Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-om1 Structured version   Visualization version   GIF version

Definition df-om1 23006
 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.)
Assertion
Ref Expression
df-om1 Ω1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝𝑗)⟩, ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩})
Distinct variable group:   𝑓,𝑗,𝑦

Detailed syntax breakdown of Definition df-om1
StepHypRef Expression
1 comi 23001 . 2 class Ω1
2 vj . . 3 setvar 𝑗
3 vy . . 3 setvar 𝑦
4 ctop 20900 . . 3 class Top
52cv 1631 . . . 4 class 𝑗
65cuni 4588 . . 3 class 𝑗
7 cnx 16056 . . . . . 6 class ndx
8 cbs 16059 . . . . . 6 class Base
97, 8cfv 6049 . . . . 5 class (Base‘ndx)
10 cc0 10128 . . . . . . . . 9 class 0
11 vf . . . . . . . . . 10 setvar 𝑓
1211cv 1631 . . . . . . . . 9 class 𝑓
1310, 12cfv 6049 . . . . . . . 8 class (𝑓‘0)
143cv 1631 . . . . . . . 8 class 𝑦
1513, 14wceq 1632 . . . . . . 7 wff (𝑓‘0) = 𝑦
16 c1 10129 . . . . . . . . 9 class 1
1716, 12cfv 6049 . . . . . . . 8 class (𝑓‘1)
1817, 14wceq 1632 . . . . . . 7 wff (𝑓‘1) = 𝑦
1915, 18wa 383 . . . . . 6 wff ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)
20 cii 22879 . . . . . . 7 class II
21 ccn 21230 . . . . . . 7 class Cn
2220, 5, 21co 6813 . . . . . 6 class (II Cn 𝑗)
2319, 11, 22crab 3054 . . . . 5 class {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}
249, 23cop 4327 . . . 4 class ⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩
25 cplusg 16143 . . . . . 6 class +g
267, 25cfv 6049 . . . . 5 class (+g‘ndx)
27 cpco 23000 . . . . . 6 class *𝑝
285, 27cfv 6049 . . . . 5 class (*𝑝𝑗)
2926, 28cop 4327 . . . 4 class ⟨(+g‘ndx), (*𝑝𝑗)⟩
30 cts 16149 . . . . . 6 class TopSet
317, 30cfv 6049 . . . . 5 class (TopSet‘ndx)
32 cxko 21566 . . . . . 6 class ^ko
335, 20, 32co 6813 . . . . 5 class (𝑗 ^ko II)
3431, 33cop 4327 . . . 4 class ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩
3524, 29, 34ctp 4325 . . 3 class {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝𝑗)⟩, ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩}
362, 3, 4, 6, 35cmpt2 6815 . 2 class (𝑗 ∈ Top, 𝑦 𝑗 ↦ {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝𝑗)⟩, ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩})
371, 36wceq 1632 1 wff Ω1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝𝑗)⟩, ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩})
 Colors of variables: wff setvar class This definition is referenced by:  om1val  23030
 Copyright terms: Public domain W3C validator