MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsumcom2 Unicode version

Theorem fsumcom2 12237
Description: Interchange order of summation. Note that  B ( j ) and 
D ( k ) are not necessarily constant expressions. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
Hypotheses
Ref Expression
fsumcom2.1  |-  ( ph  ->  A  e.  Fin )
fsumcom2.2  |-  ( ph  ->  C  e.  Fin )
fsumcom2.3  |-  ( (
ph  /\  j  e.  A )  ->  B  e.  Fin )
fsumcom2.4  |-  ( ph  ->  ( ( j  e.  A  /\  k  e.  B )  <->  ( k  e.  C  /\  j  e.  D ) ) )
fsumcom2.5  |-  ( (
ph  /\  ( j  e.  A  /\  k  e.  B ) )  ->  E  e.  CC )
Assertion
Ref Expression
fsumcom2  |-  ( ph  -> 
sum_ j  e.  A  sum_ k  e.  B  E  =  sum_ k  e.  C  sum_ j  e.  D  E
)
Distinct variable groups:    j, k, A    C, j, k    ph, j,
k    B, k    D, j
Allowed substitution hints:    B( j)    D( k)    E( j, k)

Proof of Theorem fsumcom2
Dummy variables  m  n  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4794 . . . . . . . . 9  |-  Rel  ( { j }  X.  B )
21rgenw 2610 . . . . . . . 8  |-  A. j  e.  A  Rel  ( { j }  X.  B
)
3 reliun 4806 . . . . . . . 8  |-  ( Rel  U_ j  e.  A  ( { j }  X.  B )  <->  A. j  e.  A  Rel  ( { j }  X.  B
) )
42, 3mpbir 200 . . . . . . 7  |-  Rel  U_ j  e.  A  ( {
j }  X.  B
)
5 relcnv 5051 . . . . . . 7  |-  Rel  `' U_ k  e.  C  ( { k }  X.  D )
6 ancom 437 . . . . . . . . . . . 12  |-  ( ( x  =  j  /\  y  =  k )  <->  ( y  =  k  /\  x  =  j )
)
7 vex 2791 . . . . . . . . . . . . 13  |-  x  e. 
_V
8 vex 2791 . . . . . . . . . . . . 13  |-  y  e. 
_V
97, 8opth 4245 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  =  <. j ,  k
>. 
<->  ( x  =  j  /\  y  =  k ) )
108, 7opth 4245 . . . . . . . . . . . 12  |-  ( <.
y ,  x >.  = 
<. k ,  j >.  <->  ( y  =  k  /\  x  =  j )
)
116, 9, 103bitr4i 268 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  =  <. j ,  k
>. 
<-> 
<. y ,  x >.  = 
<. k ,  j >.
)
1211a1i 10 . . . . . . . . . 10  |-  ( ph  ->  ( <. x ,  y
>.  =  <. j ,  k >.  <->  <. y ,  x >.  =  <. k ,  j
>. ) )
13 fsumcom2.4 . . . . . . . . . 10  |-  ( ph  ->  ( ( j  e.  A  /\  k  e.  B )  <->  ( k  e.  C  /\  j  e.  D ) ) )
1412, 13anbi12d 691 . . . . . . . . 9  |-  ( ph  ->  ( ( <. x ,  y >.  =  <. j ,  k >.  /\  (
j  e.  A  /\  k  e.  B )
)  <->  ( <. y ,  x >.  =  <. k ,  j >.  /\  (
k  e.  C  /\  j  e.  D )
) ) )
15142exbidv 1614 . . . . . . . 8  |-  ( ph  ->  ( E. j E. k ( <. x ,  y >.  =  <. j ,  k >.  /\  (
j  e.  A  /\  k  e.  B )
)  <->  E. j E. k
( <. y ,  x >.  =  <. k ,  j
>.  /\  ( k  e.  C  /\  j  e.  D ) ) ) )
16 eliunxp 4823 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  U_ j  e.  A  ( { j }  X.  B )  <->  E. j E. k ( <. x ,  y >.  =  <. j ,  k >.  /\  (
j  e.  A  /\  k  e.  B )
) )
177, 8opelcnv 4863 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  `' U_ k  e.  C  ( { k }  X.  D )  <->  <. y ,  x >.  e.  U_ k  e.  C  ( {
k }  X.  D
) )
18 eliunxp 4823 . . . . . . . . 9  |-  ( <.
y ,  x >.  e. 
