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Theorem List for Metamath Proof Explorer - 22801-22900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnheibor 22801* Heine-Borel theorem for complex numbers. A subset of is compact iff it is closed and bounded. (Contributed by Mario Carneiro, 14-Sep-2014.)
𝐽 = (TopOpen‘ℂfld)    &   𝑇 = (𝐽t 𝑋)       (𝑋 ⊆ ℂ → (𝑇 ∈ Comp ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥𝑋 (abs‘𝑥) ≤ 𝑟)))
 
Theoremcnllycmp 22802 The topology on the complex numbers is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
𝐽 = (TopOpen‘ℂfld)       𝐽 ∈ 𝑛-Locally Comp
 
Theoremrellycmp 22803 The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
(topGen‘ran (,)) ∈ 𝑛-Locally Comp
 
Theorembndth 22804* The Boundedness Theorem. A continuous function from a compact topological space to the reals is bounded (above). (Boundedness below is obtained by applying this theorem to -𝐹.) (Contributed by Mario Carneiro, 12-Aug-2014.)
𝑋 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥)
 
Theoremevth 22805* The Extreme Value Theorem. A continuous function from a nonempty compact topological space to the reals attains its maximum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)
𝑋 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑋 ≠ ∅)       (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
 
Theoremevth2 22806* The Extreme Value Theorem, minimum version. A continuous function from a nonempty compact topological space to the reals attains its minimum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)
𝑋 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑋 ≠ ∅)       (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑥) ≤ (𝐹𝑦))
 
Theoremlebnumlem1 22807* Lemma for lebnum 22810. The function 𝐹 measures the sum of all of the distances to escape the sets of the cover. Since by assumption it is a cover, there is at least one set which covers a given point, and since it is open, the point is a positive distance from the edge of the set. Thus, the sum is a strictly positive number. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑈 ∈ Fin)    &   (𝜑 → ¬ 𝑋𝑈)    &   𝐹 = (𝑦𝑋 ↦ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))       (𝜑𝐹:𝑋⟶ℝ+)
 
Theoremlebnumlem2 22808* Lemma for lebnum 22810. As a finite sum of point-to-set distance functions, which are continuous by metdscn 22706, the function 𝐹 is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑈 ∈ Fin)    &   (𝜑 → ¬ 𝑋𝑈)    &   𝐹 = (𝑦𝑋 ↦ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))    &   𝐾 = (topGen‘ran (,))       (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
 
Theoremlebnumlem3 22809* Lemma for lebnum 22810. By the previous lemmas, 𝐹 is continuous and positive on a compact set, so it has a positive minimum 𝑟. Then setting 𝑑 = 𝑟 / #(𝑈), since for each 𝑢𝑈 we have ball(𝑥, 𝑑) ⊆ 𝑢 iff 𝑑𝑑(𝑥, 𝑋𝑢), if ¬ ball(𝑥, 𝑑) ⊆ 𝑢 for all 𝑢 then summing over 𝑢 yields Σ𝑢𝑈𝑑(𝑥, 𝑋𝑢) = 𝐹(𝑥) < Σ𝑢𝑈𝑑 = 𝑟, in contradiction to the assumption that 𝑟 is the minimum of 𝐹. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.) (Revised by AV, 30-Sep-2020.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)    &   (𝜑𝑈 ∈ Fin)    &   (𝜑 → ¬ 𝑋𝑈)    &   𝐹 = (𝑦𝑋 ↦ Σ𝑘𝑈 inf(ran (𝑧 ∈ (𝑋𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < ))    &   𝐾 = (topGen‘ran (,))       (𝜑 → ∃𝑑 ∈ ℝ+𝑥𝑋𝑢𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
 
Theoremlebnum 22810* The Lebesgue number lemma, or Lebesgue covering lemma. If 𝑋 is a compact metric space and 𝑈 is an open cover of 𝑋, then there exists a positive real number 𝑑 such that every ball of size 𝑑 (and every subset of a ball of size 𝑑, including every subset of diameter less than 𝑑) is a subset of some member of the cover. (Contributed by Mario Carneiro, 14-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)       (𝜑 → ∃𝑑 ∈ ℝ+𝑥𝑋𝑢𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
 
