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Theorem fisumcom2 10828
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.) (Proof shortened by JJ, 2-Aug-2021.)
Hypotheses
Ref Expression
fsumcom2.1  |-  ( ph  ->  A  e.  Fin )
fsumcom2.2  |-  ( ph  ->  C  e.  Fin )
fsumcom2.3  |-  ( (
ph  /\  j  e.  A )  ->  B  e.  Fin )
fisumcom2.fi  |-  ( (
ph  /\  k  e.  C )  ->  D  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
fisumcom2  |-  ( 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 fisumcom2
Dummy variables  m  n  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4547 . . . . . . . . 9  |-  Rel  ( { j }  X.  B )
21rgenw 2430 . . . . . . . 8  |-  A. j  e.  A  Rel  ( { j }  X.  B
)
3 reliun 4558 . . . . . . . 8  |-  ( Rel  U_ j  e.  A  ( { j }  X.  B )  <->  A. j  e.  A  Rel  ( { j }  X.  B
) )
42, 3mpbir 144 . . . . . . 7  |-  Rel  U_ j  e.  A  ( {
j }  X.  B
)
5 relcnv 4810 . . . . . . 7  |-  Rel  `' U_ k  e.  C  ( { k }  X.  D )
6 ancom 262 . . . . . . . . . . . 12  |-  ( ( x  =  j  /\  y  =  k )  <->  ( y  =  k  /\  x  =  j )
)
7 vex 2622 . . . . . . . . . . . . 13  |-  x  e. 
_V
8 vex 2622 . . . . . . . . . . . . 13  |-  y  e. 
_V
97, 8opth 4064 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  =  <. j ,  k
>. 
<->  ( x  =  j  /\  y  =  k ) )
108, 7opth 4064 . . . . . . . . . . . 12  |-  ( <.
y ,  x >.  = 
<. k ,  j >.  <->  ( y  =  k  /\  x  =  j )
)
116, 9, 103bitr4i 210 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  =  <. j ,  k
>. 
<-> 
<. y ,  x >.  = 
<. k ,  j >.
)
1211a1i 9 . . . . . . . . . 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 457 . . . . . . . . 9  |-  ( ph  ->  ( ( <. x ,  y >.  =  <. j ,  k >.  /\  (
j  e.  A  /\  k  e.  B )
)  <->  ( <. y ,  x >.  =  <. k ,  j >.  /\  (
k  e.  C  /\  j  e.  D )
) ) )
15142exbidv 1796 . . . . . . . 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 4575 . . . . . . . 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 4618 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  `' U_ k  e.  C  ( { k }  X.  D )  <->  <. y ,  x >.  e.  U_ k  e.  C  ( {
k }  X.  D
) )
18 eliunxp 4575 . . . . . . . . 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 1599 . . . . . . . . 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 204 . . . . . . . 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 221 . . . . . . 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 4534 . . . . . 6  |-  ( ph  ->  U_ j  e.  A  ( { j }  X.  B )  =  `' U_ k  e.  C  ( { k }  X.  D ) )
23 nfcv 2228 . . . . . . 7  |-  F/_ m
( { j }  X.  B )
24 nfcv 2228 . . . . . . . 8  |-  F/_ j { m }
25 nfcsb1v 2963 . . . . . . . 8  |-  F/_ j [_ m  /  j ]_ B
2624, 25nfxp 4464 . . . . . . 7  |-  F/_ j
( { m }  X.  [_ m  /  j ]_ B )
27 sneq 3457 . . . . . . . 8  |-  ( j  =  m  ->  { j }  =  { m } )
28 csbeq1a 2941 . . . . . . . 8  |-  ( j  =  m  ->  B  =  [_ m  /  j ]_ B )
2927, 28xpeq12d 4463 . . . . . . 7  |-  ( j  =  m  ->  ( { j }  X.  B )  =  ( { m }  X.  [_ m  /  j ]_ B ) )
3023, 26, 29cbviun 3767 . . . . . 6  |-  U_ j  e.  A  ( {
j }  X.  B
)  =  U_ m  e.  A  ( {
m }  X.  [_ m  /  j ]_ B
)
31 nfcv 2228 . . . . . . . 8  |-  F/_ n
( { k }  X.  D )
32 nfcv 2228 . . . . . . . . 9  |-  F/_ k { n }
33 nfcsb1v 2963 . . . . . . . . 9  |-  F/_ k [_ n  /  k ]_ D
3432, 33nfxp 4464 . . . . . . . 8  |-  F/_ k
( { n }  X.  [_ n  /  k ]_ D )
35 sneq 3457 . . . . . . . . 9  |-  ( k  =  n  ->  { k }  =  { n } )
36 csbeq1a 2941 . . . . . . . . 9  |-  ( k  =  n  ->  D  =  [_ n  /  k ]_ D )
3735, 36xpeq12d 4463 . . . . . . . 8  |-  ( k  =  n  ->  ( { k }  X.  D )  =  ( { n }  X.  [_ n  /  k ]_ D ) )
3831, 34, 37cbviun 3767 . . . . . . 7  |-  U_ k  e.  C  ( {
k }  X.  D
)  =  U_ n  e.  C  ( {
n }  X.  [_ n  /  k ]_ D
)
3938cnveqi 4611 . . . . . 6  |-  `' U_ k  e.  C  ( { k }  X.  D )  =  `' U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D )
4022, 30, 393eqtr3g 2143 . . . . 5  |-  ( ph  ->  U_ m  e.  A  ( { m }  X.  [_ m  /  j ]_ B )  =  `' U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )
4140sumeq1d 10751 . . . 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 2622 . . . . . . . 8  |-  n  e. 
