Definition df-sumdc 12145

Description: Define the sum of a series with an index set of integers A. The variable 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. In both cases we have an if expression so that we only need B to be defined where k e. A. In the infinite case, we also require that the indexing set be a decidable subset of an upperset of integers (that is, membership of integers in it is decidable). These two methods of summation produce the same result on their common region of definition (i.e., finite sets of integers). 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 12314). (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 21-May-2023.)

Assertion

Ref	Expression
df-sumdc	|- sum_ k e. A B = ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) ) )

Distinct variable groups:   f, k, m, n, x, j   A, f, m, n, x, j   B, f, m, n, x, j

Allowed substitution hints:   A( k)   B( k)

Detailed syntax breakdown of Definition df-sumdc

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cB	. . 3 class B
3		vk	. . 3 setvar k
4	1, 2, 3	csu 12144	. 2 class sum_ k e. A B
5		vm	. . . . . . . . 9 setvar m
6	5	cv 1401	. . . . . . . 8 class m
7		cuz 9937	. . . . . . . 8 class ZZ>=
8	6, 7	cfv 5380	. . . . . . 7 class ( ZZ>= ` m )
9	1, 8	wss 3220	. . . . . 6 wff A C_ ( ZZ>= ` m )
10		vj	. . . . . . . . . 10 setvar j
11	10	cv 1401	. . . . . . . . 9 class j
12	11, 1	wcel 2209	. . . . . . . 8 wff j e. A
13	12	wdc 846	. . . . . . 7 wff DECID j e. A
14	13, 10, 8	wral 2528	. . . . . 6 wff A. j e. ( ZZ>= ` m )DECID j e. A
15		caddc 8188	. . . . . . . 8 class +
16		vn	. . . . . . . . 9 setvar n
17		cz 9655	. . . . . . . . 9 class ZZ
18	16	cv 1401	. . . . . . . . . . 11 class n
19	18, 1	wcel 2209	. . . . . . . . . 10 wff n e. A
20	3, 18, 2	csb 3147	. . . . . . . . . 10 class [_ n / k ]_ B
21		cc0 8185	. . . . . . . . . 10 class 0
22	19, 20, 21	cif 3638	. . . . . . . . 9 class if ( n e. A , [_ n / k ]_ B , 0 )
23	16, 17, 22	cmpt 4193	. . . . . . . 8 class ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
24	15, 23, 6	cseq 10905	. . . . . . 7 class seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
25		vx	. . . . . . . 8 setvar x
26	25	cv 1401	. . . . . . 7 class x
27		cli 12069	. . . . . . 7 class ~~>
28	24, 26, 27	wbr 4131	. . . . . 6 wff seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x
29	9, 14, 28	w3a 1009	. . . . 5 wff ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x )
30	29, 5, 17	wrex 2529	. . . 4 wff E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x )
31		c1 8186	. . . . . . . . 9 class 1
32		cfz 10428	. . . . . . . . 9 class ...
33	31, 6, 32	co 6088	. . . . . . . 8 class ( 1 ... m )
34		vf	. . . . . . . . 9 setvar f
35	34	cv 1401	. . . . . . . 8 class f
36	33, 1, 35	wf1o 5379	. . . . . . 7 wff f : ( 1 ... m ) -1-1-onto-> A
37		cn 9313	. . . . . . . . . . 11 class NN
38		cle 8367	. . . . . . . . . . . . 13 class <_
39	18, 6, 38	wbr 4131	. . . . . . . . . . . 12 wff n <_ m
40	18, 35	cfv 5380	. . . . . . . . . . . . 13 class ( f ` n )
41	3, 40, 2	csb 3147	. . . . . . . . . . . 12 class [_ ( f ` n ) / k ]_ B
42	39, 41, 21	cif 3638	. . . . . . . . . . 11 class if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
43	16, 37, 42	cmpt 4193	. . . . . . . . . 10 class ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
44	15, 43, 31	cseq 10905	. . . . . . . . 9 class seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) )
45	6, 44	cfv 5380	. . . . . . . 8 class ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
46	26, 45	wceq 1402	. . . . . . 7 wff x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
47	36, 46	wa 104	. . . . . 6 wff ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) )
48	47, 34	wex 1545	. . . . 5 wff E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) )
49	48, 5, 37	wrex 2529	. . . 4 wff E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) )
50	30, 49	wo 720	. . 3 wff ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) )
51	50, 25	cio 5338	. 2 class ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) ) )
52	4, 51	wceq 1402	1 wff sum_ k e. A B = ( iota x ( E. m e. ZZ ( A C_ ( ZZ>= ` m ) /\ A. j e. ( ZZ>= ` m )DECID j e. A /\ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ E. m e. NN E. f ( f : ( 1 ... m ) -1-1-onto-> A /\ x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m ) ) ) )

This definition is used by:  sumeq1  12146  nfsum1  12147  nfsum  12148  sumeq2  12150  cbvsum  12151  zsumdc  12176  fsum3  12179