U_ k  e.  C  ( { k }  X.  D )  <->  E. k E. j ( <. y ,  x >.  =  <. k ,  j >.  /\  (
k  e.  C  /\  j  e.  D )
) )
19 excom 1786 . . . . . . . . 9  |-  ( E. k E. j (
<. y ,  x >.  = 
<. k ,  j >.  /\  ( k  e.  C  /\  j  e.  D
) )  <->  E. j E. k ( <. y ,  x >.  =  <. k ,  j >.  /\  (
k  e.  C  /\  j  e.  D )
) )
2017, 18, 193bitri 262 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  `' U_ k  e.  C  ( { k }  X.  D )  <->  E. j E. k ( <. y ,  x >.  =  <. k ,  j >.  /\  (
k  e.  C  /\  j  e.  D )
) )
2115, 16, 203bitr4g 279 . . . . . . 7  |-  ( ph  ->  ( <. x ,  y
>.  e.  U_ j  e.  A  ( { j }  X.  B )  <->  <. x ,  y >.  e.  `' U_ k  e.  C  ( { k }  X.  D ) ) )
224, 5, 21eqrelrdv 4783 . . . . . 6  |-  ( ph  ->  U_ j  e.  A  ( { j }  X.  B )  =  `' U_ k  e.  C  ( { k }  X.  D ) )
23 nfcv 2419 . . . . . . 7  |-  F/_ m
( { j }  X.  B )
24 nfcv 2419 . . . . . . . 8  |-  F/_ j { m }
25 nfcsb1v 3113 . . . . . . . 8  |-  F/_ j [_ m  /  j ]_ B
2624, 25nfxp 4715 . . . . . . 7  |-  F/_ j
( { m }  X.  [_ m  /  j ]_ B )
27 sneq 3651 . . . . . . . 8  |-  ( j  =  m  ->  { j }  =  { m } )
28 csbeq1a 3089 . . . . . . . 8  |-  ( j  =  m  ->  B  =  [_ m  /  j ]_ B )
2927, 28xpeq12d 4714 . . . . . . 7  |-  ( j  =  m  ->  ( { j }  X.  B )  =  ( { m }  X.  [_ m  /  j ]_ B ) )
3023, 26, 29cbviun 3939 . . . . . 6  |-  U_ j  e.  A  ( {
j }  X.  B
)  =  U_ m  e.  A  ( {
m }  X.  [_ m  /  j ]_ B
)
31 nfcv 2419 . . . . . . . 8  |-  F/_ n
( { k }  X.  D )
32 nfcv 2419 . . . . . . . . 9  |-  F/_ k { n }
33 nfcsb1v 3113 . . . . . . . . 9  |-  F/_ k [_ n  /  k ]_ D
3432, 33nfxp 4715 . . . . . . . 8  |-  F/_ k
( { n }  X.  [_ n  /  k ]_ D )
35 sneq 3651 . . . . . . . . 9  |-  ( k  =  n  ->  { k }  =  { n } )
36 csbeq1a 3089 . . . . . . . . 9  |-  ( k  =  n  ->  D  =  [_ n  /  k ]_ D )
3735, 36xpeq12d 4714 . . . . . . . 8  |-  ( k  =  n  ->  ( { k }  X.  D )  =  ( { n }  X.  [_ n  /  k ]_ D ) )
3831, 34, 37cbviun 3939 . . . . . . 7  |-  U_ k  e.  C  ( {
k }  X.  D
)  =  U_ n  e.  C  ( {
n }  X.  [_ n  /  k ]_ D
)
3938cnveqi 4856 . . . . . 6  |-  `' U_ k  e.  C  ( { k }  X.  D )  =  `' U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D )
4022, 30, 393eqtr3g 2338 . . . . 5  |-  ( ph  ->  U_ m  e.  A  ( { m }  X.  [_ m  /  j ]_ B )  =  `' U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )
4140sumeq1d 12174 . . . 4  |-  ( ph  -> 
sum_ z  e.  U_  m  e.  A  ( { m }  X.  [_ m  /  j ]_ B ) [_ ( 2nd `  z )  / 
k ]_ [_ ( 1st `  z )  /  j ]_ E  =  sum_ z  e.  `'  U_ n  e.  C  ( {
n }  X.  [_ n  /  k ]_ D
) [_ ( 2nd `  z
)  /  k ]_ [_ ( 1st `  z
)  /  j ]_ E )
42 vex 2791 . . . . . . . 8  |-  n  e. 
_V
43 vex 2791 . . . . . . . 8  |-  m  e. 
