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Mirrors > Home > MPE Home > Th. List > df-gsum | Structured version Visualization version Unicode version |
Description: Define the group sum for
the structure of a
finite sequence of
elements whose values are defined by the expression and whose set
of indices is .
It may be viewed as a product (if is a
multiplication), a sum (if is an addition) or whatever. The
variable is
normally a free variable in ( i.e. can
be
thought of as ). The definition is meaningful in different
contexts, depending on the size of the index set and each
demanding different properties of .
1. If and has an identity element, then the sum equals this identity. 2. If and is any magma, then the sum is the sum of the elements, evaluated left-to-right, i.e. etc. 3. If is a finite set (or is nonzero for finitely many indices) and is a commutative monoid, then the sum adds up these elements in some order, which is then uniquely defined. 4. If is an infinite set and is a Hausdorff topological group, then there is a meaningful sum, but g cannot handle this case. See df-tsms 21930. Remark: this definition is required here because the symbol g is already used in df-prds 16108 and df-imas 16168. The related theorems are provided later, see gsumvalx 17270. (Contributed by FL, 5-Sep-2010.) (Revised by FL, 17-Oct-2011.) (Revised by Mario Carneiro, 7-Dec-2014.) |
Ref | Expression |
---|---|
df-gsum | g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cgsu 16101 | . 2 g | |
2 | vw | . . 3 | |
3 | vf | . . 3 | |
4 | cvv 3200 | . . 3 | |
5 | vo | . . . 4 | |
6 | vx | . . . . . . . . . 10 | |
7 | 6 | cv 1482 | . . . . . . . . 9 |
8 | vy | . . . . . . . . . 10 | |
9 | 8 | cv 1482 | . . . . . . . . 9 |
10 | 2 | cv 1482 | . . . . . . . . . 10 |
11 | cplusg 15941 | . . . . . . . . . 10 | |
12 | 10, 11 | cfv 5888 | . . . . . . . . 9 |
13 | 7, 9, 12 | co 6650 | . . . . . . . 8 |
14 | 13, 9 | wceq 1483 | . . . . . . 7 |
15 | 9, 7, 12 | co 6650 | . . . . . . . 8 |
16 | 15, 9 | wceq 1483 | . . . . . . 7 |
17 | 14, 16 | wa 384 | . . . . . 6 |
18 | cbs 15857 | . . . . . . 7 | |
19 | 10, 18 | cfv 5888 | . . . . . 6 |
20 | 17, 8, 19 | wral 2912 | . . . . 5 |
21 | 20, 6, 19 | crab 2916 | . . . 4 |
22 | 3 | cv 1482 | . . . . . . 7 |
23 | 22 | crn 5115 | . . . . . 6 |
24 | 5 | cv 1482 | . . . . . 6 |
25 | 23, 24 | wss 3574 | . . . . 5 |
26 | c0g 16100 | . . . . . 6 | |
27 | 10, 26 | cfv 5888 | . . . . 5 |
28 | 22 | cdm 5114 | . . . . . . 7 |
29 | cfz 12326 | . . . . . . . 8 | |
30 | 29 | crn 5115 | . . . . . . 7 |
31 | 28, 30 | wcel 1990 | . . . . . 6 |
32 | vm | . . . . . . . . . . . . 13 | |
33 | 32 | cv 1482 | . . . . . . . . . . . 12 |
34 | vn | . . . . . . . . . . . . 13 | |
35 | 34 | cv 1482 | . . . . . . . . . . . 12 |
36 | 33, 35, 29 | co 6650 | . . . . . . . . . . 11 |
37 | 28, 36 | wceq 1483 | . . . . . . . . . 10 |
38 | 12, 22, 33 | cseq 12801 | . . . . . . . . . . . 12 |
39 | 35, 38 | cfv 5888 | . . . . . . . . . . 11 |
40 | 7, 39 | wceq 1483 | . . . . . . . . . 10 |
41 | 37, 40 | wa 384 | . . . . . . . . 9 |
42 | cuz 11687 | . . . . . . . . . 10 | |
43 | 33, 42 | cfv 5888 | . . . . . . . . 9 |
44 | 41, 34, 43 | wrex 2913 | . . . . . . . 8 |
45 | 44, 32 | wex 1704 | . . . . . . 7 |
46 | 45, 6 | cio 5849 | . . . . . 6 |
47 | c1 9937 | . . . . . . . . . . . 12 | |
48 | chash 13117 | . . . . . . . . . . . . 13 | |
49 | 9, 48 | cfv 5888 | . . . . . . . . . . . 12 |
50 | 47, 49, 29 | co 6650 | . . . . . . . . . . 11 |
51 | vg | . . . . . . . . . . . 12 | |
52 | 51 | cv 1482 | . . . . . . . . . . 11 |
53 | 50, 9, 52 | wf1o 5887 | . . . . . . . . . 10 |
54 | 22, 52 | ccom 5118 | . . . . . . . . . . . . 13 |
55 | 12, 54, 47 | cseq 12801 | . . . . . . . . . . . 12 |
56 | 49, 55 | cfv 5888 | . . . . . . . . . . 11 |
57 | 7, 56 | wceq 1483 | . . . . . . . . . 10 |
58 | 53, 57 | wa 384 | . . . . . . . . 9 |
59 | 22 | ccnv 5113 | . . . . . . . . . 10 |
60 | 4, 24 | cdif 3571 | . . . . . . . . . 10 |
61 | 59, 60 | cima 5117 | . . . . . . . . 9 |
62 | 58, 8, 61 | wsbc 3435 | . . . . . . . 8 |
63 | 62, 51 | wex 1704 | . . . . . . 7 |
64 | 63, 6 | cio 5849 | . . . . . 6 |
65 | 31, 46, 64 | cif 4086 | . . . . 5 |
66 | 25, 27, 65 | cif 4086 | . . . 4 |
67 | 5, 21, 66 | csb 3533 | . . 3 |
68 | 2, 3, 4, 4, 67 | cmpt2 6652 | . 2 |
69 | 1, 68 | wceq 1483 | 1 g |
Colors of variables: wff setvar class |
This definition is referenced by: gsumvalx 17270 gsum0 17278 |
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