Theoremxlebnum 22811* Generalize lebnum 22810 to extended metrics. (Contributed by Mario Carneiro, 5-Sep-2015.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑈𝐽)    &   (𝜑𝑋 = 𝑈)       (𝜑 → ∃𝑑 ∈ ℝ+𝑥𝑋𝑢𝑈 (𝑥(ball‘𝐷)𝑑) ⊆ 𝑢)
 
Theoremlebnumii 22812* Specialize the Lebesgue number lemma lebnum 22810 to the unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.)
((𝑈 ⊆ II ∧ (0[,]1) = 𝑈) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑢𝑈 (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢)
 
12.4.12  Path homotopy
 
Syntaxchtpy 22813 Extend class notation with the class of homotopies between two continuous functions.
class Htpy
 
Syntaxcphtpy 22814 Extend class notation with the class of path homotopies between two continuous functions.
class PHtpy
 
Syntaxcphtpc 22815 Extend class notation with the path homotopy relation.
class ph
 
Definitiondf-htpy 22816* Define the function which takes topological spaces 𝑋, 𝑌 and two continuous functions 𝐹, 𝐺:𝑋𝑌 and returns the class of homotopies from 𝐹 to 𝐺. (Contributed by Mario Carneiro, 22-Feb-2015.)
Htpy = (𝑥 ∈ Top, 𝑦 ∈ Top ↦ (𝑓 ∈ (𝑥 Cn 𝑦), 𝑔 ∈ (𝑥 Cn 𝑦) ↦ { ∈ ((𝑥 ×t II) Cn 𝑦) ∣ ∀𝑠 𝑥((𝑠0) = (𝑓𝑠) ∧ (𝑠1) = (𝑔𝑠))}))
 
Definitiondf-phtpy 22817* Define the class of path homotopies between two paths 𝐹, 𝐺:II⟶𝑋; these are homotopies (in the sense of df-htpy 22816) which also preserve both endpoints of the paths throughout the homotopy. Definition of [Hatcher] p. 25. (Contributed by Jeff Madsen, 2-Sep-2009.)
PHtpy = (𝑥 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑥), 𝑔 ∈ (II Cn 𝑥) ↦ { ∈ (𝑓(II Htpy 𝑥)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}))
 
Theoremishtpy 22818* Membership in the class of homotopies between two continuous functions. (Contributed by Mario Carneiro, 22-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))))
 
Theoremhtpycn 22819 A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×t II) Cn 𝐾))
 
Theoremhtpyi 22820 A homotopy evaluated at its endpoints. (Contributed by Mario Carneiro, 22-Feb-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))       ((𝜑𝐴𝑋) → ((𝐴𝐻0) = (𝐹𝐴) ∧ (𝐴𝐻1) = (𝐺𝐴)))
 
Theoremishtpyd 22821* Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐻 ∈ ((𝐽 ×t II) Cn 𝐾))    &   ((𝜑𝑠𝑋) → (𝑠𝐻0) = (𝐹𝑠))    &   ((𝜑𝑠𝑋) → (𝑠𝐻1) = (𝐺𝑠))       (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))
 
Theoremhtpycom 22822* Given a homotopy from 𝐹 to 𝐺, produce a homotopy from 𝐺 to 𝐹. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))    &   𝑀 = (𝑥𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦)))    &   (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))       (𝜑𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐹))
 
Theoremhtpyid 22823* A homotopy from a function to itself. (Contributed by Mario Carneiro, 23-Feb-2015.)
𝐺 = (𝑥𝑋, 𝑦 ∈ (0[,]1) ↦ (𝐹𝑥))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))       (𝜑𝐺 ∈ (𝐹(𝐽 Htpy 𝐾)𝐹))
 
