Definition df-sumdc 11306

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 11474). (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 11305	. 2 class sum_ k e. A B
5		vm	. . . . . . . . 9 setvar m
6	5	cv 1347	. . . . . . . 8 class m
7		cuz 9476	. . . . . . . 8 class ZZ>=
8	6, 7	cfv 5196	. . . . . . 7 class ( ZZ>= ` m )
9	1, 8	wss 3121	. . . . . 6 wff A C_ ( ZZ>= ` m )
10		vj	. . . . . . . . . 10 setvar j
11	10	cv 1347	. . . . . . . . 9 class j
12	11, 1	wcel 2141	. . . . . . . 8 wff j e. A
13	12	wdc 829	. . . . . . 7 wff DECID j e. A
14	13, 10, 8	wral 2448	. . . . . 6 wff A. j e. ( ZZ>= ` m )DECID j e. A
15		caddc 7766	. . . . . . . 8 class +
16		vn	. . . . . . . . 9 setvar n
17		cz 9201	. . . . . . . . 9 class ZZ
18	16	cv 1347	. . . . . . . . . . 11 class n
19	18, 1	wcel 2141	. . . . . . . . . 10 wff n e. A
20	3, 18, 2	csb 3049	. . . . . . . . . 10 class [_ n / k ]_ B
21		cc0 7763	. . . . . . . . . 10 class 0
22	19, 20, 21	cif 3525	. . . . . . . . 9 class if ( n e. A , [_ n / k ]_ B , 0 )
23	16, 17, 22	cmpt 4048	. . . . . . . 8 class ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
24	15, 23, 6	cseq 10390	. . . . . . 7 class seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
25		vx	. . . . . . . 8 setvar x
26	25	cv 1347	. . . . . . 7 class x
27		cli 11230	. . . . . . 7 class ~~>
28	24, 26, 27	wbr 3987	. . . . . 6 wff seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x
29	9, 14, 28	w3a 973	. . . . 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 2449	. . . 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 7764	. . . . . . . . 9 class 1
32		cfz 9954	. . . . . . . . 9 class ...
33	31, 6, 32	co 5851	. . . . . . . 8 class ( 1 ... m )
34		vf	. . . . . . . . 9 setvar f
35	34	cv 1347	. . . . . . . 8 class f
36	33, 1, 35	wf1o 5195	. . . . . . 7 wff f : ( 1 ... m ) -1-1-onto-> A
37		cn 8867	. . . . . . . . . . 11 class NN
38		cle 7944	. . . . . . . . . . . . 13 class <_
39	18, 6, 38	wbr 3987	. . . . . . . . . . . 12 wff n <_ m
40	18, 35	cfv 5196	. . . . . . . . . . . . 13 class ( f ` n )
41	3, 40, 2	csb 3049	. . . . . . . . . . . 12 class [_ ( f ` n ) / k ]_ B
42	39, 41, 21	cif 3525	. . . . . . . . . . 11 class if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
43	16, 37, 42	cmpt 4048	. . . . . . . . . 10 class ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
44	15, 43, 31	cseq 10390	. . . . . . . . 9 class seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) )
45	6, 44	cfv 5196	. . . . . . . 8 class ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
46	26, 45	wceq 1348	. . . . . . 7 wff x = ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
47	36, 46	wa 103	. . . . . 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 1485	. . . . 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 2449	. . . 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 703	. . 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 5156	. 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 1348	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 referenced by:  sumeq1  11307  nfsum1  11308  nfsum  11309  sumeq2  11311  cbvsum  11312  zsumdc  11336  fsum3  11339