Home Metamath Proof ExplorerTheorem List (p. 229 of 429) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27903) Hilbert Space Explorer (27904-29428) Users' Mathboxes (29429-42879)

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))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
 Copyright terms: Public domain < Previous  Next >