_V
43 vex 2622 . . . . . . . 8  |-  m  e. 
_V
4442, 43op1std 5919 . . . . . . 7  |-  ( w  =  <. n ,  m >.  ->  ( 1st `  w
)  =  n )
4544csbeq1d 2939 . . . . . 6  |-  ( w  =  <. n ,  m >.  ->  [_ ( 1st `  w
)  /  k ]_ [_ ( 2nd `  w
)  /  j ]_ E  =  [_ n  / 
k ]_ [_ ( 2nd `  w )  /  j ]_ E )
4642, 43op2ndd 5920 . . . . . . . 8  |-  ( w  =  <. n ,  m >.  ->  ( 2nd `  w
)  =  m )
4746csbeq1d 2939 . . . . . . 7  |-  ( w  =  <. n ,  m >.  ->  [_ ( 2nd `  w
)  /  j ]_ E  =  [_ m  / 
j ]_ E )
4847csbeq2dv 2956 . . . . . 6  |-  ( w  =  <. n ,  m >.  ->  [_ n  /  k ]_ [_ ( 2nd `  w
)  /  j ]_ E  =  [_ n  / 
k ]_ [_ m  / 
j ]_ E )
4945, 48eqtrd 2120 . . . . 5  |-  ( w  =  <. n ,  m >.  ->  [_ ( 1st `  w
)  /  k ]_ [_ ( 2nd `  w
)  /  j ]_ E  =  [_ n  / 
k ]_ [_ m  / 
j ]_ E )
5043, 42op2ndd 5920 . . . . . . 7  |-  ( z  =  <. m ,  n >.  ->  ( 2nd `  z
)  =  n )
5150csbeq1d 2939 . . . . . 6  |-  ( z  =  <. m ,  n >.  ->  [_ ( 2nd `  z
)  /  k ]_ [_ ( 1st `  z
)  /  j ]_ E  =  [_ n  / 
k ]_ [_ ( 1st `  z )  /  j ]_ E )
5243, 42op1std 5919 . . . . . . . 8  |-  ( z  =  <. m ,  n >.  ->  ( 1st `  z
)  =  m )
5352csbeq1d 2939 . . . . . . 7  |-  ( z  =  <. m ,  n >.  ->  [_ ( 1st `  z
)  /  j ]_ E  =  [_ m  / 
j ]_ E )
5453csbeq2dv 2956 . . . . . 6  |-  ( z  =  <. m ,  n >.  ->  [_ n  /  k ]_ [_ ( 1st `  z
)  /  j ]_ E  =  [_ n  / 
k ]_ [_ m  / 
j ]_ E )
5551, 54eqtrd 2120 . . . . 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 snfig 6529 . . . . . . . . 9  |-  ( n  e.  _V  ->  { n }  e.  Fin )
5857elv 2623 . . . . . . . 8  |-  { n }  e.  Fin
59 fisumcom2.fi . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  C )  ->  D  e.  Fin )
6059ralrimiva 2446 . . . . . . . . 9  |-  ( ph  ->  A. k  e.  C  D  e.  Fin )
6133nfel1 2239 . . . . . . . . . 10  |-  F/ k
[_ n  /  k ]_ D  e.  Fin
6236eleq1d 2156 . . . . . . . . . 10  |-  ( k  =  n  ->  ( D  e.  Fin  <->  [_ n  / 
k ]_ D  e.  Fin ) )
6361, 62rspc 2716 . . . . . . . . 9  |-  ( n  e.  C  ->  ( A. k  e.  C  D  e.  Fin  ->  [_ n  /  k ]_ D  e.  Fin ) )
6460, 63mpan9 275 . . . . . . . 8  |-  ( (
ph  /\  n  e.  C )  ->  [_ n  /  k ]_ D  e.  Fin )
65 xpfi 6638 . . . . . . . 8  |-  ( ( { n }  e.  Fin  /\  [_ n  / 
k ]_ D  e.  Fin )  ->  ( { n }  X.  [_ n  / 
k ]_ D )  e. 