Theoremhtpyco1 22824* Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
𝑁 = (𝑥𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃𝑥)𝐻𝑦))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑃 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐹 ∈ (𝐾 Cn 𝐿))    &   (𝜑𝐺 ∈ (𝐾 Cn 𝐿))    &   (𝜑𝐻 ∈ (𝐹(𝐾 Htpy 𝐿)𝐺))       (𝜑𝑁 ∈ ((𝐹𝑃)(𝐽 Htpy 𝐿)(𝐺𝑃)))
 
Theoremhtpyco2 22825 Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
(𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑃 ∈ (𝐾 Cn 𝐿))    &   (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))       (𝜑 → (𝑃𝐻) ∈ ((𝑃𝐹)(𝐽 Htpy 𝐿)(𝑃𝐺)))
 
Theoremhtpycc 22826* Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
𝑁 = (𝑥𝑋, 𝑦 ∈ (0[,]1) ↦ if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐺 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐻 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐿 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))    &   (𝜑𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐻))       (𝜑𝑁 ∈ (𝐹(𝐽 Htpy 𝐾)𝐻))
 
Theoremisphtpy 22827* Membership in the class of path homotopies between two continuous functions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))       (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))))
 
Theoremphtpyhtpy 22828 A path homotopy is a homotopy. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))       (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺))
 
Theoremphtpycn 22829 A path homotopy is a continuous function. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))       (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ ((II ×t II) Cn 𝐽))
 
Theoremphtpyi 22830 Membership in the class of path homotopies between two continuous functions. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))       ((𝜑𝐴 ∈ (0[,]1)) → ((0𝐻𝐴) = (𝐹‘0) ∧ (1𝐻𝐴) = (𝐹‘1)))
 
Theoremphtpy01 22831 Two path-homotopic paths have the same start and end point. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))       (𝜑 → ((𝐹‘0) = (𝐺‘0) ∧ (𝐹‘1) = (𝐺‘1)))
 
Theoremisphtpyd 22832* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺))    &   ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘0))    &   ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘1))       (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))
 
Theoremisphtpy2d 22833* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝐻 ∈ ((II ×t II) Cn 𝐽))    &   ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐹𝑠))    &   ((𝜑𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐺𝑠))    &   ((𝜑𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘0))    &   ((𝜑𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘1))       (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))
 
Theoremphtpycom 22834* Given a homotopy from 𝐹 to 𝐺, produce a homotopy from 𝐺 to 𝐹. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦)))    &   (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))       (𝜑𝐾 ∈ (𝐺(PHtpy‘𝐽)𝐹))
 
Theoremphtpyid 22835* A homotopy from a path to itself. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
𝐺 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹𝑥))    &   (𝜑𝐹 ∈ (II Cn 𝐽))       (𝜑𝐺 ∈ (𝐹(PHtpy‘𝐽)𝐹))
 
Theoremphtpyco2 22836 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝑃 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))       (𝜑 → (𝑃𝐻) ∈ ((𝑃𝐹)(PHtpy‘𝐾)(𝑃𝐺)))
 
Theoremphtpycc 22837* Concatenate two path homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)
𝑀 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ if(𝑦 ≤ (1 / 2), (𝑥𝐾(2 · 𝑦)), (𝑥𝐿((2 · 𝑦) − 1))))    &   (𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝐻 ∈ (II Cn 𝐽))    &   (𝜑𝐾 ∈ (𝐹(PHtpy‘𝐽)𝐺))    &   (𝜑𝐿 ∈ (𝐺(PHtpy‘𝐽)𝐻))       (𝜑𝑀 ∈ (𝐹(PHtpy‘𝐽)𝐻))
 
Definitiondf-phtpc 22838* Define the function which takes a topology and returns the path homotopy relation on that topology. Definition of [Hatcher] p. 25. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)
ph = (𝑥 ∈ Top ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)})
 
Theoremphtpcrel 22839 The path homotopy relation is a relation. (Contributed by Mario Carneiro, 7-Jun-2014.) (Revised by Mario Carneiro, 7-Aug-2014.)
Rel ( ≃ph𝐽)
 
Theoremisphtpc 22840 The relation "is path homotopic to". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Sep-2015.)
(𝐹( ≃ph𝐽)𝐺 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))
 
