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Mirrors > Home > MPE Home > Th. List > df-om1 | Structured version Visualization version Unicode version |
Description: Define the loop space of a topological space, with a magma structure on it given by concatenation of loops. This structure is not a group, but the operation is compatible with homotopy, which allows the homotopy groups to be defined based on this operation. (Contributed by Mario Carneiro, 10-Jul-2015.) |
Ref | Expression |
---|---|
df-om1 | TopSet |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | comi 22801 | . 2 | |
2 | vj | . . 3 | |
3 | vy | . . 3 | |
4 | ctop 20698 | . . 3 | |
5 | 2 | cv 1482 | . . . 4 |
6 | 5 | cuni 4436 | . . 3 |
7 | cnx 15854 | . . . . . 6 | |
8 | cbs 15857 | . . . . . 6 | |
9 | 7, 8 | cfv 5888 | . . . . 5 |
10 | cc0 9936 | . . . . . . . . 9 | |
11 | vf | . . . . . . . . . 10 | |
12 | 11 | cv 1482 | . . . . . . . . 9 |
13 | 10, 12 | cfv 5888 | . . . . . . . 8 |
14 | 3 | cv 1482 | . . . . . . . 8 |
15 | 13, 14 | wceq 1483 | . . . . . . 7 |
16 | c1 9937 | . . . . . . . . 9 | |
17 | 16, 12 | cfv 5888 | . . . . . . . 8 |
18 | 17, 14 | wceq 1483 | . . . . . . 7 |
19 | 15, 18 | wa 384 | . . . . . 6 |
20 | cii 22678 | . . . . . . 7 | |
21 | ccn 21028 | . . . . . . 7 | |
22 | 20, 5, 21 | co 6650 | . . . . . 6 |
23 | 19, 11, 22 | crab 2916 | . . . . 5 |
24 | 9, 23 | cop 4183 | . . . 4 |
25 | cplusg 15941 | . . . . . 6 | |
26 | 7, 25 | cfv 5888 | . . . . 5 |
27 | cpco 22800 | . . . . . 6 | |
28 | 5, 27 | cfv 5888 | . . . . 5 |
29 | 26, 28 | cop 4183 | . . . 4 |
30 | cts 15947 | . . . . . 6 TopSet | |
31 | 7, 30 | cfv 5888 | . . . . 5 TopSet |
32 | cxko 21364 | . . . . . 6 | |
33 | 5, 20, 32 | co 6650 | . . . . 5 |
34 | 31, 33 | cop 4183 | . . . 4 TopSet |
35 | 24, 29, 34 | ctp 4181 | . . 3 TopSet |
36 | 2, 3, 4, 6, 35 | cmpt2 6652 | . 2 TopSet |
37 | 1, 36 | wceq 1483 | 1 TopSet |
Colors of variables: wff setvar class |
This definition is referenced by: om1val 22830 |
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