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Theorem fsumdvdsmul 20383
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
StepHypRef Expression
1 fzfid 10987 . . . 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 20292 . . . . . 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 3183 . . . 4  |-  ( ph  ->  X  C_  ( 1 ... M ) )
7 ssfi 7037 . . . 4  |-  ( ( ( 1 ... M
)  e.  Fin  /\  X  C_  ( 1 ... M ) )  ->  X  e.  Fin )
81, 6, 7syl2anc 645 . . 3  |-  ( ph  ->  X  e.  Fin )
9 fzfid 10987 . . . . 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 20292 . . . . . . 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 3183 . . . . 5  |-  ( ph  ->  Y  C_  ( 1 ... N ) )
15 ssfi 7037 . . . . 5  |-  ( ( ( 1 ... N
)  e.  Fin  /\  Y  C_  ( 1 ... N ) )  ->  Y  e.  Fin )
169, 14, 15syl2anc 645 . . . 4  |-  ( ph  ->  Y  e.  Fin )
17 fsumdvdsmul.5 . . . 4  |-  ( (
ph  /\  k  e.  Y )  ->  B  e.  CC )
1816, 17fsumcl 12157 . . 3  |-  ( ph  -> 
sum_ k  e.  Y  B  e.  CC )
19 fsumdvdsmul.4 . . 3  |-  ( (
ph  /\  j  e.  X )  ->  A  e.  CC )
208, 18, 19fsummulc1 12198 . 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 453 . . . . 5  |-  ( (
ph  /\  j  e.  X )  ->  Y  e.  Fin )
2217adantlr 698 . . . . 5  |-  ( ( ( ph  /\  j  e.  X )  /\  k  e.  Y )  ->  B  e.  CC )
2321, 19, 22fsummulc2 12197 . . . 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 632 . . . . 5  |-  ( ( ( ph  /\  j  e.  X )  /\  k  e.  Y )  ->  ( A  x.  B )  =  D )
2625sumeq2dv 12127 . . . 4  |-  ( (
ph  /\  j  e.  X )  ->  sum_ k  e.  Y  ( A  x.  B )  =  sum_ k  e.  Y  D
)
2723, 26eqtrd 2288 . . 3  |-  ( (
ph  /\  j  e.  X )  ->  ( A  x.  sum_ k  e.  Y  B )  = 
sum_ k  e.  Y  D )
2827sumeq2dv 12127 . 2  |-  ( ph  -> 
sum_ j  e.  X  ( A  x.  sum_ k  e.  Y  B )  =  sum_ j  e.  X  sum_ k  e.  Y  D
)
29 fveq2 5444 . . . . . . 7  |-  ( z  =  <. j ,  k
>.  ->  (  x.  `  z )  =  (  x.  `  <. j ,  k >. )
)
30 df-ov 5781 . . . . . . 7  |-  ( j  x.  k )  =  (  x.  `  <. j ,  k >. )
3129, 30syl6eqr 2306 . . . . . 6  |-  ( z  =  <. j ,  k
>.  ->  (  x.  `  z )  =  ( j  x.  k ) )
3231csbeq1d 3048 . . . . 5  |-  ( z  =  <. j ,  k
>.  ->  [_ (  x.  `  z )  /  i ]_ C  =  [_ (
j  x.  k )  /  i ]_ C
)
33 ovex 5803 . . . . . 6  |-  ( j  x.  k )  e. 
_V
34 nfcv 2392 . . . . . 6  |-  F/_ i D
35 fsumdvdsmul.7 . . . . . 6  |-  ( i  =  ( j  x.  k )  ->  C  =  D )
3633, 34, 35csbief 3083 . . . . 5  |-  [_ (
j  x.  k )  /  i ]_ C  =  D
3732, 36syl6eq 2304 . . . 4  |-  ( z  =  <. j ,  k
>.  ->  [_ (  x.  `  z )  /  i ]_ C  =  D
)
3819adantrr 700 . . . . . 6  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  A  e.  CC )
3917adantrl 699 . . . . . 6  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  B  e.  CC )
4038, 39mulcld 8809 . . . . 5  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  -> 
( A  x.  B
)  e.  CC )
4124, 40eqeltrrd 2331 . . . 4  |-  ( (
ph  /\  ( j  e.  X  /\  k  e.  Y ) )  ->  D  e.  CC )
4237, 8, 16, 41fsumxp 12186 . . 3  |-  ( ph  -> 
sum_ j  e.  X  sum_ k  e.  Y  D  =  sum_ z  e.  ( X  X.  Y )
[_ (  x.  `  z )  /  i ]_ C )
43 nfcv 2392 . . . . 5  |-  F/_ w C
44 nfcsb1v 3074 . . . . 5  |-  F/_ i [_ w  /  i ]_ C
45 csbeq1a 3050 . . . . 5  |-  ( i  =  w  ->  C  =  [_ w  /  i ]_ C )
4643, 44, 45cbvsumi 12121 . . . 4  |-  sum_ i  e.  Z  C  =  sum_ w  e.  Z  [_ w  /  i ]_ C
47 csbeq1 3045 . . . . 5  |-  ( w  =  (  x.  `  z )  ->  [_ w  /  i ]_ C  =  [_ (  x.  `  z )  /  i ]_ C )
48 xpfi 7082 . . . . . 6  |-  ( ( X  e.  Fin  /\  Y  e.  Fin )  ->  ( X  X.  Y
)  e.  Fin )
498, 16, 48syl2anc 645 . . . . 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 20382 . . . . 5  |-  ( ph  ->  (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -1-1-onto-> Z )
53 fvres 5461 . . . . . 6  |-  ( z  e.  ( X  X.  Y )  ->  (
(  x.  |`  ( X  X.  Y ) ) `
 z )  =  (  x.  `  z
) )
