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Mirrors > Home > MPE Home > Th. List > df-pi1 | Structured version Visualization version GIF version |
Description: Define the fundamental group, whose operation is given by concatenation of homotopy classes of loops. Definition of [Hatcher] p. 26. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
df-pi1 | ⊢ π1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpi1 24154 | . 2 class π1 | |
2 | vj | . . 3 setvar 𝑗 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | ctop 22030 | . . 3 class Top | |
5 | 2 | cv 1538 | . . . 4 class 𝑗 |
6 | 5 | cuni 4840 | . . 3 class ∪ 𝑗 |
7 | 3 | cv 1538 | . . . . 5 class 𝑦 |
8 | comi 24152 | . . . . 5 class Ω1 | |
9 | 5, 7, 8 | co 7268 | . . . 4 class (𝑗 Ω1 𝑦) |
10 | cphtpc 24120 | . . . . 5 class ≃ph | |
11 | 5, 10 | cfv 6427 | . . . 4 class ( ≃ph‘𝑗) |
12 | cqus 17204 | . . . 4 class /s | |
13 | 9, 11, 12 | co 7268 | . . 3 class ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗)) |
14 | 2, 3, 4, 6, 13 | cmpo 7270 | . 2 class (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗))) |
15 | 1, 14 | wceq 1539 | 1 wff π1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph‘𝑗))) |
Colors of variables: wff setvar class |
This definition is referenced by: pi1val 24188 |
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