Definition df-sumdc 11281

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 11449). (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 11280	. 2 class sum_ k e. A B
5		vm	. . . . . . . . 9 setvar m
6	5	cv 1341	. . . . . . . 8 class m
7		cuz 9457	. . . . . . . 8 class ZZ>=
8	6, 7	cfv 5182	. . . . . . 7 class ( ZZ>= ` m )
9	1, 8	wss 3111	. . . . . 6 wff A C_ ( ZZ>= ` m )
10		vj	. . . . . . . . . 10 setvar j
11	10	cv 1341	. . . . . . . . 9 class j
12	11, 1	wcel 2135	. . . . . . . 8 wff j e. A
13	12	wdc 824	. . . . . . 7 wff DECID j e. A
14	13, 10, 8	wral 2442	. . . . . 6 wff A. j e. ( ZZ>= ` m )DECID j e. A
15		caddc 7747	. . . . . . . 8 class +
16		vn	. . . . . . . . 9 setvar n
17		cz 9182	. . . . . . . . 9 class ZZ
18	16	cv 1341	. . . . . . . . . . 11 class n
19	18, 1	wcel 2135	. . . . . . . . . 10 wff n e. A
20	3, 18, 2	csb 3040	. . . . . . . . . 10 class [_ n / k ]_ B
21		cc0 7744	. . . . . . . . . 10 class 0
22	19, 20, 21	cif 3515	. . . . . . . . 9 class if ( n e. A , [_ n / k ]_ B , 0 )
23	16, 17, 22	cmpt 4037	. . . . . . . 8 class ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) )
24	15, 23, 6	cseq 10370	. . . . . . 7 class seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) )
25		vx	. . . . . . . 8 setvar x
26	25	cv 1341	. . . . . . 7 class x
27		cli 11205	. . . . . . 7 class ~~>
28	24, 26, 27	wbr 3976	. . . . . 6 wff seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x
29	9, 14, 28	w3a 967	. . . . 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 2443	. . . 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 7745	. . . . . . . . 9 class 1
32		cfz 9935	. . . . . . . . 9 class ...
33	31, 6, 32	co 5836	. . . . . . . 8 class ( 1 ... m )
34		vf	. . . . . . . . 9 setvar f
35	34	cv 1341	. . . . . . . 8 class f
36	33, 1, 35	wf1o 5181	. . . . . . 7 wff f : ( 1 ... m ) -1-1-onto-> A
37		cn 8848	. . . . . . . . . . 11 class NN
38		cle 7925	. . . . . . . . . . . . 13 class <_
39	18, 6, 38	wbr 3976	. . . . . . . . . . . 12 wff n <_ m
40	18, 35	cfv 5182	. . . . . . . . . . . . 13 class ( f ` n )
41	3, 40, 2	csb 3040	. . . . . . . . . . . 12 class [_ ( f ` n ) / k ]_ B
42	39, 41, 21	cif 3515	. . . . . . . . . . 11 class if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 )
43	16, 37, 42	cmpt 4037	. . . . . . . . . 10 class ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) )
44	15, 43, 31	cseq 10370	. . . . . . . . 9 class seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) )
45	6, 44	cfv 5182	. . . . . . . 8 class ( seq 1 ( + , ( n e. NN |-> if ( n <_ m , [_ ( f ` n ) / k ]_ B , 0 ) ) ) ` m )
46	26, 45	wceq 1342	. . . . . . 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 1479	. . . . 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 2443	. . . 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 698	. . 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 5145	. 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 1342	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  11282  nfsum1  11283  nfsum  11284  sumeq2  11286  cbvsum  11287  zsumdc  11311  fsum3  11314