_V
4442, 43op1std 6130 . . . . . . 7  |-  ( w  =  <. n ,  m >.  ->  ( 1st `  w
)  =  n )
4544csbeq1d 3087 . . . . . 6  |-  ( w  =  <. n ,  m >.  ->  [_ ( 1st `  w
)  /  k ]_ [_ ( 2nd `  w
)  /  j ]_ E  =  [_ n  / 
k ]_ [_ ( 2nd `  w )  /  j ]_ E )
4642, 43op2ndd 6131 . . . . . . . 8  |-  ( w  =  <. n ,  m >.  ->  ( 2nd `  w
)  =  m )
4746csbeq1d 3087 . . . . . . 7  |-  ( w  =  <. n ,  m >.  ->  [_ ( 2nd `  w
)  /  j ]_ E  =  [_ m  / 
j ]_ E )
4847csbeq2dv 3106 . . . . . 6  |-  ( w  =  <. n ,  m >.  ->  [_ n  /  k ]_ [_ ( 2nd `  w
)  /  j ]_ E  =  [_ n  / 
k ]_ [_ m  / 
j ]_ E )
4945, 48eqtrd 2315 . . . . 5  |-  ( w  =  <. n ,  m >.  ->  [_ ( 1st `  w
)  /  k ]_ [_ ( 2nd `  w
)  /  j ]_ E  =  [_ n  / 
k ]_ [_ m  / 
j ]_ E )
5043, 42op2ndd 6131 . . . . . . 7  |-  ( z  =  <. m ,  n >.  ->  ( 2nd `  z
)  =  n )
5150csbeq1d 3087 . . . . . 6  |-  ( z  =  <. m ,  n >.  ->  [_ ( 2nd `  z
)  /  k ]_ [_ ( 1st `  z
)  /  j ]_ E  =  [_ n  / 
k ]_ [_ ( 1st `  z )  /  j ]_ E )
5243, 42op1std 6130 . . . . . . . 8  |-  ( z  =  <. m ,  n >.  ->  ( 1st `  z
)  =  m )
5352csbeq1d 3087 . . . . . . 7  |-  ( z  =  <. m ,  n >.  ->  [_ ( 1st `  z
)  /  j ]_ E  =  [_ m  / 
j ]_ E )
5453csbeq2dv 3106 . . . . . 6  |-  ( z  =  <. m ,  n >.  ->  [_ n  /  k ]_ [_ ( 1st `  z
)  /  j ]_ E  =  [_ n  / 
k ]_ [_ m  / 
j ]_ E )
5551, 54eqtrd 2315 . . . . 5  |-  ( z  =  <. m ,  n >.  ->  [_ ( 2nd `  z
)  /  k ]_ [_ ( 1st `  z
)  /  j ]_ E  =  [_ n  / 
k ]_ [_ m  / 
j ]_ E )
56 fsumcom2.2 . . . . . 6  |-  ( ph  ->  C  e.  Fin )
57 snfi 6941 . . . . . . . 8  |-  { n }  e.  Fin
58 fsumcom2.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  Fin )
5958adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  C )  ->  A  e.  Fin )
6033nfel2 2431 . . . . . . . . . . . . . . . . . 18  |-  F/ k  m  e.  [_ n  /  k ]_ D
61 id 19 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  n  ->  k  =  n )
62 vex 2791 . . . . . . . . . . . . . . . . . . . . . . 23  |-  k  e. 
_V
6362snid 3667 . . . . . . . . . . . . . . . . . . . . . 22  |-  k  e. 
{ k }
6461, 63syl6eqelr 2372 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  n  ->  n  e.  { k } )
6564biantrurd 494 . . . . . . . . . . . . . . . . . . . 20  |-  ( k  =  n  ->  (
m  e.  D  <->  ( n  e.  { k }  /\  m  e.  D )
) )
66 opelxp 4719 . . . . . . . . . . . . . . . . . . . 20  |-  ( <.
n ,  m >.  e.  ( { k }  X.  D )  <->  ( n  e.  { k }  /\  m  e.  D )
)
6765, 66syl6rbbr 255 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  n  ->  ( <. n ,  m >.  e.  ( { k }  X.  D )  <->  m  e.  D ) )
6836eleq2d 2350 . . . . . . . . . . . . . . . . . . 19  |-  ( k  =  n  ->  (
m  e.  D  <->  m  e.  [_ n  /  k ]_ D ) )
6967, 68bitrd 244 . . . . . . . . . . . . . . . . . 18  |-  ( k  =  n  ->  ( <. n ,  m >.  e.  ( { k }  X.  D )  <->  m  e.  [_ n  /  k ]_ D ) )
7060, 69rspce 2879 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  C  /\  m  e.  [_ n  / 
k ]_ D )  ->  E. k  e.  C  <. n ,  m >.  e.  ( { k }  X.  D ) )
71 eliun 3909 . . . . . . . . . . . . . . . . 17  |-  ( <.
n ,  m >.  e. 
U_ k  e.  C  ( { k }  X.  D )  <->  E. k  e.  C  <. n ,  m >.  e.  ( { k }  X.  D ) )
7270, 71sylibr 203 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  C  /\  m  e.  [_ n  / 
k ]_ D )  ->  <. n ,  m >.  e. 
