Metamath Proof Explorer


Definition df-om1

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
Assertion df-om1 Ω1 = ( 𝑗 ∈ Top , 𝑦 𝑗 ↦ { ⟨ ( Base ‘ ndx ) , { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } ⟩ , ⟨ ( +g ‘ ndx ) , ( *𝑝𝑗 ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( 𝑗ko II ) ⟩ } )

Detailed syntax breakdown

Step Hyp Ref Expression
0 comi Ω1
1 vj 𝑗
2 ctop Top
3 vy 𝑦
4 1 cv 𝑗
5 4 cuni 𝑗
6 cbs Base
7 cnx ndx
8 7 6 cfv ( Base ‘ ndx )
9 vf 𝑓
10 cii II
11 ccn Cn
12 10 4 11 co ( II Cn 𝑗 )
13 9 cv 𝑓
14 cc0 0
15 14 13 cfv ( 𝑓 ‘ 0 )
16 3 cv 𝑦
17 15 16 wceq ( 𝑓 ‘ 0 ) = 𝑦
18 c1 1
19 18 13 cfv ( 𝑓 ‘ 1 )
20 19 16 wceq ( 𝑓 ‘ 1 ) = 𝑦
21 17 20 wa ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 )
22 21 9 12 crab { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) }
23 8 22 cop ⟨ ( Base ‘ ndx ) , { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } ⟩
24 cplusg +g
25 7 24 cfv ( +g ‘ ndx )
26 cpco *𝑝
27 4 26 cfv ( *𝑝𝑗 )
28 25 27 cop ⟨ ( +g ‘ ndx ) , ( *𝑝𝑗 ) ⟩
29 cts TopSet
30 7 29 cfv ( TopSet ‘ ndx )
31 cxko ko
32 4 10 31 co ( 𝑗ko II )
33 30 32 cop ⟨ ( TopSet ‘ ndx ) , ( 𝑗ko II ) ⟩
34 23 28 33 ctp { ⟨ ( Base ‘ ndx ) , { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } ⟩ , ⟨ ( +g ‘ ndx ) , ( *𝑝𝑗 ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( 𝑗ko II ) ⟩ }
35 1 3 2 5 34 cmpo ( 𝑗 ∈ Top , 𝑦 𝑗 ↦ { ⟨ ( Base ‘ ndx ) , { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } ⟩ , ⟨ ( +g ‘ ndx ) , ( *𝑝𝑗 ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( 𝑗ko II ) ⟩ } )
36 0 35 wceq Ω1 = ( 𝑗 ∈ Top , 𝑦 𝑗 ↦ { ⟨ ( Base ‘ ndx ) , { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } ⟩ , ⟨ ( +g ‘ ndx ) , ( *𝑝𝑗 ) ⟩ , ⟨ ( TopSet ‘ ndx ) , ( 𝑗ko II ) ⟩ } )