Fin )
6658, 64, 65sylancr 405 . . . . . . 7  |-  ( (
ph  /\  n  e.  C )  ->  ( { n }  X.  [_ n  /  k ]_ D )  e.  Fin )
6766ralrimiva 2446 . . . . . 6  |-  ( ph  ->  A. n  e.  C  ( { n }  X.  [_ n  /  k ]_ D )  e.  Fin )
68 disjsnxp 6002 . . . . . . 7  |- Disj  n  e.  C  ( { n }  X.  [_ n  / 
k ]_ D )
6968a1i 9 . . . . . 6  |-  ( ph  -> Disj  n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )
70 iunfidisj 6653 . . . . . 6  |-  ( ( C  e.  Fin  /\  A. n  e.  C  ( { n }  X.  [_ n  /  k ]_ D )  e.  Fin  /\ Disj  n  e.  C  ( { n }  X.  [_ n  /  k ]_ D
) )  ->  U_ n  e.  C  ( {
n }  X.  [_ n  /  k ]_ D
)  e.  Fin )
7156, 67, 69, 70syl3anc 1174 . . . . 5  |-  ( ph  ->  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D )  e.  Fin )
72 reliun 4558 . . . . . . 7  |-  ( Rel  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D )  <->  A. n  e.  C  Rel  ( { n }  X.  [_ n  /  k ]_ D
) )
73 relxp 4547 . . . . . . . 8  |-  Rel  ( { n }  X.  [_ n  /  k ]_ D )
7473a1i 9 . . . . . . 7  |-  ( n  e.  C  ->  Rel  ( { n }  X.  [_ n  /  k ]_ D ) )
7572, 74mprgbir 2433 . . . . . 6  |-  Rel  U_ n  e.  C  ( {
n }  X.  [_ n  /  k ]_ D
)
7675a1i 9 . . . . 5  |-  ( ph  ->  Rel  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )
77 csbeq1 2936 . . . . . . . 8  |-  ( m  =  ( 2nd `  w
)  ->  [_ m  / 
j ]_ E  =  [_ ( 2nd `  w )  /  j ]_ E
)
7877csbeq2dv 2956 . . . . . . 7  |-  ( m  =  ( 2nd `  w
)  ->  [_ ( 1st `  w )  /  k ]_ [_ m  /  j ]_ E  =  [_ ( 1st `  w )  / 
k ]_ [_ ( 2nd `  w )  /  j ]_ E )
7978eleq1d 2156 . . . . . 6  |-  ( m  =  ( 2nd `  w
)  ->  ( [_ ( 1st `  w )  /  k ]_ [_ m  /  j ]_ E  e.  CC  <->  [_ ( 1st `  w
)  /  k ]_ [_ ( 2nd `  w
)  /  j ]_ E  e.  CC )
)
80 csbeq1 2936 . . . . . . . 8  |-  ( n  =  ( 1st `  w
)  ->  [_ n  / 
k ]_ D  =  [_ ( 1st `  w )  /  k ]_ D
)
81 csbeq1 2936 . . . . . . . . 9  |-  ( n  =  ( 1st `  w
)  ->  [_ n  / 
k ]_ [_ m  / 
j ]_ E  =  [_ ( 1st `  w )  /  k ]_ [_ m  /  j ]_ E
)
8281eleq1d 2156 . . . . . . . 8  |-  ( n  =  ( 1st `  w
)  ->  ( [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC  <->  [_ ( 1st `  w
)  /  k ]_ [_ m  /  j ]_ E  e.  CC )
)
8380, 82raleqbidv 2574 . . . . . . 7  |-  ( 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 )
)
84 simpl 107 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  ph )
8543, 42opelcnv 4618 . . . . . . . . . . . . . . 15  |-  ( <.
m ,  n >.  e.  `' U_ k  e.  C  ( { k }  X.  D )  <->  <. n ,  m >.  e.  U_ k  e.  C  ( {
k }  X.  D
) )
8633, 36opeliunxp2f 6003 . . . . . . . . . . . . . . 15  |-  ( <.
n ,  m >.  e. 