U_ k  e.  C  ( { k }  X.  D ) )
7343, 42opelcnv 4863 . . . . . . . . . . . . . . . 16  |-  ( <.
m ,  n >.  e.  `' U_ k  e.  C  ( { k }  X.  D )  <->  <. n ,  m >.  e.  U_ k  e.  C  ( {
k }  X.  D
) )
7472, 73sylibr 203 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  C  /\  m  e.  [_ n  / 
k ]_ D )  ->  <. m ,  n >.  e.  `' U_ k  e.  C  ( { k }  X.  D ) )
7574adantl 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  <. m ,  n >.  e.  `' U_ k  e.  C  ( { k }  X.  D ) )
7622adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  U_ j  e.  A  ( { j }  X.  B )  =  `' U_ k  e.  C  ( { k }  X.  D ) )
7775, 76eleqtrrd 2360 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  <. m ,  n >.  e. 
U_ j  e.  A  ( { j }  X.  B ) )
78 eliun 3909 . . . . . . . . . . . . 13  |-  ( <.
m ,  n >.  e. 
U_ j  e.  A  ( { j }  X.  B )  <->  E. j  e.  A  <. m ,  n >.  e.  ( { j }  X.  B ) )
7977, 78sylib 188 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  E. j  e.  A  <. m ,  n >.  e.  ( { j }  X.  B ) )
80 simpr 447 . . . . . . . . . . . . . . . . 17  |-  ( ( j  e.  A  /\  <.
m ,  n >.  e.  ( { j }  X.  B ) )  ->  <. m ,  n >.  e.  ( { j }  X.  B ) )
81 opelxp 4719 . . . . . . . . . . . . . . . . 17  |-  ( <.
m ,  n >.  e.  ( { j }  X.  B )  <->  ( m  e.  { j }  /\  n  e.  B )
)
8280, 81sylib 188 . . . . . . . . . . . . . . . 16  |-  ( ( j  e.  A  /\  <.
m ,  n >.  e.  ( { j }  X.  B ) )  ->  ( m  e. 
{ j }  /\  n  e.  B )
)
8382simpld 445 . . . . . . . . . . . . . . 15  |-  ( ( j  e.  A  /\  <.
m ,  n >.  e.  ( { j }  X.  B ) )  ->  m  e.  {
j } )
84 elsni 3664 . . . . . . . . . . . . . . 15  |-  ( m  e.  { j }  ->  m  =  j )
8583, 84syl 15 . . . . . . . . . . . . . 14  |-  ( ( j  e.  A  /\  <.
m ,  n >.  e.  ( { j }  X.  B ) )  ->  m  =  j )
86 simpl 443 . . . . . . . . . . . . . 14  |-  ( ( j  e.  A  /\  <.
m ,  n >.  e.  ( { j }  X.  B ) )  ->  j  e.  A
)
8785, 86eqeltrd 2357 . . . . . . . . . . . . 13  |-  ( ( j  e.  A  /\  <.
m ,  n >.  e.  ( { j }  X.  B ) )  ->  m  e.  A
)
8887rexlimiva 2662 . . . . . . . . . . . 12  |-  ( E. j  e.  A  <. m ,  n >.  e.  ( { j }  X.  B )  ->  m  e.  A )
8979, 88syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  m  e.  A )
9089expr 598 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  C )  ->  (
m  e.  [_ n  /  k ]_ D  ->  m  e.  A ) )
9190ssrdv 3185 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  C )  ->  [_ n  /  k ]_ D  C_  A )
92 ssfi 7083 . . . . . . . . 9  |-  ( ( A  e.  Fin  /\  [_ n  /  k ]_ D  C_  A )  ->  [_ n  /  k ]_ D  e.  Fin )
9359, 91, 92syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  n  e.  C )  ->  [_ n  /  k ]_ D  e.  Fin )
94 xpfi 7128 . . . . . . . 8  |-  ( ( { n }  e.  Fin  /\  [_ n  / 
k ]_ D  e.  Fin )  ->  ( { n }  X.  [_ n  / 
k ]_ D )  e. 
