Y )  ->  (
(  x.  |`  ( X  X.  Y ) ) `
 z )  =  (  x.  `  z
) )
5453adantl 463 . . . . 5  |-  ( (
ph  /\  z  e.  ( X  X.  Y
) )  ->  (
(  x.  |`  ( X  X.  Y ) ) `
 z )  =  (  x.  `  z
) )
5541ralrimivva 2798 . . . . . . . 8  |-  ( ph  ->  A. j  e.  X  A. k  e.  Y  D  e.  CC )
5637eleq1d 2499 . . . . . . . . 9  |-  ( z  =  <. j ,  k
>.  ->  ( [_ (  x.  `  z )  / 
i ]_ C  e.  CC  <->  D  e.  CC ) )
5756ralxp 4968 . . . . . . . 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 9350 . . . . . . . . . 10  |-  x.  :
( CC  X.  CC )
--> CC
60 ffn 5547 . . . . . . . . . 10  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
6159, 60ax-mp 5 . . . . . . . . 9  |-  x.  Fn  ( CC  X.  CC )
62 ssrab2 3425 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  M }  C_  NN
632, 62eqsstri 3374 . . . . . . . . . . 11  |-  X  C_  NN
64 nnsscn 10315 . . . . . . . . . . 11  |-  NN  C_  CC
6563, 64sstri 3353 . . . . . . . . . 10  |-  X  C_  CC
66 ssrab2 3425 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  N }  C_  NN
6710, 66eqsstri 3374 . . . . . . . . . . 11  |-  Y  C_  NN
6867, 64sstri 3353 . . . . . . . . . 10  |-  Y  C_  CC
69 xpss12 4932 . . . . . . . . . 10  |-  ( ( X  C_  CC  /\  Y  C_  CC )  ->  ( X  X.  Y )  C_  ( CC  X.  CC ) )
7065, 68, 69mp2an 665 . . . . . . . . 9  |-  ( X  X.  Y )  C_  ( CC  X.  CC )
7147eleq1d 2499 . . . . . . . . . 10  |-  ( w  =  (  x.  `  z )  ->  ( [_ w  /  i ]_ C  e.  CC  <->  [_ (  x.  `  z
)  /  i ]_ C  e.  CC )
)
7271ralima 5944 . . . . . . . . 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 665 . . . . . . . 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 4840 . . . . . . . . . 10  |-  (  x.  " ( X  X.  Y ) )  =  ran  (  x.  |`  ( X  X.  Y ) )
75 f1ofo 5636 . . . . . . . . . . 11  |-  ( (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y
)
-1-1-onto-> Z  ->  (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -onto-> Z )
76 forn 5611 . . . . . . . . . . 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 2477 . . . . . . . . 9  |-  ( ph  ->  (  x.  " ( X  X.  Y ) )  =  Z )
7978raleqdv 2913 . . . . . . . 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 2804 . . . . 5  |-  ( (
ph  /\  w  e.  Z )  ->  [_ w  /  i ]_ C  e.  CC )
8347, 49, 52, 54, 82fsumf1o 13184 . . . 4  |-  ( ph  -> 
sum_ w  e.  Z  [_ w  /  i ]_ C  =  sum_ z  e.  ( X  X.  Y
) [_ (  x.  `  z )  /  i ]_ C )
8446, 83syl5eq 2477 . . 3  |-  ( ph  -> 
sum_ i  e.  Z  C  =  sum_ z  e.  ( X  X.  Y
) [_ (  x.  `  z )  /  i ]_ C )
8542, 84eqtr4d 2468 . 2  |-  ( ph  -> 
sum_ j  e.  X  sum_ k  e.  Y  D  =  sum_ i  e.  Z  C )
8620, 28, 853eqtrd 2469 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 1362    e. wcel 1755   A.wral 2705   {crab 2709   [_csb 3276    C_ wss 3316   <.cop 3871   class class class wbr 4280    X. cxp 4825   ran crn 4828    |` cres 4829   "cima 4830    Fn wfn 5401   -->wf 5402   -onto->wfo 5404   -1-1-onto->wf1o 5405   ` cfv 5406  (class class class)co 6080   Fincfn 7298   CCcc 9268   1c1 9271    x. cmul 9275   NNcn 10310   ...cfz 11424   sum_csu 13147    || cdivides 13518    gcd cgcd 13673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-sum 13148  df-dvds 13519  df-gcd 13674
This theorem is referenced by:  sgmmul  22425  dchrisum0fmul  22640
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