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Theorem List for Metamath Proof Explorer - 22501-22600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-phtpy 22501* Define the class of path homotopies between two paths 𝐹, 𝐺:II⟶𝑋; these are homotopies (in the sense of df-htpy 22500) 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 22502* 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 22503 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 22504 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 22505* 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 22506* 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 22507* 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 22508* 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 22509 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 22510* 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 22511* 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 22512 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 22513 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 22514 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 22515 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 22516* 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 22517* 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 22518* 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 22519* 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 22520 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 22521* 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 22522* 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 22523 The path homotopy relation is a relation. (Contributed by Mario Carneiro, 7-Jun-2014.) (Revised by Mario Carneiro, 7-Aug-2014.)
Rel ( ≃ph𝐽)

Theoremisphtpc 22524 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 22525 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)

TheoremphtpcerOLD 22526 Obsolete proof of phtpcer 22525 as of 1-May-2021. 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 modification is discouraged.) (New usage is discouraged.)
( ≃ph𝐽) Er (II Cn 𝐽)

Theoremphtpc01 22527 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 22528* Lemma for reparpht 22529. (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 22529 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 22530 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 22531 Extend class notation with the concatenation operation for paths in a topological space.
class *𝑝

Syntaxcomi 22532 Extend class notation with the loop space.
class Ω1

Syntaxcomn 22533 Extend class notation with the higher loop spaces.
class Ω𝑛

Syntaxcpi1 22534 Extend class notation with the fundamental group.
class π1

Syntaxcpin 22535 Extend class notation with the higher homotopy groups.
class πn

Definitiondf-pco 22536* 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 22537* 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 22538* 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 22539* 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 22540* 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 22541* 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 22542* 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 22543 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 22544 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 22545 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 22546 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 22547 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 22548 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 22549 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 22550* Lemma for pcohtpy 22551. (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 22551 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 22552 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 22553 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 22554 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 22555* 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 22556* 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 22557* Lemma for pcorev 22558. 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 22558* 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 22559* 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 22560* 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 22561* 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 22562* 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 22563 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 22564 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 22565 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 22566 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 22567 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 22568 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 22569 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 22570 Lemma for pi1buni 22571. (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 22571 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 22572 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 22573 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 22574 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 22575 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 22576* 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 22577 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 22578 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 22579 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 22580 Lemma for pi1grp 22581. (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 22581 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 22582 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 22583* 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 22584* 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 22585* 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))

Theorempi1xfr 22586* Given a path 𝐹 and its inverse 𝐼 between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.)
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))    &   𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))       (𝜑𝐺 ∈ (𝑃 GrpHom 𝑄))

Theorempi1xfrcnvlem 22587* Given a path 𝐹 between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened 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))    &   𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    &   𝐻 = ran ( (Base‘𝑄) ↦ ⟨[]( ≃ph𝐽), [(𝐹(*𝑝𝐽)((*𝑝𝐽)𝐼))]( ≃ph𝐽)⟩)       (𝜑𝐺𝐻)

Theorempi1xfrcnv 22588* Given a path 𝐹 between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened 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))    &   𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    &   𝐻 = ran ( (Base‘𝑄) ↦ ⟨[]( ≃ph𝐽), [(𝐹(*𝑝𝐽)((*𝑝𝐽)𝐼))]( ≃ph𝐽)⟩)       (𝜑 → (𝐺 = 𝐻𝐺 ∈ (𝑄 GrpHom 𝑃)))

Theorempi1xfrgim 22589* The mapping 𝐺 between fundamental groups is an isomorphism. (Contributed by Mario Carneiro, 12-Feb-2015.)
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))    &   𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))       (𝜑𝐺 ∈ (𝑃 GrpIso 𝑄))

Theorempi1cof 22590* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝑃 = (𝐽 π1 𝐴)    &   𝑄 = (𝐾 π1 𝐵)    &   𝑉 = (Base‘𝑃)    &   𝐺 = ran (𝑔 𝑉 ↦ ⟨[𝑔]( ≃ph𝐽), [(𝐹𝑔)]( ≃ph𝐾)⟩)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝐹𝐴) = 𝐵)       (𝜑𝐺:𝑉⟶(Base‘𝑄))

Theorempi1coval 22591* The value of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 10-Aug-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
𝑃 = (𝐽 π1 𝐴)    &   𝑄 = (𝐾 π1 𝐵)    &   𝑉 = (Base‘𝑃)    &   𝐺 = ran (𝑔 𝑉 ↦ ⟨[𝑔]( ≃ph𝐽), [(𝐹𝑔)]( ≃ph𝐾)⟩)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝐹𝐴) = 𝐵)       ((𝜑𝑇 𝑉) → (𝐺‘[𝑇]( ≃ph𝐽)) = [(𝐹𝑇)]( ≃ph𝐾))

Theorempi1coghm 22592* The mapping 𝐺 between fundamental groups is a group homomorphism. (Contributed by Mario Carneiro, 10-Aug-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
𝑃 = (𝐽 π1 𝐴)    &   𝑄 = (𝐾 π1 𝐵)    &   𝑉 = (Base‘𝑃)    &   𝐺 = ran (𝑔 𝑉 ↦ ⟨[𝑔]( ≃ph𝐽), [(𝐹𝑔)]( ≃ph𝐾)⟩)    &   (𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝐹𝐴) = 𝐵)       (𝜑𝐺 ∈ (𝑃 GrpHom 𝑄))

12.5  Complex metric vector spaces

12.5.1  Complex left modules

Syntaxcclm 22593 Complex module.
class ℂMod

Definitiondf-clm 22594* Define a complex module, which is just a left module over a subring of , which allows us to use conventional addition, multiplication, etc. in the left module theorems. (Contributed by Mario Carneiro, 16-Oct-2015.)
ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}

Theoremisclm 22595 A complex module is a left module over a subring of the complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))

Theoremclmsca 22596 A complex module is a left module over a subring of the complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂMod → 𝐹 = (ℂflds 𝐾))

Theoremclmsubrg 22597 A complex module is a left module over a subring of the complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld))

Theoremclmlmod 22598 A complex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ LMod)

Theoremclmgrp 22599 A complex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ Grp)

Theoremclmabl 22600 A complex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ Abel)

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