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Definition df-pin 23618
Description: Define the n-th homotopy group, which is formed by taking the 𝑛-th loop space and forming the quotient under the relation of path homotopy equivalence in the base space of the 𝑛-th loop space, which is the 𝑛 − 1-th loop space. For 𝑛 = 0, since this is not well-defined we replace this relation with the path-connectedness relation, so that the 0-th homotopy group is the set of path components of 𝑋. (Since the 0-th loop space does not have a group operation, neither does the 0-th homotopy group, but the rest are genuine groups.) (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
df-pin πn = (𝑗 ∈ Top, 𝑝 𝑗 ↦ (𝑛 ∈ ℕ0 ↦ ((1st ‘((𝑗 Ω𝑛 𝑝)‘𝑛)) /s if(𝑛 = 0, {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}, ( ≃ph‘(TopOpen‘(1st ‘((𝑗 Ω𝑛 𝑝)‘(𝑛 − 1)))))))))
Distinct variable group:   𝑓,𝑗,𝑛,𝑝,𝑥,𝑦

Detailed syntax breakdown of Definition df-pin
StepHypRef Expression
1 cpin 23613 . 2 class πn
2 vj . . 3 setvar 𝑗
3 vp . . 3 setvar 𝑝
4 ctop 21502 . . 3 class Top
52cv 1537 . . . 4 class 𝑗
65cuni 4803 . . 3 class 𝑗
7 vn . . . 4 setvar 𝑛
8 cn0 11889 . . . 4 class 0
97cv 1537 . . . . . . 7 class 𝑛
103cv 1537 . . . . . . . 8 class 𝑝
11 comn 23611 . . . . . . . 8 class Ω𝑛
125, 10, 11co 7139 . . . . . . 7 class (𝑗 Ω𝑛 𝑝)
139, 12cfv 6328 . . . . . 6 class ((𝑗 Ω𝑛 𝑝)‘𝑛)
14 c1st 7673 . . . . . 6 class 1st
1513, 14cfv 6328 . . . . 5 class (1st ‘((𝑗 Ω𝑛 𝑝)‘𝑛))
16 cc0 10530 . . . . . . 7 class 0
179, 16wceq 1538 . . . . . 6 wff 𝑛 = 0
18 vf . . . . . . . . . . . 12 setvar 𝑓
1918cv 1537 . . . . . . . . . . 11 class 𝑓
2016, 19cfv 6328 . . . . . . . . . 10 class (𝑓‘0)
21 vx . . . . . . . . . . 11 setvar 𝑥
2221cv 1537 . . . . . . . . . 10 class 𝑥
2320, 22wceq 1538 . . . . . . . . 9 wff (𝑓‘0) = 𝑥
24 c1 10531 . . . . . . . . . . 11 class 1
2524, 19cfv 6328 . . . . . . . . . 10 class (𝑓‘1)
26 vy . . . . . . . . . . 11 setvar 𝑦
2726cv 1537 . . . . . . . . . 10 class 𝑦
2825, 27wceq 1538 . . . . . . . . 9 wff (𝑓‘1) = 𝑦
2923, 28wa 399 . . . . . . . 8 wff ((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)
30 cii 23484 . . . . . . . . 9 class II
31 ccn 21833 . . . . . . . . 9 class Cn
3230, 5, 31co 7139 . . . . . . . 8 class (II Cn 𝑗)
3329, 18, 32wrex 3110 . . . . . . 7 wff 𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)
3433, 21, 26copab 5095 . . . . . 6 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}
35 cmin 10863 . . . . . . . . . . 11 class
369, 24, 35co 7139 . . . . . . . . . 10 class (𝑛 − 1)
3736, 12cfv 6328 . . . . . . . . 9 class ((𝑗 Ω𝑛 𝑝)‘(𝑛 − 1))
3837, 14cfv 6328 . . . . . . . 8 class (1st ‘((𝑗 Ω𝑛 𝑝)‘(𝑛 − 1)))
39 ctopn 16691 . . . . . . . 8 class TopOpen
4038, 39cfv 6328 . . . . . . 7 class (TopOpen‘(1st ‘((𝑗 Ω𝑛 𝑝)‘(𝑛 − 1))))
41 cphtpc 23578 . . . . . . 7 class ph
4240, 41cfv 6328 . . . . . 6 class ( ≃ph‘(TopOpen‘(1st ‘((𝑗 Ω𝑛 𝑝)‘(𝑛 − 1)))))
4317, 34, 42cif 4428 . . . . 5 class if(𝑛 = 0, {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}, ( ≃ph‘(TopOpen‘(1st ‘((𝑗 Ω𝑛 𝑝)‘(𝑛 − 1))))))
44 cqus 16774 . . . . 5 class /s
4515, 43, 44co 7139 . . . 4 class ((1st ‘((𝑗 Ω𝑛 𝑝)‘𝑛)) /s if(𝑛 = 0, {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}, ( ≃ph‘(TopOpen‘(1st ‘((𝑗 Ω𝑛 𝑝)‘(𝑛 − 1)))))))
467, 8, 45cmpt 5113 . . 3 class (𝑛 ∈ ℕ0 ↦ ((1st ‘((𝑗 Ω𝑛 𝑝)‘𝑛)) /s if(𝑛 = 0, {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}, ( ≃ph‘(TopOpen‘(1st ‘((𝑗 Ω𝑛 𝑝)‘(𝑛 − 1))))))))
472, 3, 4, 6, 46cmpo 7141 . 2 class (𝑗 ∈ Top, 𝑝 𝑗 ↦ (𝑛 ∈ ℕ0 ↦ ((1st ‘((𝑗 Ω𝑛 𝑝)‘𝑛)) /s if(𝑛 = 0, {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}, ( ≃ph‘(TopOpen‘(1st ‘((𝑗 Ω𝑛 𝑝)‘(𝑛 − 1)))))))))
481, 47wceq 1538 1 wff πn = (𝑗 ∈ Top, 𝑝 𝑗 ↦ (𝑛 ∈ ℕ0 ↦ ((1st ‘((𝑗 Ω𝑛 𝑝)‘𝑛)) /s if(𝑛 = 0, {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 ∈ (II Cn 𝑗)((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)}, ( ≃ph‘(TopOpen‘(1st ‘((𝑗 Ω𝑛 𝑝)‘(𝑛 − 1)))))))))
Colors of variables: wff setvar class
This definition is referenced by: (None)
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