U_ k  e.  C  ( { k }  X.  D )  <->  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )
8785, 86sylbbr 134 . . . . . . . . . . . . . 14  |-  ( ( n  e.  C  /\  m  e.  [_ n  / 
k ]_ D )  ->  <. m ,  n >.  e.  `' U_ k  e.  C  ( { k }  X.  D ) )
8887adantl 271 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  <. m ,  n >.  e.  `' U_ k  e.  C  ( { k }  X.  D ) )
8922adantr 270 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  U_ j  e.  A  ( { j }  X.  B )  =  `' U_ k  e.  C  ( { k }  X.  D ) )
9088, 89eleqtrrd 2167 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  <. m ,  n >.  e. 
U_ j  e.  A  ( { j }  X.  B ) )
91 eliun 3734 . . . . . . . . . . . 12  |-  ( <.
m ,  n >.  e. 
U_ j  e.  A  ( { j }  X.  B )  <->  E. j  e.  A  <. m ,  n >.  e.  ( { j }  X.  B ) )
9290, 91sylib 120 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  E. j  e.  A  <. m ,  n >.  e.  ( { j }  X.  B ) )
93 simpr 108 . . . . . . . . . . . . . . . 16  |-  ( ( j  e.  A  /\  <.
m ,  n >.  e.  ( { j }  X.  B ) )  ->  <. m ,  n >.  e.  ( { j }  X.  B ) )
94 opelxp 4467 . . . . . . . . . . . . . . . 16  |-  ( <.
m ,  n >.  e.  ( { j }  X.  B )  <->  ( m  e.  { j }  /\  n  e.  B )
)
9593, 94sylib 120 . . . . . . . . . . . . . . 15  |-  ( ( j  e.  A  /\  <.
m ,  n >.  e.  ( { j }  X.  B ) )  ->  ( m  e. 
{ j }  /\  n  e.  B )
)
9695simpld 110 . . . . . . . . . . . . . 14  |-  ( ( j  e.  A  /\  <.
m ,  n >.  e.  ( { j }  X.  B ) )  ->  m  e.  {
j } )
97 elsni 3464 . . . . . . . . . . . . . 14  |-  ( m  e.  { j }  ->  m  =  j )
9896, 97syl 14 . . . . . . . . . . . . 13  |-  ( ( j  e.  A  /\  <.
m ,  n >.  e.  ( { j }  X.  B ) )  ->  m  =  j )
99 simpl 107 . . . . . . . . . . . . 13  |-  ( ( j  e.  A  /\  <.
m ,  n >.  e.  ( { j }  X.  B ) )  ->  j  e.  A
)
10098, 99eqeltrd 2164 . . . . . . . . . . . 12  |-  ( ( j  e.  A  /\  <.
m ,  n >.  e.  ( { j }  X.  B ) )  ->  m  e.  A
)
101100rexlimiva 2484 . . . . . . . . . . 11  |-  ( E. j  e.  A  <. m ,  n >.  e.  ( { j }  X.  B )  ->  m  e.  A )
10292, 101syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  m  e.  A )
10325nfcri 2222 . . . . . . . . . . . 12  |-  F/ j  n  e.  [_ m  /  j ]_ B
10497equcomd 1640 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  { j }  ->  j  =  m )
105104, 28syl 14 . . . . . . . . . . . . . . . 16  |-  ( m  e.  { j }  ->  B  =  [_ m  /  j ]_ B
)
106105eleq2d 2157 . . . . . . . . . . . . . . 15  |-  ( m  e.  { j }  ->  ( n  e.  B  <->  n  e.  [_ m  /  j ]_ B
) )
107106biimpa 290 . . . . . . . . . . . . . 14  |-  ( ( m  e.  { j }  /\  n  e.  B )  ->  n  e.  [_ m  /  j ]_ B )
10894, 107sylbi 119 . . . . . . . . . . . . 13  |-  ( <.
