P. 233 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	phtpyco2 23201	Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺))    ⇒   ⊢ (𝜑 → (𝑃 ∘ 𝐻) ∈ ((𝑃 ∘ 𝐹)(PHtpy‘𝐾)(𝑃 ∘ 𝐺)))
Theorem	phtpycc 23202*	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))))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐾 ∈ (𝐹(PHtpy‘𝐽)𝐺))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐺(PHtpy‘𝐽)𝐻))    ⇒   ⊢ (𝜑 → 𝑀 ∈ (𝐹(PHtpy‘𝐽)𝐻))
Definition	df-phtpc 23203*	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) ≠ ∅)})
Theorem	phtpcrel 23204	The path homotopy relation is a relation. (Contributed by Mario Carneiro, 7-Jun-2014.) (Revised by Mario Carneiro, 7-Aug-2014.)
⊢ Rel ( ≃ph‘𝐽)
Theorem	isphtpc 23205	The relation "is path homotopic to". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Sep-2015.)
⊢ (𝐹( ≃ph‘𝐽)𝐺 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐺 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐺) ≠ ∅))
Theorem	phtpcer 23206	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.)
⊢ ( ≃ph‘𝐽) Er (II Cn 𝐽)
Theorem	phtpc01 23207	Path homotopic paths have the same endpoints. (Contributed by Mario Carneiro, 24-Feb-2015.)
⊢ (𝐹( ≃ph‘𝐽)𝐺 → ((𝐹‘0) = (𝐺‘0) ∧ (𝐹‘1) = (𝐺‘1)))
Theorem	reparphti 23208*	Lemma for reparpht 23209. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn II))    &   ⊢ (𝜑 → (𝐺‘0) = 0)    &   ⊢ (𝜑 → (𝐺‘1) = 1)    &   ⊢ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))))    ⇒   ⊢ (𝜑 → 𝐻 ∈ ((𝐹 ∘ 𝐺)(PHtpy‘𝐽)𝐹))
Theorem	reparpht 23209	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.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn II))    &   ⊢ (𝜑 → (𝐺‘0) = 0)    &   ⊢ (𝜑 → (𝐺‘1) = 1)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺)( ≃ph‘𝐽)𝐹)
Theorem	phtpcco2 23210	Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 6-Jul-2015.)
⊢ (𝜑 → 𝐹( ≃ph‘𝐽)𝐺)    &   ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑃 ∘ 𝐹)( ≃ph‘𝐾)(𝑃 ∘ 𝐺))
12.4.13  The fundamental group
Syntax	cpco 23211	Extend class notation with the concatenation operation for paths in a topological space.
class *𝑝
Syntax	comi 23212	Extend class notation with the loop space.
class Ω1
Syntax	comn 23213	Extend class notation with the higher loop spaces.
class Ω𝑛
Syntax	cpi1 23214	Extend class notation with the fundamental group.
class π1
Syntax	cpin 23215	Extend class notation with the higher homotopy groups.
class πn
Definition	df-pco 23216*	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))))))
Definition	df-om1 23217*	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), (*𝑝‘𝑗)⟩, ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩})
Definition	df-omn 23218*	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‘(1st ‘𝑥)) Ω1 (2nd ‘𝑥)), ((0[,]1) × {(2nd ‘𝑥)})⟩) ∘ 1st ), ⟨{⟨(Base‘ndx), ∪ 𝑗⟩, ⟨(TopSet‘ndx), 𝑗⟩}, 𝑦⟩))
Definition	df-pi1 23219*	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 ( ≃ph‘𝑗)))
Definition	df-pin 23220*	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)))))))))
Theorem	pcofval 23221*	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.)