5453adantl 454 . . . . 5  |-  ( (
ph  /\  z  e.  ( X  X.  Y
) )  ->  (
(  x.  |`  ( X  X.  Y ) ) `
 z )  =  (  x.  `  z
) )
5541ralrimivva 2608 . . . . . . . 8  |-  ( ph  ->  A. j  e.  X  A. k  e.  Y  D  e.  CC )
5637eleq1d 2322 . . . . . . . . 9  |-  ( z  =  <. j ,  k
>.  ->  ( [_ (  x.  `  z )  / 
i ]_ C  e.  CC  <->  D  e.  CC ) )
5756ralxp 4801 . . . . . . . 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 205 . . . . . . 7  |-  ( ph  ->  A. z  e.  ( X  X.  Y )
[_ (  x.  `  z )  /  i ]_ C  e.  CC )
59 ax-mulf 8771 . . . . . . . . . 10  |-  x.  :
( CC  X.  CC )
--> CC
60 ffn 5313 . . . . . . . . . 10  |-  (  x.  : ( CC  X.  CC ) --> CC  ->  x.  Fn  ( CC  X.  CC ) )
6159, 60ax-mp 10 . . . . . . . . 9  |-  x.  Fn  ( CC  X.  CC )
62 ssrab2 3219 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  M }  C_  NN
632, 62eqsstri 3169 . . . . . . . . . . 11  |-  X  C_  NN
64 nnsscn 9705 . . . . . . . . . . 11  |-  NN  C_  CC
6563, 64sstri 3149 . . . . . . . . . 10  |-  X  C_  CC
66 ssrab2 3219 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  N }  C_  NN
6710, 66eqsstri 3169 . . . . . . . . . . 11  |-  Y  C_  NN
6867, 64sstri 3149 . . . . . . . . . 10  |-  Y  C_  CC
69 xpss12 4766 . . . . . . . . . 10  |-  ( ( X  C_  CC  /\  Y  C_  CC )  ->  ( X  X.  Y )  C_  ( CC  X.  CC ) )
7065, 68, 69mp2an 656 . . . . . . . . 9  |-  ( X  X.  Y )  C_  ( CC  X.  CC )
7147eleq1d 2322 . . . . . . . . . 10  |-  ( w  =  (  x.  `  z )  ->  ( [_ w  /  i ]_ C  e.  CC  <->  [_ (  x.  `  z
)  /  i ]_ C  e.  CC )
)
7271ralima 5678 . . . . . . . . 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 656 . . . . . . . 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 4668 . . . . . . . . . 10  |-  (  x.  " ( X  X.  Y ) )  =  ran  (  x.  |`  ( X  X.  Y ) )
75 f1ofo 5403 . . . . . . . . . . 11  |-  ( (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y
)
-1-1-onto-> Z  ->  (  x.  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -onto-> Z )
76 forn 5378 . . . . . . . . . . 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 2300 . . . . . . . . 9  |-  ( ph  ->  (  x.  " ( X  X.  Y ) )  =  Z )
7978raleqdv 2713 . . . . . . . 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 252 . . . . . . 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 203 . . . . . 6  |-  ( ph  ->  A. w  e.  Z  [_ w  /  i ]_ C  e.  CC )
8281r19.21bi 2614 . . . . 5  |-  ( (
ph  /\  w  e.  Z )  ->  [_ w  /  i ]_ C  e.  CC )
8347, 49, 52, 54, 82fsumf1o 12147 . . . 4  |-  ( ph  -> 
sum_ w  e.  Z  [_ w  /  i ]_ C  =  sum_ z  e.  ( X  X.  Y
) [_ (  x.  `  z )  /  i ]_ C )
8446, 83syl5eq 2300 . . 3  |-  ( ph  -> 
sum_ i  e.  Z  C  =  sum_ z  e.  ( X  X.  Y
) [_ (  x.  `  z )  /  i ]_ C )
8542, 84eqtr4d 2291 . 2  |-  ( ph  -> 
sum_ j  e.  X  sum_ k  e.  Y  D  =  sum_ i  e.  Z  C )
8620, 28, 853eqtrd 2292 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   {crab 2520   [_csb 3042    C_ wss 3113   <.cop 3603   class class class wbr 3983    X. cxp 4645   ran crn 4648    |` cres 4649   "cima 4650    Fn wfn 4654   -->wf 4655   -onto->wfo 4657   -1-1-onto->wf1o 4658   ` cfv 4659  (class class class)co 5778   Fincfn 6817   CCcc 8689   1c1 8692    x. cmul 8696   NNcn 9700   ...cfz 10734   sum_csu 12109    || cdivides 12479    gcd cgcd 12633
This theorem is referenced by:  sgmmul  20388  dchrisum0fmul  20603
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-mulf 8771
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-sup 7148  df-oi 7179  df-card 7526  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-n0 9919  df-z 9978  df-uz 10184  df-rp 10308  df-fz 10735  df-fzo 10823  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-hash 11290  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-clim 11913  df-sum 12110  df-divides 12480  df-gcd 12634
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