Fin )
9557, 93, 94sylancr 644 . . . . . . 7  |-  ( (
ph  /\  n  e.  C )  ->  ( { n }  X.  [_ n  /  k ]_ D )  e.  Fin )
9695ralrimiva 2626 . . . . . 6  |-  ( ph  ->  A. n  e.  C  ( { n }  X.  [_ n  /  k ]_ D )  e.  Fin )
97 iunfi 7144 . . . . . 6  |-  ( ( C  e.  Fin  /\  A. n  e.  C  ( { n }  X.  [_ n  /  k ]_ D )  e.  Fin )  ->  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D )  e.  Fin )
9856, 96, 97syl2anc 642 . . . . 5  |-  ( ph  ->  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D )  e.  Fin )
99 reliun 4806 . . . . . . 7  |-  ( Rel  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D )  <->  A. n  e.  C  Rel  ( { n }  X.  [_ n  /  k ]_ D
) )
100 relxp 4794 . . . . . . . 8  |-  Rel  ( { n }  X.  [_ n  /  k ]_ D )
101100a1i 10 . . . . . . 7  |-  ( n  e.  C  ->  Rel  ( { n }  X.  [_ n  /  k ]_ D ) )
10299, 101mprgbir 2613 . . . . . 6  |-  Rel  U_ n  e.  C  ( {
n }  X.  [_ n  /  k ]_ D
)
103102a1i 10 . . . . 5  |-  ( ph  ->  Rel  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )
104 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )  ->  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  / 
k ]_ D ) )
105 eliun 3909 . . . . . . . 8  |-  ( w  e.  U_ n  e.  C  ( { n }  X.  [_ n  / 
k ]_ D )  <->  E. n  e.  C  w  e.  ( { n }  X.  [_ n  /  k ]_ D ) )
106104, 105sylib 188 . . . . . . 7  |-  ( (
ph  /\  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )  ->  E. n  e.  C  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )
107 xp2nd 6150 . . . . . . . . . 10  |-  ( w  e.  ( { n }  X.  [_ n  / 
k ]_ D )  -> 
( 2nd `  w
)  e.  [_ n  /  k ]_ D
)
108107adantl 452 . . . . . . . . 9  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  ( 2nd `  w )  e. 
[_ n  /  k ]_ D )
109 xp1st 6149 . . . . . . . . . . . 12  |-  ( w  e.  ( { n }  X.  [_ n  / 
k ]_ D )  -> 
( 1st `  w
)  e.  { n } )
110109adantl 452 . . . . . . . . . . 11  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  ( 1st `  w )  e. 
{ n } )
111 elsni 3664 . . . . . . . . . . 11  |-  ( ( 1st `  w )  e.  { n }  ->  ( 1st `  w
)  =  n )
112110, 111syl 15 . . . . . . . . . 10  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  ( 1st `  w )  =  n )
113112csbeq1d 3087 . . . . . . . . 9  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  [_ ( 1st `  w )  / 
k ]_ D  =  [_ n  /  k ]_ D
)
114108, 113eleqtrrd 2360 . . . . . . . 8  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  ( 2nd `  w )  e. 
[_ ( 1st `  w
)  /  k ]_ D )
115114rexlimiva 2662 . . . . . . 7  |-  ( E. n  e.  C  w  e.  ( { n }  X.  [_ n  / 
k ]_ D )  -> 
( 2nd `  w
)  e.  [_ ( 1st `  w )  / 
k ]_ D )
116106, 115syl 15 . . . . . 6  |-  ( (
ph  /\  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )  -> 
( 2nd `  w
)  e.  [_ ( 1st `  w )  / 
k ]_ D )
117 simpl 443 . . . . . . . . . 10  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  n  e.  C )
118112, 117eqeltrd 2357 . . . . . . . . 9  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  ( 1st `  w )  e.  C )
119118rexlimiva 2662 . . . . . . . 8  |-  ( E. n  e.  C  w  e.  ( { n }  X.  [_ n  / 
k ]_ D )  -> 
( 1st `  w
)  e.  C )
120106, 119syl 15 . . . . . . 7  |-  ( (
ph  /\  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )  -> 
( 1st `  w
)  e.  C )
121 simpl 443 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  ph )
12225nfel2 2431 . . . . . . . . . . . 12  |-  F/ j  n  e.  [_ m  /  j ]_ B
12384eqcomd 2288 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  { j }  ->  j  =  m )
124123, 28syl 15 . . . . . . . . . . . . . . . 16  |-  ( m  e.  { j }  ->  B  =  [_ m  /  j ]_ B
)
125124eleq2d 2350 . . . . . . . . . . . . . . 15  |-  ( m  e.  { j }  ->  ( n  e.  B  <->  n  e.  [_ m  /  j ]_ B
) )
126125biimpa 470 . . . . . . . . . . . . . 14  |-  ( ( m  e.  { j }  /\  n  e.  B )  ->  n  e.  [_ m  /  j ]_ B )
12781, 126sylbi 187 . . . . . . . . . . . . 13  |-  ( <.
