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Definition df-sum 12472
Description: Define the sum of a series with an index set of integers  A.  k is normally a free variable in  B, i.e.  B can be thought of as  B ( k ). This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. These two methods of summation produce the same result on their common region of definition (i.e. finite subsets of the upper integers) by summo 12503. Examples:  sum_ k  e. 
{ 1 ,  2 ,  4 }  k means  1  +  2  + 
4  =  7, and  sum_ k  e.  NN  ( 1  / 
( 2 ^ k
) )  =  1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 12651). (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
Assertion
Ref Expression
df-sum  |-  sum_ k  e.  A  B  =  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )
Distinct variable groups:    f, k, m, n, x    A, f, m, n, x    B, f, m, n, x
Allowed substitution hints:    A( k)    B( k)

Detailed syntax breakdown of Definition df-sum
StepHypRef Expression
1 cA . . 3  class  A
2 cB . . 3  class  B
3 vk . . 3  set  k
41, 2, 3csu 12471 . 2  class  sum_ k  e.  A  B
5 vm . . . . . . . . 9  set  m
65cv 1651 . . . . . . . 8  class  m
7 cuz 10480 . . . . . . . 8  class  ZZ>=
86, 7cfv 5446 . . . . . . 7  class  ( ZZ>= `  m )
91, 8wss 3312 . . . . . 6  wff  A  C_  ( ZZ>= `  m )
10 caddc 8985 . . . . . . . 8  class  +
11 cz 10274 . . . . . . . . 9  class  ZZ
123cv 1651 . . . . . . . . . . 11  class  k
1312, 1wcel 1725 . . . . . . . . . 10  wff  k  e.  A
14 cc0 8982 . . . . . . . . . 10  class  0
1513, 2, 14cif 3731 . . . . . . . . 9  class  if ( k  e.  A ,  B ,  0 )
163, 11, 15cmpt 4258 . . . . . . . 8  class  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
1710, 16, 6cseq 11315 . . . . . . 7  class  seq  m
(  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )
18 vx . . . . . . . 8  set  x
1918cv 1651 . . . . . . 7  class  x
20 cli 12270 . . . . . . 7  class  ~~>
2117, 19, 20wbr 4204 . . . . . 6  wff  seq  m
(  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  x
229, 21wa 359 . . . . 5  wff  ( A 
C_  ( ZZ>= `  m
)  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  x )
2322, 5, 11wrex 2698 . . . 4  wff  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  x )
24 c1 8983 . . . . . . . . 9  class  1
25 cfz 11035 . . . . . . . . 9  class  ...
2624, 6, 25co 6073 . . . . . . . 8  class  ( 1 ... m )
27 vf . . . . . . . . 9  set  f
2827cv 1651 . . . . . . . 8  class  f
2926, 1, 28wf1o 5445 . . . . . . 7  wff  f : ( 1 ... m
)
-1-1-onto-> A
30 vn . . . . . . . . . . 11  set  n
31 cn 9992 . . . . . . . . . . 11  class  NN
3230cv 1651 . . . . . . . . . . . . 13  class  n
3332, 28cfv 5446 . . . . . . . . . . . 12  class  ( f `
 n )
343, 33, 2csb 3243 . . . . . . . . . . 11  class  [_ (
f `  n )  /  k ]_ B
3530, 31, 34cmpt 4258 . . . . . . . . . 10  class  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
)
3610, 35, 24cseq 11315 . . . . . . . . 9  class  seq  1
(  +  ,  ( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
376, 36cfv 5446 . . . . . . . 8  class  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
3819, 37wceq 1652 . . . . . . 7  wff  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
) ) `  m
)
3929, 38wa 359 . . . . . 6  wff  ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
) ) `  m
) )
4039, 27wex 1550 . . . . 5  wff  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) )
4140, 5, 31wrex 2698 . . . 4  wff  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) )
4223, 41wo 358 . . 3  wff  ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) )
4342, 18cio 5408 . 2  class  ( iota
x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  x )  \/ 
E. m  e.  NN  E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) ) ) )
444, 43wceq 1652 1  wff  sum_ k  e.  A  B  =  ( iota x ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  x )  \/  E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  sumex  12473  sumeq1f  12474  nfsum1  12476  nfsum  12477  sumeq2w  12478  sumeq2ii  12479  cbvsum  12481  zsum  12504  fsum  12506
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