Theoremphtpcer 22841 Path homotopy is an equivalence relation. Proposition 1.2 of [Hatcher] p. 26. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 6-Jul-2015.) (Proof shortened by AV, 1-May-2021.)
( ≃ph𝐽) Er (II Cn 𝐽)
 
Theoremphtpc01 22842 Path homotopic paths have the same endpoints. (Contributed by Mario Carneiro, 24-Feb-2015.)
(𝐹( ≃ph𝐽)𝐺 → ((𝐹‘0) = (𝐺‘0) ∧ (𝐹‘1) = (𝐺‘1)))
 
Theoremreparphti 22843* Lemma for reparpht 22844. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn II))    &   (𝜑 → (𝐺‘0) = 0)    &   (𝜑 → (𝐺‘1) = 1)    &   𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺𝑥)) + (𝑦 · 𝑥))))       (𝜑𝐻 ∈ ((𝐹𝐺)(PHtpy‘𝐽)𝐹))
 
Theoremreparpht 22844 Reparametrization lemma. The reparametrization of a path by any continuous map 𝐺:II⟶II with 𝐺(0) = 0 and 𝐺(1) = 1 is path-homotopic to the original path. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Feb-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn II))    &   (𝜑 → (𝐺‘0) = 0)    &   (𝜑 → (𝐺‘1) = 1)       (𝜑 → (𝐹𝐺)( ≃ph𝐽)𝐹)
 
Theoremphtpcco2 22845 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 6-Jul-2015.)
(𝜑𝐹( ≃ph𝐽)𝐺)    &   (𝜑𝑃 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝑃𝐹)( ≃ph𝐾)(𝑃𝐺))
 
12.4.13  The fundamental group
 
Syntaxcpco 22846 Extend class notation with the concatenation operation for paths in a topological space.
class *𝑝
 
Syntaxcomi 22847 Extend class notation with the loop space.
class Ω1
 
Syntaxcomn 22848 Extend class notation with the higher loop spaces.
class Ω𝑛
 
Syntaxcpi1 22849 Extend class notation with the fundamental group.
class π1
 
Syntaxcpin 22850 Extend class notation with the higher homotopy groups.
class πn
 
Definitiondf-pco 22851* Define the concatenation of two paths in a topological space 𝐽. For simplicity of definition, we define it on all paths, not just those whose endpoints line up. Definition of [Hatcher] p. 26. Hatcher denotes path concatenation with a square dot; other authors, such as Munkres, use a star. (Contributed by Jeff Madsen, 15-Jun-2010.)
*𝑝 = (𝑗 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑗), 𝑔 ∈ (II Cn 𝑗) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1))))))
 
Definitiondf-om1 22852* 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.)
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)⟩})
 
Definitiondf-omn 22853* Define the n-th iterated loop space of a topological space. Unlike Ω1 this is actually a pointed topological space, which is to say a tuple of a topological space (a member of TopSp, not Top) and a point in the space. Higher loop spaces select the constant loop at the point from the lower loop space for the distinguished point. (Contributed by Mario Carneiro, 10-Jul-2015.)
Ω𝑛 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ seq0(((𝑥 ∈ V, 𝑝 ∈ V ↦ ⟨((TopOpen‘(1st𝑥)) Ω1 (2nd𝑥)), ((0[,]1) × {(2nd𝑥)})⟩) ∘ 1st ), ⟨{⟨(Base‘ndx), 𝑗⟩, ⟨(TopSet‘ndx), 𝑗⟩}, 𝑦⟩))
 
Definitiondf-pi1 22854* 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.)
π1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ ((𝑗 Ω1 𝑦) /s ( ≃ph𝑗)))
 
Definitiondf-pin 22855* 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.)
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)))))))))
 
Theorempcofval 22856* The value of the path concatenation function on a topological space. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)
(*𝑝𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1)))))
 
Theorempcoval 22857* The concatenation of two paths. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Aug-2014.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))       (𝜑 → (𝐹(*𝑝𝐽)𝐺) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1)))))
 