m ,  n >.  e.  ( { j }  X.  B )  ->  n  e.  [_ m  / 
j ]_ B )
128127a1i 10 . . . . . . . . . . . 12  |-  ( j  e.  A  ->  ( <. m ,  n >.  e.  ( { j }  X.  B )  ->  n  e.  [_ m  / 
j ]_ B ) )
129122, 128rexlimi 2660 . . . . . . . . . . 11  |-  ( E. j  e.  A  <. m ,  n >.  e.  ( { j }  X.  B )  ->  n  e.  [_ m  /  j ]_ B )
13079, 129syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  n  e.  [_ m  / 
j ]_ B )
131 fsumcom2.5 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( j  e.  A  /\  k  e.  B ) )  ->  E  e.  CC )
132131ralrimivva 2635 . . . . . . . . . . . . 13  |-  ( ph  ->  A. j  e.  A  A. k  e.  B  E  e.  CC )
133 nfcsb1v 3113 . . . . . . . . . . . . . . . 16  |-  F/_ j [_ m  /  j ]_ E
134133nfel1 2429 . . . . . . . . . . . . . . 15  |-  F/ j
[_ m  /  j ]_ E  e.  CC
13525, 134nfral 2596 . . . . . . . . . . . . . 14  |-  F/ j A. k  e.  [_  m  /  j ]_ B [_ m  /  j ]_ E  e.  CC
136 csbeq1a 3089 . . . . . . . . . . . . . . . 16  |-  ( j  =  m  ->  E  =  [_ m  /  j ]_ E )
137136eleq1d 2349 . . . . . . . . . . . . . . 15  |-  ( j  =  m  ->  ( E  e.  CC  <->  [_ m  / 
j ]_ E  e.  CC ) )
13828, 137raleqbidv 2748 . . . . . . . . . . . . . 14  |-  ( j  =  m  ->  ( A. k  e.  B  E  e.  CC  <->  A. k  e.  [_  m  /  j ]_ B [_ m  / 
j ]_ E  e.  CC ) )
139135, 138rspc 2878 . . . . . . . . . . . . 13  |-  ( m  e.  A  ->  ( A. j  e.  A  A. k  e.  B  E  e.  CC  ->  A. k  e.  [_  m  /  j ]_ B [_ m  /  j ]_ E  e.  CC ) )
140132, 139mpan9 455 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  A )  ->  A. k  e.  [_  m  /  j ]_ B [_ m  / 
j ]_ E  e.  CC )
141 nfcsb1v 3113 . . . . . . . . . . . . . 14  |-  F/_ k [_ n  /  k ]_ [_ m  /  j ]_ E
142141nfel1 2429 . . . . . . . . . . . . 13  |-  F/ k
[_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC
143 csbeq1a 3089 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  [_ m  /  j ]_ E  =  [_ n  /  k ]_ [_ m  /  j ]_ E )
144143eleq1d 2349 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  ( [_ m  /  j ]_ E  e.  CC  <->  [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC )
)
145142, 144rspc 2878 . . . . . . . . . . . 12  |-  ( n  e.  [_ m  / 
j ]_ B  ->  ( A. k  e.  [_  m  /  j ]_ B [_ m  /  j ]_ E  e.  CC  ->  [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC ) )
146140, 145syl5com 26 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  A )  ->  (
n  e.  [_ m  /  j ]_ B  ->  [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC ) )
147146impr 602 . . . . . . . . . 10  |-  ( (
ph  /\  ( m  e.  A  /\  n  e.  [_ m  /  j ]_ B ) )  ->  [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC )
148121, 89, 130, 147syl12anc 1180 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC )
149148ralrimivva 2635 . . . . . . . 8  |-  ( ph  ->  A. n  e.  C  A. m  e.  [_  n  /  k ]_ D [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC )
150149adantr 451 . . . . . . 7  |-  ( (
ph  /\  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )  ->  A. n  e.  C  A. m  e.  [_  n  /  k ]_ D [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC )
151 csbeq1 3084 . . . . . . . . 9  |-  ( n  =  ( 1st `  w
)  ->  [_ n  / 
k ]_ D  =  [_ ( 1st `  w )  /  k ]_ D
)
