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Definition df-sum 12175
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 12206. 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 12354). (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 12174 . 2  class  sum_ k  e.  A  B
5 vm . . . . . . . . 9  set  m
65cv 1631 . . . . . . . 8  class  m
7 cuz 10246 . . . . . . . 8  class  ZZ>=
86, 7cfv 5271 . . . . . . 7  class  ( ZZ>= `  m )
91, 8wss 3165 . . . . . 6  wff  A  C_  ( ZZ>= `  m )
10 caddc 8756 . . . . . . . 8  class  +
11 cz 10040 . . . . . . . . 9  class  ZZ
123cv 1631 . . . . . . . . . . 11  class  k
1312, 1wcel 1696 . . . . . . . . . 10  wff  k  e.  A
14 cc0 8753 . . . . . . . . . 10  class  0
1513, 2, 14cif 3578 . . . . . . . . 9  class  if ( k  e.  A ,  B ,  0 )
163, 11, 15cmpt 4093 . . . . . . . 8  class  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
1710, 16, 6cseq 11062 . . . . . . 7  class  seq  m
(  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )
18 vx . . . . . . . 8  set  x
1918cv 1631 . . . . . . 7  class  x
20 cli 11974 . . . . . . 7  class  ~~>
2117, 19, 20wbr 4039 . . . . . 6  wff  seq  m
(  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  x
229, 21wa 358 . . . . 5  wff  ( A 
C_  ( ZZ>= `  m
)  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  x )
2322, 5, 11wrex 2557 . . . 4  wff  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) ) )  ~~>  x )
24 c1 8754 . . . . . . . . 9  class  1
25 cfz 10798 . . . . . . . . 9  class  ...
2624, 6, 25co 5874 . . . . . . . 8  class  ( 1 ... m )
27 vf . . . . . . . . 9  set  f
2827cv 1631 . . . . . . . 8  class  f
2926, 1, 28wf1o 5270 . . . . . . 7  wff  f : ( 1 ... m
)
-1-1-onto-> A
30 vn . . . . . . . . . . 11  set  n
31 cn 9762 . . . . . . . . . . 11  class  NN
3230cv 1631 . . . . . . . . . . . . 13  class  n
3332, 28cfv 5271 . . . . . . . . . . . 12  class  ( f `
 n )
343, 33, 2csb 3094 . . . . . . . . . . 11  class  [_ (
f `  n )  /  k ]_ B
3530, 31, 34cmpt 4093 . . . . . . . . . 10  class  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
)
3610, 35, 24cseq 11062 . . . . . . . . 9  class  seq  1
(  +  ,  ( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) )
376, 36cfv 5271 . . . . . . . 8  class  (  seq  1 (  +  , 
( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B ) ) `  m )
3819, 37wceq 1632 . . . . . . 7  wff  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
) ) `  m
)
3929, 38wa 358 . . . . . 6  wff  ( f : ( 1 ... m ) -1-1-onto-> A  /\  x  =  (  seq  1 (  +  ,  ( n  e.  NN  |->  [_ (
f `  n )  /  k ]_ B
) ) `  m
) )
4039, 27wex 1531 . . . . 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 2557 . . . 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 357 . . 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 5233 . 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 1632 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  12176  sumeq1f  12177  nfsum1  12179  nfsum  12180  sumeq2w  12181  sumeq2ii  12182  cbvsum  12184  zsum  12207  fsum  12209
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