Theorempcovalg 22858 Evaluate the concatenation of two paths. (Contributed by Mario Carneiro, 7-Jun-2014.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))       ((𝜑𝑋 ∈ (0[,]1)) → ((𝐹(*𝑝𝐽)𝐺)‘𝑋) = if(𝑋 ≤ (1 / 2), (𝐹‘(2 · 𝑋)), (𝐺‘((2 · 𝑋) − 1))))
 
Theorempcoval1 22859 Evaluate the concatenation of two paths on the first half. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))       ((𝜑𝑋 ∈ (0[,](1 / 2))) → ((𝐹(*𝑝𝐽)𝐺)‘𝑋) = (𝐹‘(2 · 𝑋)))
 
Theorempco0 22860 The starting point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))       (𝜑 → ((𝐹(*𝑝𝐽)𝐺)‘0) = (𝐹‘0))
 
Theorempco1 22861 The ending point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))       (𝜑 → ((𝐹(*𝑝𝐽)𝐺)‘1) = (𝐺‘1))
 
Theorempcoval2 22862 Evaluate the concatenation of two paths on the second half. (Contributed by Jeff Madsen, 15-Jun-2010.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑 → (𝐹‘1) = (𝐺‘0))       ((𝜑𝑋 ∈ ((1 / 2)[,]1)) → ((𝐹(*𝑝𝐽)𝐺)‘𝑋) = (𝐺‘((2 · 𝑋) − 1)))
 
Theorempcocn 22863 The concatenation of two paths is a path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑 → (𝐹‘1) = (𝐺‘0))       (𝜑 → (𝐹(*𝑝𝐽)𝐺) ∈ (II Cn 𝐽))
 
Theoremcopco 22864 The composition of a concatenation of paths with a continuous function. (Contributed by Mario Carneiro, 9-Jul-2015.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑 → (𝐹‘1) = (𝐺‘0))    &   (𝜑𝐻 ∈ (𝐽 Cn 𝐾))       (𝜑 → (𝐻 ∘ (𝐹(*𝑝𝐽)𝐺)) = ((𝐻𝐹)(*𝑝𝐾)(𝐻𝐺)))
 
Theorempcohtpylem 22865* Lemma for pcohtpy 22866. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)
(𝜑 → (𝐹‘1) = (𝐺‘0))    &   (𝜑𝐹( ≃ph𝐽)𝐻)    &   (𝜑𝐺( ≃ph𝐽)𝐾)    &   𝑃 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)))    &   (𝜑𝑀 ∈ (𝐹(PHtpy‘𝐽)𝐻))    &   (𝜑𝑁 ∈ (𝐺(PHtpy‘𝐽)𝐾))       (𝜑𝑃 ∈ ((𝐹(*𝑝𝐽)𝐺)(PHtpy‘𝐽)(𝐻(*𝑝𝐽)𝐾)))
 
Theorempcohtpy 22866 Homotopy invariance of path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)
(𝜑 → (𝐹‘1) = (𝐺‘0))    &   (𝜑𝐹( ≃ph𝐽)𝐻)    &   (𝜑𝐺( ≃ph𝐽)𝐾)       (𝜑 → (𝐹(*𝑝𝐽)𝐺)( ≃ph𝐽)(𝐻(*𝑝𝐽)𝐾))
 
Theorempcoptcl 22867 A constant function is a path from 𝑌 to itself. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 19-Mar-2015.)
𝑃 = ((0[,]1) × {𝑌})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
 
Theorempcopt 22868 Concatenation with a point does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
𝑃 = ((0[,]1) × {𝑌})       ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑃(*𝑝𝐽)𝐹)( ≃ph𝐽)𝐹)
 
Theorempcopt2 22869 Concatenation with a point does not affect homotopy class. (Contributed by Mario Carneiro, 12-Feb-2015.)
𝑃 = ((0[,]1) × {𝑌})       ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹(*𝑝𝐽)𝑃)( ≃ph𝐽)𝐹)
 
