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Theorem fsumdvdsmul 20980
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 11312 . . . 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 20889 . . . . . 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 3392 . . . 4  |-  ( ph  ->  X  C_  ( 1 ... M ) )
7 ssfi 7329 . . . 4  |-  ( ( ( 1 ... M
)  e.  Fin  /\  X  C_  ( 1 ... M ) )  ->  X  e.  Fin )
81, 6, 7syl2anc 643 . . 3  |-  ( ph  ->  X  e.  Fin )
9 fzfid 11312 . . . . 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 20889 . . . . . . 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 3392 . . . . 5  |-  ( ph  ->  Y  C_  ( 1 ... N ) )
15 ssfi 7329 . . . . 5  |-  ( ( ( 1 ... N
)  e.  Fin  /\  Y  C_  ( 1 ... N ) )  ->  Y  e.  Fin )
169, 14, 15syl2anc 643 . . . 4  |-  ( ph  ->  Y  e.  Fin )
17 fsumdvdsmul.5 . . . 4  |-  ( (
ph  /\  k  e.  Y )  ->  B  e.  CC )
1816, 17fsumcl 12527 . . 3  |-  ( ph  -> 
sum_ k  e.  Y  B  e.  CC )
19 fsumdvdsmul.4 . . 3  |-  ( (
ph  /\  j  e.  X )  ->  A  e.  CC )
208, 18, 19fsummulc1 12568 . 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 452 . . . . 5  |-  ( (
ph  /\  j  e.  X )  ->  Y  e.  Fin )
2217adantlr 696 . . . . 5  |-  ( ( ( ph  /\  j  e.  X )  /\  k  e.  Y )  ->  B  e.  CC )
2321, 19, 22fsummulc2 12567 . . . 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 630 . . . . 5  |-  ( ( ( ph  /\  j  e.  X )  /\  k  e.  Y )  ->  ( A  x.  B )  =  D )
2625sumeq2dv 12497 . . . 4  |-  ( (
ph  /\  j  e.  X )  ->  sum_ k  e.  Y  ( A  x.  B )  =  sum_ k  e.  Y  D
)
2723, 26eqtrd 2468 . . 3  |-  ( (
ph  /\  j  e.  X )  ->  ( A  x.  sum_ k  e.  Y  B )  = 
sum_ k  e.  Y  D )
2827sumeq2dv 12497 . 2  |-  ( ph  -> 
sum_ j  e.  X  ( A  x.  sum_ k  e.  Y  B )  =  sum_ j  e.  X  sum_ k  e.  Y  D
)
29 fveq2 5728 . . . . . . 7  |-  ( z  =  <. j ,  k
>.  ->  (  x.  `  z )  =  (  x.  `  <. j ,  k >. )
)
30 df-ov 6084 . . . . . . 7  |-  ( j  x.  k )  =  (  x.  `  <. j ,  k >. )
3129, 30syl6eqr 2486 . . . . . 6  |-  ( z  =  <. j ,  k
>.  ->  (  x.  `  z )  =  ( j  x.  k ) )
3231csbeq1d 3257 . . . . 5  |-  ( z  =  <. j ,  k
>.  ->  [_ (  x.  `  z )  /  i ]_ C  =  [_ (
j  x.  k )  /  i ]_ C
)
33 ovex 6106 . . . . . 6  |-  ( j  x.  k )  e. 
_V
34 nfcv 2572 . . . . . 6  |-  F/_ i D
35 fsumdvdsmul.7 . . . . . 6  |-  ( i  =  ( j  x.  k )  ->  C  =  D )
3633, 34, 35csbief 3292 . . . . 5  |-  [_ (
j  x.  k )  /  i ]_ C  =  D
3732, 36syl6eq 2484 . . . 4  |-  ( z  =  <. j ,  k
>.  ->  [_ (  x.  `  z )  /  i ]_ C  =  D
)
3819adantrr 698 . . . . . 6  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  A  e.  CC )
3917adantrl 697 . . . . . 6  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  B  e.  CC )
4038, 39mulcld 9108 . . . . 5  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  -> 
( A  x.  B
)  e.  CC )
4124, 40eqeltrrd 2511 . . . 4  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  D  e.  CC )
4237, 8, 16, 41fsumxp 12556 . . 3  |-  ( ph  -> 
sum_ j  e.  X  sum_ k  e.  Y  D  =  sum_ z  e.  ( X  X.  Y )
[_ (  x.  `  z )  /  i ]_ C )
43 nfcv 2572 . . . . 5  |-  F/_ w C
44 nfcsb1v 3283 . . . . 5  |-  F/_ i [_ w  /  i ]_ C
45 csbeq1a 3259 . . . . 5  |-  ( i  =  w  ->  C  =  [_ w  /  i ]_ C )
4643, 44, 45cbvsumi 12491 . . . 4  |-  sum_ i  e.  Z  C  =  sum_ w  e.  Z  [_ w  /  i ]_ C
47 csbeq1 3254 . . . . 5  |-  ( w  =  (  x.  `  z )  ->  [_ w  /  i ]_ C  =  [_ (  x.  `  z )  /  i ]_ C )
48 xpfi 7378 . . . . . 6  |-  ( ( X  e.  Fin  /\  Y  e.  Fin )  ->  ( X  X.  Y
)  e.  Fin )
498, 16, 48syl2anc 643 . . . . 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 20979 . . . . 5  |-  ( ph  ->  (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -1-1-onto-> Z )