⊢ (*𝑝‘𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1)))))
Theorem	pcoval 23222*	The concatenation of two paths. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    ⇒   ⊢ (𝜑 → (𝐹(*𝑝‘𝐽)𝐺) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐺‘((2 · 𝑥) − 1)))))
Theorem	pcovalg 23223	Evaluate the concatenation of two paths. (Contributed by Mario Carneiro, 7-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,]1)) → ((𝐹(*𝑝‘𝐽)𝐺)‘𝑋) = if(𝑋 ≤ (1 / 2), (𝐹‘(2 · 𝑋)), (𝐺‘((2 · 𝑋) − 1))))
Theorem	pcoval1 23224	Evaluate the concatenation of two paths on the first half. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (0[,](1 / 2))) → ((𝐹(*𝑝‘𝐽)𝐺)‘𝑋) = (𝐹‘(2 · 𝑋)))
Theorem	pco0 23225	The starting point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    ⇒   ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘0) = (𝐹‘0))
Theorem	pco1 23226	The ending point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    ⇒   ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)‘1) = (𝐺‘1))
Theorem	pcoval2 23227	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.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘1) = (𝐺‘0))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ ((1 / 2)[,]1)) → ((𝐹(*𝑝‘𝐽)𝐺)‘𝑋) = (𝐺‘((2 · 𝑋) − 1)))
Theorem	pcocn 23228	The concatenation of two paths is a path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘1) = (𝐺‘0))    ⇒   ⊢ (𝜑 → (𝐹(*𝑝‘𝐽)𝐺) ∈ (II Cn 𝐽))
Theorem	copco 23229	The composition of a concatenation of paths with a continuous function. (Contributed by Mario Carneiro, 9-Jul-2015.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘1) = (𝐺‘0))    &   ⊢ (𝜑 → 𝐻 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝐻 ∘ (𝐹(*𝑝‘𝐽)𝐺)) = ((𝐻 ∘ 𝐹)(*𝑝‘𝐾)(𝐻 ∘ 𝐺)))
Theorem	pcohtpylem 23230*	Lemma for pcohtpy 23231. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → (𝐹‘1) = (𝐺‘0))    &   ⊢ (𝜑 → 𝐹( ≃ph‘𝐽)𝐻)    &   ⊢ (𝜑 → 𝐺( ≃ph‘𝐽)𝐾)    &   ⊢ 𝑃 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)))    &   ⊢ (𝜑 → 𝑀 ∈ (𝐹(PHtpy‘𝐽)𝐻))    &   ⊢ (𝜑 → 𝑁 ∈ (𝐺(PHtpy‘𝐽)𝐾))    ⇒   ⊢ (𝜑 → 𝑃 ∈ ((𝐹(*𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(*𝑝‘𝐽)𝐾)))
Theorem	pcohtpy 23231	Homotopy invariance of path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)
⊢ (𝜑 → (𝐹‘1) = (𝐺‘0))    &   ⊢ (𝜑 → 𝐹( ≃ph‘𝐽)𝐻)    &   ⊢ (𝜑 → 𝐺( ≃ph‘𝐽)𝐾)    ⇒   ⊢ (𝜑 → (𝐹(*𝑝‘𝐽)𝐺)( ≃ph‘𝐽)(𝐻(*𝑝‘𝐽)𝐾))
Theorem	pcoptcl 23232	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) ∧ 𝑌 ∈ 𝑋) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
Theorem	pcopt 23233	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(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)𝐹)
Theorem	pcopt2 23234	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(*𝑝‘𝐽)𝑃)( ≃ph‘𝐽)𝐹)
Theorem	pcoass 23235*	Order of concatenation does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐻 ∈ (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(*𝑝‘𝐽)𝐺)(*𝑝‘𝐽)𝐻)( ≃ph‘𝐽)(𝐹(*𝑝‘𝐽)(𝐺(*𝑝‘𝐽)𝐻)))
Theorem	pcorevcl 23236*	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)))
Theorem	pcorevlem 23237*	Lemma for pcorev 23238. 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(*𝑝‘𝐽)𝐹)(PHtpy‘𝐽)𝑃) ∧ (𝐺(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)𝑃))
Theorem	pcorev 23238*	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(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)𝑃)
Theorem	pcorev2 23239*	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(*𝑝‘𝐽)𝐺)( ≃ph‘𝐽)𝑃)
Theorem	pcophtb 23240*	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 𝐹 − 𝐺 formed by concatenating 𝐹 with the inverse of 𝐺. Thus, all the homotopy information in ≃ph‘𝐽 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)})    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0))    &   ⊢ (𝜑 → (𝐹‘1) = (𝐺‘1))    ⇒   ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐻)( ≃ph‘𝐽)𝑃 ↔ 𝐹( ≃ph‘𝐽)𝐺))