Theorempcoass 22870* Order of concatenation does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)
(𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑𝐻 ∈ (II Cn 𝐽))    &   (𝜑 → (𝐹‘1) = (𝐺‘0))    &   (𝜑 → (𝐺‘1) = (𝐻‘0))    &   𝑃 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), if(𝑥 ≤ (1 / 4), (2 · 𝑥), (𝑥 + (1 / 4))), ((𝑥 / 2) + (1 / 2))))       (𝜑 → ((𝐹(*𝑝𝐽)𝐺)(*𝑝𝐽)𝐻)( ≃ph𝐽)(𝐹(*𝑝𝐽)(𝐺(*𝑝𝐽)𝐻)))
 
Theorempcorevcl 22871* Closure for a reversed path. (Contributed by Mario Carneiro, 12-Feb-2015.)
𝐺 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))       (𝐹 ∈ (II Cn 𝐽) → (𝐺 ∈ (II Cn 𝐽) ∧ (𝐺‘0) = (𝐹‘1) ∧ (𝐺‘1) = (𝐹‘0)))
 
Theorempcorevlem 22872* Lemma for pcorev 22873. Prove continuity of the homotopy function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)
𝐺 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    &   𝑃 = ((0[,]1) × {(𝐹‘1)})    &   𝐻 = (𝑠 ∈ (0[,]1), 𝑡 ∈ (0[,]1) ↦ (𝐹‘if(𝑠 ≤ (1 / 2), (1 − ((1 − 𝑡) · (2 · 𝑠))), (1 − ((1 − 𝑡) · (1 − ((2 · 𝑠) − 1)))))))       (𝐹 ∈ (II Cn 𝐽) → (𝐻 ∈ ((𝐺(*𝑝𝐽)𝐹)(PHtpy‘𝐽)𝑃) ∧ (𝐺(*𝑝𝐽)𝐹)( ≃ph𝐽)𝑃))
 
Theorempcorev 22873* Concatenation with the reverse path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
𝐺 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    &   𝑃 = ((0[,]1) × {(𝐹‘1)})       (𝐹 ∈ (II Cn 𝐽) → (𝐺(*𝑝𝐽)𝐹)( ≃ph𝐽)𝑃)
 
Theorempcorev2 22874* Concatenation with the reverse path. (Contributed by Mario Carneiro, 12-Feb-2015.)
𝐺 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    &   𝑃 = ((0[,]1) × {(𝐹‘0)})       (𝐹 ∈ (II Cn 𝐽) → (𝐹(*𝑝𝐽)𝐺)( ≃ph𝐽)𝑃)
 
Theorempcophtb 22875* The path homotopy equivalence relation on two paths 𝐹, 𝐺 with the same start and end point can be written in terms of the loop 𝐹𝐺 formed by concatenating 𝐹 with the inverse of 𝐺. Thus, all the homotopy information in ph𝐽 is available if we restrict our attention to closed loops, as in the definition of the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.)
𝐻 = (𝑥 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑥)))    &   𝑃 = ((0[,]1) × {(𝐹‘0)})    &   (𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐺 ∈ (II Cn 𝐽))    &   (𝜑 → (𝐹‘0) = (𝐺‘0))    &   (𝜑 → (𝐹‘1) = (𝐺‘1))       (𝜑 → ((𝐹(*𝑝𝐽)𝐻)( ≃ph𝐽)𝑃𝐹( ≃ph𝐽)𝐺))
 
Theoremom1val 22876* The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
𝑂 = (𝐽 Ω1 𝑌)    &   (𝜑𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})    &   (𝜑+ = (*𝑝𝐽))    &   (𝜑𝐾 = (𝐽 ^ko II))    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)       (𝜑𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
 
Theoremom1bas 22877* The base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
𝑂 = (𝐽 Ω1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   (𝜑𝐵 = (Base‘𝑂))       (𝜑𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
 
Theoremom1elbas 22878 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
𝑂 = (𝐽 Ω1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   (𝜑𝐵 = (Base‘𝑂))       (𝜑 → (𝐹𝐵 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)))
 