53 fvres 5745 . . . . . 6  |-  ( z  e.  ( X  X.  Y )  ->  (
(  x.  |`  ( X  X.  Y ) ) `
 z )  =  (  x.  `  z
) )
5453adantl 453 . . . . 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 2502 . . . . . . . . 9  |-  ( z  =  <. j ,  k
>.  ->  ( [_ (  x.  `  z )  / 
i ]_ C  e.  CC  <->  D  e.  CC ) )
5756ralxp 5016 . . . . . . . 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 204 . . . . . . 7  |-  ( ph  ->  A. z  e.  ( X  X.  Y )
[_ (  x.  `  z )  /  i ]_ C  e.  CC )
59 ax-mulf 9070 . . . . . . . . . 10  |-  x.  :
( CC  X.  CC )
--> CC
60 ffn 5591 . . . . . . . . . 10  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
6159, 60ax-mp 8 . . . . . . . . 9  |-  x.  Fn  ( CC  X.  CC )
62 ssrab2 3428 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  M }  C_  NN
632, 62eqsstri 3378 . . . . . . . . . . 11  |-  X  C_  NN
64 nnsscn 10005 . . . . . . . . . . 11  |-  NN  C_  CC
6563, 64sstri 3357 . . . . . . . . . 10  |-  X  C_  CC
66 ssrab2 3428 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  N }  C_  NN
6710, 66eqsstri 3378 . . . . . . . . . . 11  |-  Y  C_  NN
6867, 64sstri 3357 . . . . . . . . . 10  |-  Y  C_  CC
69 xpss12 4981 . . . . . . . . . 10  |-  ( ( X  C_  CC  /\  Y  C_  CC )  ->  ( X  X.  Y )  C_  ( CC  X.  CC ) )
7065, 68, 69mp2an 654 . . . . . . . . 9  |-  ( X  X.  Y )  C_  ( CC  X.  CC )
7147eleq1d 2502 . . . . . . . . . 10  |-  ( w  =  (  x.  `  z )  ->  ( [_ w  /  i ]_ C  e.  CC  <->  [_ (  x.  `  z
)  /  i ]_ C  e.  CC )
)
7271ralima 5978 . . . . . . . . 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 654 . . . . . . . 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 4891 . . . . . . . . . 10  |-  (  x.  " ( X  X.  Y ) )  =  ran  (  x.  |`  ( X  X.  Y ) )
75 f1ofo 5681 . . . . . . . . . . 11  |-  ( (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y
)
-1-1-onto-> Z  ->  (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -onto-> Z )
76 forn 5656 . . . . . . . . . . 11  |-  ( (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y
) -onto-> Z  ->  ran  (  x.  |`  ( X  X.  Y ) )  =  Z )
7752, 75, 763syl 19 . . . . . . . . . 10  |-  ( ph  ->  ran  (  x.  |`  ( X  X.  Y ) )  =  Z )
7874, 77syl5eq 2480 . . . . . . . . 9  |-  ( ph  ->  (  x.  " ( X  X.  Y ) )  =  Z )
7978raleqdv 2910 . . . . . . . 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 251 . . . . . . 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 202 . . . . . 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 12517 . . . 4  |-  ( ph  -> 
sum_ w  e.  Z  [_ w  /  i ]_ C  =  sum_ z  e.  ( X  X.  Y
) [_ (  x.  `  z )  /  i ]_ C )
8446, 83syl5eq 2480 . . 3  |-  ( ph  -> 
sum_ i  e.  Z  C  =  sum_ z  e.  ( X  X.  Y
) [_ (  x.  `  z )  /  i ]_ C )
8542, 84eqtr4d 2471 . 2  |-  ( ph  -> 
sum_ j  e.  X  sum_ k  e.  Y  D  =  sum_ i  e.  Z  C )
8620, 28, 853eqtrd 2472 1  |-  ( ph  ->  ( sum_ j  e.  X  A  x.  sum_ k  e.  Y  B )  = 
sum_ i  e.  Z  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   [_csb 3251    C_ wss 3320   <.cop 3817   class class class wbr 4212    X. cxp 4876   ran crn 4879    |` cres 4880   "cima 4881    Fn wfn 5449   -->wf 5450   -onto->wfo 5452   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   Fincfn 7109   CCcc 8988   1c1 8991    x. cmul 8995   NNcn 10000   ...cfz 11043   sum_csu 12479    || cdivides 12852    gcd cgcd 13006
This theorem is referenced by:  sgmmul  20985  dchrisum0fmul  21200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-dvds 12853  df-gcd 13007
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