152 csbeq1 3084 . . . . . . . . . 10  |-  ( n  =  ( 1st `  w
)  ->  [_ n  / 
k ]_ [_ m  / 
j ]_ E  =  [_ ( 1st `  w )  /  k ]_ [_ m  /  j ]_ E
)
153152eleq1d 2349 . . . . . . . . 9  |-  ( n  =  ( 1st `  w
)  ->  ( [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC  <->  [_ ( 1st `  w
)  /  k ]_ [_ m  /  j ]_ E  e.  CC )
)
154151, 153raleqbidv 2748 . . . . . . . 8  |-  ( n  =  ( 1st `  w
)  ->  ( A. m  e.  [_  n  / 
k ]_ D [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC  <->  A. m  e.  [_  ( 1st `  w )  /  k ]_ D [_ ( 1st `  w
)  /  k ]_ [_ m  /  j ]_ E  e.  CC )
)
155154rspcv 2880 . . . . . . 7  |-  ( ( 1st `  w )  e.  C  ->  ( A. n  e.  C  A. m  e.  [_  n  /  k ]_ D [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC  ->  A. m  e.  [_  ( 1st `  w )  /  k ]_ D [_ ( 1st `  w
)  /  k ]_ [_ m  /  j ]_ E  e.  CC )
)
156120, 150, 155sylc 56 . . . . . 6  |-  ( (
ph  /\  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )  ->  A. m  e.  [_  ( 1st `  w )  / 
k ]_ D [_ ( 1st `  w )  / 
k ]_ [_ m  / 
j ]_ E  e.  CC )
157 csbeq1 3084 . . . . . . . . 9  |-  ( m  =  ( 2nd `  w
)  ->  [_ m  / 
j ]_ E  =  [_ ( 2nd `  w )  /  j ]_ E
)
158157csbeq2dv 3106 . . . . . . . 8  |-  ( m  =  ( 2nd `  w
)  ->  [_ ( 1st `  w )  /  k ]_ [_ m  /  j ]_ E  =  [_ ( 1st `  w )  / 
k ]_ [_ ( 2nd `  w )  /  j ]_ E )
159158eleq1d 2349 . . . . . . 7  |-  ( m  =  ( 2nd `  w
)  ->  ( [_ ( 1st `  w )  /  k ]_ [_ m  /  j ]_ E  e.  CC  <->  [_ ( 1st `  w
)  /  k ]_ [_ ( 2nd `  w
)  /  j ]_ E  e.  CC )
)
160159rspcv 2880 . . . . . 6  |-  ( ( 2nd `  w )  e.  [_ ( 1st `  w )  /  k ]_ D  ->  ( A. m  e.  [_  ( 1st `  w )  /  k ]_ D [_ ( 1st `  w )  /  k ]_ [_ m  /  j ]_ E  e.  CC  ->  [_ ( 1st `  w
)  /  k ]_ [_ ( 2nd `  w
)  /  j ]_ E  e.  CC )
)
161116, 156, 160sylc 56 . . . . 5  |-  ( (
ph  /\  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )  ->  [_ ( 1st `  w
)  /  k ]_ [_ ( 2nd `  w
)  /  j ]_ E  e.  CC )
16249, 55, 98, 103, 161fsumcnv 12236 . . . 4  |-  ( ph  -> 
sum_ w  e.  U_  n  e.  C  ( {
n }  X.  [_ n  /  k ]_ D
) [_ ( 1st `  w
)  /  k ]_ [_ ( 2nd `  w
)  /  j ]_ E  =  sum_ z  e.  `'  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) [_ ( 2nd `  z )  / 
k ]_ [_ ( 1st `  z )  /  j ]_ E )
16341, 162eqtr4d 2318 . . 3  |-  ( ph  -> 
sum_ z  e.  U_  m  e.  A  ( { m }  X.  [_ m  /  j ]_ B ) [_ ( 2nd `  z )  / 
k ]_ [_ ( 1st `  z )  /  j ]_ E  =  sum_ w  e.  U_  n  e.  C  ( { n }  X.  [_ n  / 
k ]_ D ) [_ ( 1st `  w )  /  k ]_ [_ ( 2nd `  w )  / 
j ]_ E )
164 fsumcom2.3 . . . . . 6  |-  ( (
ph  /\  j  e.  A )  ->  B  e.  Fin )
165164ralrimiva 2626 . . . . 5  |-  ( ph  ->  A. j  e.  A  B  e.  Fin )
16625nfel1 2429 . . . . . 6  |-  F/ j
[_ m  /  j ]_ B  e.  Fin
16728eleq1d 2349 . . . . . 6  |-  ( j  =  m  ->  ( B  e.  Fin  <->  [_ m  / 
j ]_ B  e.  Fin ) )
168166, 167rspc 2878 . . . . 5  |-  ( m  e.  A  ->  ( A. j  e.  A  B  e.  Fin  ->  [_ m  /  j ]_ B  e.  Fin ) )
169165, 168mpan9 455 . . . 4  |-  ( (
ph  /\  m  e.  A )  ->  [_ m  /  j ]_ B  e.  Fin )
17055, 58, 169, 147fsum2d 12234 . . 3  |-  ( ph  -> 
sum_ m  e.  A  sum_ n  e.  [_  m  /  j ]_ B [_ n  /  k ]_ [_ m  /  j ]_ E  =  sum_ z  e.  U_  m  e.  A  ( { m }  X.  [_ m  / 