Theoremom1addcl 22879 Closure of the group operation of the loop space. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 5-Sep-2015.)
𝑂 = (𝐽 Ω1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   (𝜑𝐵 = (Base‘𝑂))    &   (𝜑𝐻𝐵)    &   (𝜑𝐾𝐵)       (𝜑 → (𝐻(*𝑝𝐽)𝐾) ∈ 𝐵)
 
Theoremom1plusg 22880 The group operation (which isn't much more than a magma) of the loop space. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝑂 = (𝐽 Ω1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)       (𝜑 → (*𝑝𝐽) = (+g𝑂))
 
Theoremom1tset 22881 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
𝑂 = (𝐽 Ω1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)       (𝜑 → (𝐽 ^ko II) = (TopSet‘𝑂))
 
Theoremom1opn 22882 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
𝑂 = (𝐽 Ω1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   𝐾 = (TopOpen‘𝑂)    &   (𝜑𝐵 = (Base‘𝑂))       (𝜑𝐾 = ((𝐽 ^ko II) ↾t 𝐵))
 
Theorempi1val 22883 The definition of the fundamental group. (Contributed by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
𝐺 = (𝐽 π1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   𝑂 = (𝐽 Ω1 𝑌)       (𝜑𝐺 = (𝑂 /s ( ≃ph𝐽)))
 
Theorempi1bas 22884 The base set of the fundamental group of a topological space at a given base point. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
𝐺 = (𝐽 π1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   𝑂 = (𝐽 Ω1 𝑌)    &   (𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐾 = (Base‘𝑂))       (𝜑𝐵 = (𝐾 / ( ≃ph𝐽)))
 
Theorempi1blem 22885 Lemma for pi1buni 22886. (Contributed by Mario Carneiro, 10-Jul-2015.)
𝐺 = (𝐽 π1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   𝑂 = (𝐽 Ω1 𝑌)    &   (𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐾 = (Base‘𝑂))       (𝜑 → ((( ≃ph𝐽) “ 𝐾) ⊆ 𝐾𝐾 ⊆ (II Cn 𝐽)))
 
Theorempi1buni 22886 Another way to write the loop space base in terms of the base of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
𝐺 = (𝐽 π1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   𝑂 = (𝐽 Ω1 𝑌)    &   (𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐾 = (Base‘𝑂))       (𝜑 𝐵 = 𝐾)
 
Theorempi1bas2 22887 The base set of the fundamental group, written self-referentially. (Contributed by Mario Carneiro, 10-Jul-2015.)
𝐺 = (𝐽 π1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   (𝜑𝐵 = (Base‘𝐺))       (𝜑𝐵 = ( 𝐵 / ( ≃ph𝐽)))
 
Theorempi1eluni 22888 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
𝐺 = (𝐽 π1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   (𝜑𝐵 = (Base‘𝐺))       (𝜑 → (𝐹 𝐵 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)))
 
Theorempi1bas3 22889 The base set of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
𝐺 = (𝐽 π1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   (𝜑𝐵 = (Base‘𝐺))    &   𝑅 = (( ≃ph𝐽) ∩ ( 𝐵 × 𝐵))       (𝜑𝐵 = ( 𝐵 / 𝑅))
 
Theorempi1cpbl 22890 The group operation, loop concatenation, is compatible with homotopy equivalence. (Contributed by Mario Carneiro, 10-Jul-2015.)
𝐺 = (𝐽 π1 𝑌)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   (𝜑𝐵 = (Base‘𝐺))    &   𝑅 = (( ≃ph𝐽) ∩ ( 𝐵 × 𝐵))    &   𝑂 = (𝐽 Ω1 𝑌)    &    + = (+g𝑂)       (𝜑 → ((𝑀𝑅𝑁𝑃𝑅𝑄) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄)))
 
Theoremelpi1 22891* The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
𝐺 = (𝐽 π1 𝑌)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)       (𝜑 → (𝐹𝐵 ↔ ∃𝑓 ∈ (II Cn 𝐽)(((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ 𝐹 = [𝑓]( ≃ph𝐽))))
 