m ,  n >.  e.  ( { j }  X.  B )  ->  n  e.  [_ m  / 
j ]_ B )
109108a1i 9 . . . . . . . . . . . 12  |-  ( j  e.  A  ->  ( <. m ,  n >.  e.  ( { j }  X.  B )  ->  n  e.  [_ m  / 
j ]_ B ) )
110103, 109rexlimi 2482 . . . . . . . . . . 11  |-  ( E. j  e.  A  <. m ,  n >.  e.  ( { j }  X.  B )  ->  n  e.  [_ m  /  j ]_ B )
11192, 110syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  n  e.  [_ m  / 
j ]_ B )
112 fsumcom2.5 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( j  e.  A  /\  k  e.  B ) )  ->  E  e.  CC )
113112ralrimivva 2455 . . . . . . . . . . . . 13  |-  ( ph  ->  A. j  e.  A  A. k  e.  B  E  e.  CC )
114 nfcsb1v 2963 . . . . . . . . . . . . . . . 16  |-  F/_ j [_ m  /  j ]_ E
115114nfel1 2239 . . . . . . . . . . . . . . 15  |-  F/ j
[_ m  /  j ]_ E  e.  CC
11625, 115nfralxy 2414 . . . . . . . . . . . . . 14  |-  F/ j A. k  e.  [_  m  /  j ]_ B [_ m  /  j ]_ E  e.  CC
117 csbeq1a 2941 . . . . . . . . . . . . . . . 16  |-  ( j  =  m  ->  E  =  [_ m  /  j ]_ E )
118117eleq1d 2156 . . . . . . . . . . . . . . 15  |-  ( j  =  m  ->  ( E  e.  CC  <->  [_ m  / 
j ]_ E  e.  CC ) )
11928, 118raleqbidv 2574 . . . . . . . . . . . . . 14  |-  ( j  =  m  ->  ( A. k  e.  B  E  e.  CC  <->  A. k  e.  [_  m  /  j ]_ B [_ m  / 
j ]_ E  e.  CC ) )
120116, 119rspc 2716 . . . . . . . . . . . . 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 ) )
121113, 120mpan9 275 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  A )  ->  A. k  e.  [_  m  /  j ]_ B [_ m  / 
j ]_ E  e.  CC )
122 nfcsb1v 2963 . . . . . . . . . . . . . 14  |-  F/_ k [_ n  /  k ]_ [_ m  /  j ]_ E
123122nfel1 2239 . . . . . . . . . . . . 13  |-  F/ k
[_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC
124 csbeq1a 2941 . . . . . . . . . . . . . 14  |-  ( k  =  n  ->  [_ m  /  j ]_ E  =  [_ n  /  k ]_ [_ m  /  j ]_ E )
125124eleq1d 2156 . . . . . . . . . . . . 13  |-  ( k  =  n  ->  ( [_ m  /  j ]_ E  e.  CC  <->  [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC )
)
126123, 125rspc 2716 . . . . . . . . . . . 12  |-  ( n  e.  [_ m  / 
j ]_ B  ->  ( A. k  e.  [_  m  /  j ]_ B [_ m  /  j ]_ E  e.  CC  ->  [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC ) )
127121, 126syl5com 29 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  A )  ->  (
n  e.  [_ m  /  j ]_ B  ->  [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC ) )
128127impr 371 . . . . . . . . . 10  |-  ( (
ph  /\  ( m  e.  A  /\  n  e.  [_ m  /  j ]_ B ) )  ->  [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC )
12984, 102, 111, 128syl12anc 1172 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  C  /\  m  e.  [_ n  /  k ]_ D ) )  ->  [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC )
130129ralrimivva 2455 . . . . . . . 8  |-  ( ph  ->  A. n  e.  C  A. m  e.  [_  n  /  k ]_ D [_ n  /  k ]_ [_ m  /  j ]_ E  e.  CC )
131130adantr 270 . . . . . . 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 )
132 simpr 108 . . . . . . . . 9  |-  ( (
ph  /\  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )  ->  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  / 
k ]_ D ) )
133 eliun 3734 . . . . . . . . 9  |-  ( w  e.  U_ n  e.  C  ( { n }  X.  [_ n  / 
k ]_ D )  <->  E. n  e.  C  w  e.  ( { n }  X.  [_ n  /  k ]_ D ) )
134132, 133sylib 120 . . . . . . . 8  |-  ( (
ph  /\  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )  ->  E. n  e.  C  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )
135 xp1st 5936 . . . . . . . . . . . 12  |-  ( w  e.  ( { n }  X.  [_ n  / 
k ]_ D )  -> 
( 1st `  w
)  e.  { n } )
136135adantl 271 . . . . . . . . . . 11  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  ( 1st `  w )  e. 
