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Mirrors > Home > ILE Home > Th. List > df-sum | 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. These two methods of summation produce the same result on their common region of definition (i.e. finite subsets of the upper integers). Examples: means , and means 1/2 + 1/4 + 1/8 + ... = 1. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
Ref | Expression |
---|---|
df-sum |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 | |
2 | cB | . . 3 | |
3 | vk | . . 3 | |
4 | 1, 2, 3 | csu 10190 | . 2 |
5 | vm | . . . . . . . . 9 | |
6 | 5 | cv 1283 | . . . . . . . 8 |
7 | cuz 8619 | . . . . . . . 8 | |
8 | 6, 7 | cfv 4922 | . . . . . . 7 |
9 | 1, 8 | wss 2973 | . . . . . 6 |
10 | caddc 6984 | . . . . . . . 8 | |
11 | cc 6979 | . . . . . . . 8 | |
12 | vn | . . . . . . . . 9 | |
13 | cz 8351 | . . . . . . . . 9 | |
14 | 12 | cv 1283 | . . . . . . . . . . 11 |
15 | 14, 1 | wcel 1433 | . . . . . . . . . 10 |
16 | 3, 14, 2 | csb 2908 | . . . . . . . . . 10 |
17 | cc0 6981 | . . . . . . . . . 10 | |
18 | 15, 16, 17 | cif 3351 | . . . . . . . . 9 |
19 | 12, 13, 18 | cmpt 3839 | . . . . . . . 8 |
20 | 10, 11, 19, 6 | cseq 9431 | . . . . . . 7 |
21 | vx | . . . . . . . 8 | |
22 | 21 | cv 1283 | . . . . . . 7 |
23 | cli 10117 | . . . . . . 7 | |
24 | 20, 22, 23 | wbr 3785 | . . . . . 6 |
25 | 9, 24 | wa 102 | . . . . 5 |
26 | 25, 5, 13 | wrex 2349 | . . . 4 |
27 | c1 6982 | . . . . . . . . 9 | |
28 | cfz 9029 | . . . . . . . . 9 | |
29 | 27, 6, 28 | co 5532 | . . . . . . . 8 |
30 | vf | . . . . . . . . 9 | |
31 | 30 | cv 1283 | . . . . . . . 8 |
32 | 29, 1, 31 | wf1o 4921 | . . . . . . 7 |
33 | cn 8039 | . . . . . . . . . . 11 | |
34 | 14, 31 | cfv 4922 | . . . . . . . . . . . 12 |
35 | 3, 34, 2 | csb 2908 | . . . . . . . . . . 11 |
36 | 12, 33, 35 | cmpt 3839 | . . . . . . . . . 10 |
37 | 10, 11, 36, 27 | cseq 9431 | . . . . . . . . 9 |
38 | 6, 37 | cfv 4922 | . . . . . . . 8 |
39 | 22, 38 | wceq 1284 | . . . . . . 7 |
40 | 32, 39 | wa 102 | . . . . . 6 |
41 | 40, 30 | wex 1421 | . . . . 5 |
42 | 41, 5, 33 | wrex 2349 | . . . 4 |
43 | 26, 42 | wo 661 | . . 3 |
44 | 43, 21 | cio 4885 | . 2 |
45 | 4, 44 | wceq 1284 | 1 |
Colors of variables: wff set class |
This definition is referenced by: sumeq1 10192 nfsum1 10193 nfsum 10194 |
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