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Theorem List for Metamath Proof Explorer - 22601-22700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisphtpc 22601 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 22602 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 22603 Obsolete proof of phtpcer 22602 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 22604 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 22605* Lemma for reparpht 22606. (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 22606 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 22607 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 22608 Extend class notation with the concatenation operation for paths in a topological space.
class *𝑝

Syntaxcomi 22609 Extend class notation with the loop space.
class Ω1

Syntaxcomn 22610 Extend class notation with the higher loop spaces.
class Ω𝑛

Syntaxcpi1 22611 Extend class notation with the fundamental group.
class π1

Syntaxcpin 22612 Extend class notation with the higher homotopy groups.
class πn

Definitiondf-pco 22613* 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 22614* 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 22615* 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 22616* 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 22617* 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 22618* 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 22619* 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 22620 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 22621 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 22622 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 22623 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 22624 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 22625 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 22626 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 22627* Lemma for pcohtpy 22628. (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 22628 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 22629 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 22630 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 22631 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 22632* 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 22633* 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 22634* Lemma for pcorev 22635. 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 22635* 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 22636* 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 22637* 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 22638* 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 22639* 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 22640 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 22641 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 22642 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 22643 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 22644 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 22645 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 22646 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 22647 Lemma for pi1buni 22648. (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 22648 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 22649 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 22650 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 22651 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 22652 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 22653* 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 22654 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 22655 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 22656 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 22657 Lemma for pi1grp 22658. (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 22658 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 22659 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 22660* 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 22661* 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 22662* 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 22663* 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 22664* 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 22665* 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 22666* 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 22667* 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 22668* 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 22669* 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 22670 Complex module.
class ℂMod

Definitiondf-clm 22671* 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 22672 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 22673 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 22674 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 22675 A complex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ LMod)

Theoremclmgrp 22676 A complex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ Grp)

Theoremclmabl 22677 A complex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ Abel)

Theoremclmring 22678 The scalar ring of a complex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 𝐹 ∈ Ring)

Theoremclmfgrp 22679 The scalar ring of a complex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)

Theoremclm0 22680 The zero of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 0 = (0g𝐹))

Theoremclm1 22681 The identity of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 1 = (1r𝐹))

Theoremclmadd 22682 The addition of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → + = (+g𝐹))

Theoremclmmul 22683 The multiplication of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → · = (.r𝐹))

Theoremclmcj 22684 The conjugation of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → ∗ = (*𝑟𝐹))

Theoremisclmi 22685 Reverse direction of isclm 22672. (Contributed by Mario Carneiro, 30-Oct-2015.)
𝐹 = (Scalar‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)

Theoremclmzss 22686 The scalar ring of a complex module contains the integers. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂMod → ℤ ⊆ 𝐾)

Theoremclmsscn 22687 The scalar ring of a complex module is a subset of the complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)

Theoremclmsub 22688 Subtraction in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝐴𝐾𝐵𝐾) → (𝐴𝐵) = (𝐴(-g𝐹)𝐵))

Theoremclmneg 22689 Negation in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝐴𝐾) → -𝐴 = ((invg𝐹)‘𝐴))

Theoremclmneg1 22690 Minus one is in the scalar ring of a complex module. (Contributed by AV, 28-Sep-2021.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂMod → -1 ∈ 𝐾)

Theoremclmabs 22691 Norm in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝐴𝐾) → (abs‘𝐴) = ((norm‘𝐹)‘𝐴))

Theoremclmacl 22692 Closure of ring addition for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)

Theoremclmmcl 22693 Closure of ring multiplication for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 · 𝑌) ∈ 𝐾)

Theoremclmsubcl 22694 Closure of ring subtraction for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋𝑌) ∈ 𝐾)

Theoremlmhmclm 22695 The domain of a linear operator is a complex module iff the range is. (Contributed by Mario Carneiro, 21-Oct-2015.)
(𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ ℂMod ↔ 𝑇 ∈ ℂMod))

Theoremclmvscl 22696 Closure of scalar product for a complex module. (lmodvscl 18703 analog.) (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑄𝐾𝑋𝑉) → (𝑄 · 𝑋) ∈ 𝑉)

Theoremclmvsass 22697 Associative law for scalar product. (lmodvsass 18711 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 · 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))

Theoremclmvscom 22698 Commutative law for the scalar product. (Contributed by NM, 14-Feb-2008.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → (𝑄 · (𝑅 · 𝑋)) = (𝑅 · (𝑄 · 𝑋)))

Theoremclmvsdir 22699 Distributive law for scalar product (right-distributivity). (lmodvsdir 18710 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 + 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))

Theoremclmvsdi 22700 Distributive law for scalar product (left-distributivity). (lmodvsdi 18709 analog.) (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ (𝐴𝐾𝑋𝑉𝑌𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌)))

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