{ n } )
137 elsni 3464 . . . . . . . . . . 11  |-  ( ( 1st `  w )  e.  { n }  ->  ( 1st `  w
)  =  n )
138136, 137syl 14 . . . . . . . . . 10  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  ( 1st `  w )  =  n )
139 simpl 107 . . . . . . . . . 10  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  n  e.  C )
140138, 139eqeltrd 2164 . . . . . . . . 9  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  ( 1st `  w )  e.  C )
141140rexlimiva 2484 . . . . . . . 8  |-  ( E. n  e.  C  w  e.  ( { n }  X.  [_ n  / 
k ]_ D )  -> 
( 1st `  w
)  e.  C )
142134, 141syl 14 . . . . . . 7  |-  ( (
ph  /\  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )  -> 
( 1st `  w
)  e.  C )
14383, 131, 142rspcdva 2727 . . . . . 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 )
144 xp2nd 5937 . . . . . . . . . 10  |-  ( w  e.  ( { n }  X.  [_ n  / 
k ]_ D )  -> 
( 2nd `  w
)  e.  [_ n  /  k ]_ D
)
145144adantl 271 . . . . . . . . 9  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  ( 2nd `  w )  e. 
[_ n  /  k ]_ D )
146138csbeq1d 2939 . . . . . . . . 9  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  [_ ( 1st `  w )  / 
k ]_ D  =  [_ n  /  k ]_ D
)
147145, 146eleqtrrd 2167 . . . . . . . 8  |-  ( ( n  e.  C  /\  w  e.  ( {
n }  X.  [_ n  /  k ]_ D
) )  ->  ( 2nd `  w )  e. 
[_ ( 1st `  w
)  /  k ]_ D )
148147rexlimiva 2484 . . . . . . 7  |-  ( E. n  e.  C  w  e.  ( { n }  X.  [_ n  / 
k ]_ D )  -> 
( 2nd `  w
)  e.  [_ ( 1st `  w )  / 
k ]_ D )
149134, 148syl 14 . . . . . 6  |-  ( (
ph  /\  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )  -> 
( 2nd `  w
)  e.  [_ ( 1st `  w )  / 
k ]_ D )
15079, 143, 149rspcdva 2727 . . . . 5  |-  ( (
ph  /\  w  e.  U_ n  e.  C  ( { n }  X.  [_ n  /  k ]_ D ) )  ->  [_ ( 1st `  w
)  /  k ]_ [_ ( 2nd `  w
)  /  j ]_ E  e.  CC )
15149, 55, 71, 76, 150fsumcnv 10827 . . . 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 )
15241, 151eqtr4d 2123 . . 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 )
153 fsumcom2.1 . . . 4  |-  ( ph  ->  A  e.  Fin )
154 fsumcom2.3 . . . . . 6  |-  ( (
ph  /\  j  e.  A )  ->  B  e.  Fin )
155154ralrimiva 2446 . . . . 5  |-  ( ph  ->  A. j  e.  A  B  e.  Fin )
15625nfel1 2239 . . . . . 6  |-  F/ j
[_ m  /  j ]_ B  e.  Fin
15728eleq1d 2156 . . . . . 6  |-  ( j  =  m  ->  ( B  e.  Fin  <->  [_ m  / 
j ]_ B  e.  Fin ) )
158156, 157rspc 2716 . . . . 5  |-  ( m  e.  A  ->  ( A. j  e.  A  B  e.  Fin  ->  [_ m  /  j ]_ B  e.  Fin ) )
159155, 158mpan9 275 . . . 4  |-  ( (
ph  /\  m  e.  A )  ->  [_ m  /  j ]_ B  e.  Fin )
16055, 153, 159, 128fsum2d 10825 . . 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 )
16149, 56, 64, 129fsum2d 10825 . . 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 )
162152, 160, 1613eqtr4d 2130 . 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 )
163 nfcv 2228 . . 3  |-  F/_ m sum_ k  e.  B  E
164 nfcv 2228 . . . . 5  |-  F/_ j