j ]_ B ) [_ ( 2nd `  z )  /  k ]_ [_ ( 1st `  z )  / 
j ]_ E )
17149, 56, 93, 148fsum2d 12234 . . 3  |-  ( ph  -> 
sum_ n  e.  C  sum_ m  e.  [_  n  /  k ]_ D [_ n  /  k ]_ [_ m  /  j ]_ E  =  sum_ w  e.  U_  n  e.  C  ( { n }  X.  [_ n  / 
k ]_ D ) [_ ( 1st `  w )  /  k ]_ [_ ( 2nd `  w )  / 
j ]_ E )
172163, 170, 1713eqtr4d 2325 . 2  |-  ( ph  -> 
sum_ m  e.  A  sum_ n  e.  [_  m  /  j ]_ B [_ n  /  k ]_ [_ m  /  j ]_ E  =  sum_ n  e.  C  sum_ m  e.  [_  n  /  k ]_ D [_ n  / 
k ]_ [_ m  / 
j ]_ E )
173 nfcv 2419 . . 3  |-  F/_ m sum_ k  e.  B  E
174 nfcv 2419 . . . . 5  |-  F/_ j
n
175174, 133nfcsb 3115 . . . 4  |-  F/_ j [_ n  /  k ]_ [_ m  /  j ]_ E
17625, 175nfsum 12164 . . 3  |-  F/_ j sum_ n  e.  [_  m  /  j ]_ B [_ n  /  k ]_ [_ m  /  j ]_ E
177 nfcv 2419 . . . . 5  |-  F/_ n E
178 nfcsb1v 3113 . . . . 5  |-  F/_ k [_ n  /  k ]_ E
179 csbeq1a 3089 . . . . 5  |-  ( k  =  n  ->  E  =  [_ n  /  k ]_ E )
180177, 178, 179cbvsumi 12170 . . . 4  |-  sum_ k  e.  B  E  =  sum_ n  e.  B  [_ n  /  k ]_ E
181136csbeq2dv 3106 . . . . . 6  |-  ( j  =  m  ->  [_ n  /  k ]_ E  =  [_ n  /  k ]_ [_ m  /  j ]_ E )
182181adantr 451 . . . . 5  |-  ( ( j  =  m  /\  n  e.  B )  ->  [_ n  /  k ]_ E  =  [_ n  /  k ]_ [_ m  /  j ]_ E
)
18328, 182sumeq12dv 12179 . . . 4  |-  ( j  =  m  ->  sum_ n  e.  B  [_ n  / 
k ]_ E  =  sum_ n  e.  [_  m  / 
j ]_ B [_ n  /  k ]_ [_ m  /  j ]_ E
)
184180, 183syl5eq 2327 . . 3  |-  ( j  =  m  ->  sum_ k  e.  B  E  =  sum_ n  e.  [_  m  /  j ]_ B [_ n  /  k ]_ [_ m  /  j ]_ E )
185173, 176, 184cbvsumi 12170 . 2  |-  sum_ j  e.  A  sum_ k  e.  B  E  =  sum_ m  e.  A  sum_ n  e.  [_  m  /  j ]_ B [_ n  / 
k ]_ [_ m  / 
j ]_ E
186 nfcv 2419 . . 3  |-  F/_ n sum_ j  e.  D  E
18733, 141nfsum 12164 . . 3  |-  F/_ k sum_ m  e.  [_  n  /  k ]_ D [_ n  /  k ]_ [_ m  /  j ]_ E
188 nfcv 2419 . . . . 5  |-  F/_ m E
189188, 133, 136cbvsumi 12170 . . . 4  |-  sum_ j  e.  D  E  =  sum_ m  e.  D  [_ m  /  j ]_ E
190143adantr 451 . . . . 5  |-  ( ( k  =  n  /\  m  e.  D )  ->  [_ m  /  j ]_ E  =  [_ n  /  k ]_ [_ m  /  j ]_ E
)
19136, 190sumeq12dv 12179 . . . 4  |-  ( k  =  n  ->  sum_ m  e.  D  [_ m  / 
j ]_ E  =  sum_ m  e.  [_  n  / 
k ]_ D [_ n  /  k ]_ [_ m  /  j ]_ E
)
192189, 191syl5eq 2327 . . 3  |-  ( k  =  n  ->  sum_ j  e.  D  E  =  sum_ m  e.  [_  n  /  k ]_ D [_ n  /  k ]_ [_ m  /  j ]_ E )
193186, 187, 192cbvsumi 12170 . 2  |-  sum_ k  e.  C  sum_ j  e.  D  E  =  sum_ n  e.  C  sum_ m  e.  [_  n  /  k ]_ D [_ n  / 
k ]_ [_ m  / 
j ]_ E
194172, 185, 1933eqtr4g 2340 1  |-  ( ph  -> 
sum_ j  e.  A  sum_ k  e.  B  E  =  sum_ k  e.  C  sum_ j  e.  D  E
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   [_csb 3081    C_ wss 3152   {csn 3640   <.cop 3643   U_ciun 3905    X. cxp 4687   `'ccnv 4688   Rel wrel 4694   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   Fincfn 6863   CCcc 8735   sum_csu 12158
This theorem is referenced by:  fsumcom  12238  fsum0diag  12240  fsumdvdsdiag  20424  dvdsflsumcom  20428  fsumfldivdiag  20430  logfac2  20456  chpchtsum  20458  logfaclbnd  20461  dchrisum0lem1  20665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159
  Copyright terms: Public domain W3C validator