Theoremelpi1i 22892 The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
𝐺 = (𝐽 π1 𝑌)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   (𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑 → (𝐹‘0) = 𝑌)    &   (𝜑 → (𝐹‘1) = 𝑌)       (𝜑 → [𝐹]( ≃ph𝐽) ∈ 𝐵)
 
Theorempi1addf 22893 The group operation of π1 is a binary operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
𝐺 = (𝐽 π1 𝑌)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &    + = (+g𝐺)       (𝜑+ :(𝐵 × 𝐵)⟶𝐵)
 
Theorempi1addval 22894 The concatenation of two path-homotopy classes in the fundamental group. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
𝐺 = (𝐽 π1 𝑌)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &    + = (+g𝐺)    &   (𝜑𝑀 𝐵)    &   (𝜑𝑁 𝐵)       (𝜑 → ([𝑀]( ≃ph𝐽) + [𝑁]( ≃ph𝐽)) = [(𝑀(*𝑝𝐽)𝑁)]( ≃ph𝐽))
 
Theorempi1grplem 22895 Lemma for pi1grp 22896. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Aug-2015.)
𝐺 = (𝐽 π1 𝑌)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &    0 = ((0[,]1) × {𝑌})       (𝜑 → (𝐺 ∈ Grp ∧ [ 0 ]( ≃ph𝐽) = (0g𝐺)))
 
Theorempi1grp 22896 The fundamental group is a group. Proposition 1.3 of [Hatcher] p. 26. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.) (Revised by Mario Carneiro, 10-Aug-2015.)
𝐺 = (𝐽 π1 𝑌)       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → 𝐺 ∈ Grp)
 
Theorempi1id 22897 The identity element of the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
𝐺 = (𝐽 π1 𝑌)    &    0 = ((0[,]1) × {𝑌})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌𝑋) → [ 0 ]( ≃ph𝐽) = (0g𝐺))
 
Theorempi1inv 22898* An inverse in the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
𝐺 = (𝐽 π1 𝑌)    &   𝑁 = (invg𝐺)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝑌𝑋)    &   (𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑 → (𝐹‘0) = 𝑌)    &   (𝜑 → (𝐹‘1) = 𝑌)    &   𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))       (𝜑 → (𝑁‘[𝐹]( ≃ph𝐽)) = [𝐼]( ≃ph𝐽))
 
Theorempi1xfrf 22899* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝑃 = (𝐽 π1 (𝐹‘0))    &   𝑄 = (𝐽 π1 (𝐹‘1))    &   𝐵 = (Base‘𝑃)    &   𝐺 = ran (𝑔 𝐵 ↦ ⟨[𝑔]( ≃ph𝐽), [(𝐼(*𝑝𝐽)(𝑔(*𝑝𝐽)𝐹))]( ≃ph𝐽)⟩)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐼 ∈ (II Cn 𝐽))    &   (𝜑 → (𝐹‘1) = (𝐼‘0))    &   (𝜑 → (𝐼‘1) = (𝐹‘0))       (𝜑𝐺:𝐵⟶(Base‘𝑄))
 
Theorempi1xfrval 22900* The value of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
𝑃 = (𝐽 π1 (𝐹‘0))    &   𝑄 = (𝐽 π1 (𝐹‘1))    &   𝐵 = (Base‘𝑃)    &   𝐺 = ran (𝑔 𝐵 ↦ ⟨[𝑔]( ≃ph𝐽), [(𝐼(*𝑝𝐽)(𝑔(*𝑝𝐽)𝐹))]( ≃ph𝐽)⟩)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (II Cn 𝐽))    &   (𝜑𝐼 ∈ (II Cn 𝐽))    &   (𝜑 → (𝐹‘1) = (𝐼‘0))    &   (𝜑 → (𝐼‘1) = (𝐹‘0))    &   (𝜑𝐴 𝐵)       (𝜑 → (𝐺‘[𝐴]( ≃ph𝐽)) = [(𝐼(*𝑝𝐽)(𝐴(*𝑝𝐽)𝐹))]( ≃ph𝐽))
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