n
165164, 114nfcsb 2965 . . . 4  |-  F/_ j [_ n  /  k ]_ [_ m  /  j ]_ E
16625, 165nfsum 10742 . . 3  |-  F/_ j sum_ n  e.  [_  m  /  j ]_ B [_ n  /  k ]_ [_ m  /  j ]_ E
167 nfcv 2228 . . . . 5  |-  F/_ n E
168 nfcsb1v 2963 . . . . 5  |-  F/_ k [_ n  /  k ]_ E
169 csbeq1a 2941 . . . . 5  |-  ( k  =  n  ->  E  =  [_ n  /  k ]_ E )
170167, 168, 169cbvsumi 10747 . . . 4  |-  sum_ k  e.  B  E  =  sum_ n  e.  B  [_ n  /  k ]_ E
171117csbeq2dv 2956 . . . . . 6  |-  ( j  =  m  ->  [_ n  /  k ]_ E  =  [_ n  /  k ]_ [_ m  /  j ]_ E )
172171adantr 270 . . . . 5  |-  ( ( j  =  m  /\  n  e.  B )  ->  [_ n  /  k ]_ E  =  [_ n  /  k ]_ [_ m  /  j ]_ E
)
17328, 172sumeq12dv 10757 . . . 4  |-  ( j  =  m  ->  sum_ n  e.  B  [_ n  / 
k ]_ E  =  sum_ n  e.  [_  m  / 
j ]_ B [_ n  /  k ]_ [_ m  /  j ]_ E
)
174170, 173syl5eq 2132 . . 3  |-  ( j  =  m  ->  sum_ k  e.  B  E  =  sum_ n  e.  [_  m  /  j ]_ B [_ n  /  k ]_ [_ m  /  j ]_ E )
175163, 166, 174cbvsumi 10747 . 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
176 nfcv 2228 . . 3  |-  F/_ n sum_ j  e.  D  E
17733, 122nfsum 10742 . . 3  |-  F/_ k sum_ m  e.  [_  n  /  k ]_ D [_ n  /  k ]_ [_ m  /  j ]_ E
178 nfcv 2228 . . . . 5  |-  F/_ m E
179178, 114, 117cbvsumi 10747 . . . 4  |-  sum_ j  e.  D  E  =  sum_ m  e.  D  [_ m  /  j ]_ E
180124adantr 270 . . . . 5  |-  ( ( k  =  n  /\  m  e.  D )  ->  [_ m  /  j ]_ E  =  [_ n  /  k ]_ [_ m  /  j ]_ E
)
18136, 180sumeq12dv 10757 . . . 4  |-  ( k  =  n  ->  sum_ m  e.  D  [_ m  / 
j ]_ E  =  sum_ m  e.  [_  n  / 
k ]_ D [_ n  /  k ]_ [_ m  /  j ]_ E
)
182179, 181syl5eq 2132 . . 3  |-  ( k  =  n  ->  sum_ j  e.  D  E  =  sum_ m  e.  [_  n  /  k ]_ D [_ n  /  k ]_ [_ m  /  j ]_ E )
183176, 177, 182cbvsumi 10747 . 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
184162, 175, 1833eqtr4g 2145 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    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   A.wral 2359   E.wrex 2360   _Vcvv 2619   [_csb 2933   {csn 3446   <.cop 3449   U_ciun 3730  Disj wdisj 3822    X. cxp 4436   `'ccnv 4437   Rel wrel 4443   ` cfv 5015   1stc1st 5909   2ndc2nd 5910   Fincfn 6455   CCcc 7346   sum_csu 10738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462  ax-caucvg 7463
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-disj 3823  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-frec 6156  df-1o 6181  df-oadd 6185  df-er 6290  df-en 6456  df-dom 6457  df-fin 6458  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-2 8479  df-3 8480  df-4 8481  df-n0 8672  df-z 8749  df-uz 9018  df-q 9103  df-rp 9133  df-fz 9423  df-fzo 9550  df-iseq 9849  df-seq3 9850  df-exp 9951  df-ihash 10180  df-cj 10272  df-re 10273  df-im 10274  df-rsqrt 10427  df-abs 10428  df-clim 10663  df-isum 10739
This theorem is referenced by:  fsumcom  10829  fisum0diag  10831
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