Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > df-sumdc | Unicode version |
Description: Define the sum of a series with an index set of integers . is normally a free variable in , i.e. can be thought of as . 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 expression so that we only need to be defined where . 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: means , and means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 11284). (Contributed by NM, 11-Dec-2005.) (Revised by Jim Kingdon, 21-May-2023.) |
Ref | Expression |
---|---|
df-sumdc | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 | |
2 | cB | . . 3 | |
3 | vk | . . 3 | |
4 | 1, 2, 3 | csu 11115 | . 2 |
5 | vm | . . . . . . . . 9 | |
6 | 5 | cv 1330 | . . . . . . . 8 |
7 | cuz 9319 | . . . . . . . 8 | |
8 | 6, 7 | cfv 5118 | . . . . . . 7 |
9 | 1, 8 | wss 3066 | . . . . . 6 |
10 | vj | . . . . . . . . . 10 | |
11 | 10 | cv 1330 | . . . . . . . . 9 |
12 | 11, 1 | wcel 1480 | . . . . . . . 8 |
13 | 12 | wdc 819 | . . . . . . 7 DECID |
14 | 13, 10, 8 | wral 2414 | . . . . . 6 DECID |
15 | caddc 7616 | . . . . . . . 8 | |
16 | vn | . . . . . . . . 9 | |
17 | cz 9047 | . . . . . . . . 9 | |
18 | 16 | cv 1330 | . . . . . . . . . . 11 |
19 | 18, 1 | wcel 1480 | . . . . . . . . . 10 |
20 | 3, 18, 2 | csb 2998 | . . . . . . . . . 10 |
21 | cc0 7613 | . . . . . . . . . 10 | |
22 | 19, 20, 21 | cif 3469 | . . . . . . . . 9 |
23 | 16, 17, 22 | cmpt 3984 | . . . . . . . 8 |
24 | 15, 23, 6 | cseq 10211 | . . . . . . 7 |
25 | vx | . . . . . . . 8 | |
26 | 25 | cv 1330 | . . . . . . 7 |
27 | cli 11040 | . . . . . . 7 | |
28 | 24, 26, 27 | wbr 3924 | . . . . . 6 |
29 | 9, 14, 28 | w3a 962 | . . . . 5 DECID |
30 | 29, 5, 17 | wrex 2415 | . . . 4 DECID |
31 | c1 7614 | . . . . . . . . 9 | |
32 | cfz 9783 | . . . . . . . . 9 | |
33 | 31, 6, 32 | co 5767 | . . . . . . . 8 |
34 | vf | . . . . . . . . 9 | |
35 | 34 | cv 1330 | . . . . . . . 8 |
36 | 33, 1, 35 | wf1o 5117 | . . . . . . 7 |
37 | cn 8713 | . . . . . . . . . . 11 | |
38 | cle 7794 | . . . . . . . . . . . . 13 | |
39 | 18, 6, 38 | wbr 3924 | . . . . . . . . . . . 12 |
40 | 18, 35 | cfv 5118 | . . . . . . . . . . . . 13 |
41 | 3, 40, 2 | csb 2998 | . . . . . . . . . . . 12 |
42 | 39, 41, 21 | cif 3469 | . . . . . . . . . . 11 |
43 | 16, 37, 42 | cmpt 3984 | . . . . . . . . . 10 |
44 | 15, 43, 31 | cseq 10211 | . . . . . . . . 9 |
45 | 6, 44 | cfv 5118 | . . . . . . . 8 |
46 | 26, 45 | wceq 1331 | . . . . . . 7 |
47 | 36, 46 | wa 103 | . . . . . 6 |
48 | 47, 34 | wex 1468 | . . . . 5 |
49 | 48, 5, 37 | wrex 2415 | . . . 4 |
50 | 30, 49 | wo 697 | . . 3 DECID |
51 | 50, 25 | cio 5081 | . 2 DECID |
52 | 4, 51 | wceq 1331 | 1 DECID |
Colors of variables: wff set class |
This definition is referenced by: sumeq1 11117 nfsum1 11118 nfsum 11119 sumeq2 11121 cbvsum 11122 zsumdc 11146 fsum3 11149 |
Copyright terms: Public domain | W3C validator |