Theorem	om1val 23241*	The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})    &   ⊢ (𝜑 → + = (*𝑝‘𝐽))    &   ⊢ (𝜑 → 𝐾 = (𝐽 ^ko II))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
Theorem	om1bas 23242*	The base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑂))    ⇒   ⊢ (𝜑 → 𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
Theorem	om1elbas 23243	Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑂))    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝐵 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)))
Theorem	om1addcl 23244	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)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑂))    &   ⊢ (𝜑 → 𝐻 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻(*𝑝‘𝐽)𝐾) ∈ 𝐵)
Theorem	om1plusg 23245	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)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (*𝑝‘𝐽) = (+g‘𝑂))
Theorem	om1tset 23246	The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐽 ^ko II) = (TopSet‘𝑂))
Theorem	om1opn 23247	The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ 𝐾 = (TopOpen‘𝑂)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑂))    ⇒   ⊢ (𝜑 → 𝐾 = ((𝐽 ^ko II) ↾t 𝐵))
Theorem	pi1val 23248	The definition of the fundamental group. (Contributed by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ 𝑂 = (𝐽 Ω1 𝑌)    ⇒   ⊢ (𝜑 → 𝐺 = (𝑂 /s ( ≃ph‘𝐽)))
Theorem	pi1bas 23249	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)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → 𝐾 = (Base‘𝑂))    ⇒   ⊢ (𝜑 → 𝐵 = (𝐾 / ( ≃ph‘𝐽)))
Theorem	pi1blem 23250	Lemma for pi1buni 23251. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → 𝐾 = (Base‘𝑂))    ⇒   ⊢ (𝜑 → ((( ≃ph‘𝐽) “ 𝐾) ⊆ 𝐾 ∧ 𝐾 ⊆ (II Cn 𝐽)))
Theorem	pi1buni 23251	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)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → 𝐾 = (Base‘𝑂))    ⇒   ⊢ (𝜑 → ∪ 𝐵 = 𝐾)
Theorem	pi1bas2 23252	The base set of the fundamental group, written self-referentially. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    ⇒   ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / ( ≃ph‘𝐽)))
Theorem	pi1eluni 23253	Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    ⇒   ⊢ (𝜑 → (𝐹 ∈ ∪ 𝐵 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)))
Theorem	pi1bas3 23254	The base set of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ 𝑅 = (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))    ⇒   ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / 𝑅))
Theorem	pi1cpbl 23255	The group operation, loop concatenation, is compatible with homotopy equivalence. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ 𝑅 = (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))    &   ⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ + = (+g‘𝑂)    ⇒   ⊢ (𝜑 → ((𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄)))
Theorem	elpi1 23256*	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)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝐵 ↔ ∃𝑓 ∈ (II Cn 𝐽)(((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ 𝐹 = [𝑓]( ≃ph‘𝐽))))
Theorem	elpi1i 23257	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)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘0) = 𝑌)    &   ⊢ (𝜑 → (𝐹‘1) = 𝑌)    ⇒   ⊢ (𝜑 → [𝐹]( ≃ph‘𝐽) ∈ 𝐵)
Theorem	pi1addf 23258	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)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐵)
Theorem	pi1addval 23259	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)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ∪ 𝐵)    ⇒   ⊢ (𝜑 → ([𝑀]( ≃ph‘𝐽) + [𝑁]( ≃ph‘𝐽)) = [(𝑀(*𝑝‘𝐽)𝑁)]( ≃ph‘𝐽))
Theorem	pi1grplem 23260	Lemma for pi1grp 23261. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Aug-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ 0 = ((0[,]1) × {𝑌})    ⇒   ⊢ (𝜑 → (𝐺 ∈ Grp ∧ [ 0 ]( ≃ph‘𝐽) = (0g‘𝐺)))
Theorem	pi1grp 23261	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) ∧ 𝑌 ∈ 𝑋) → 𝐺 ∈ Grp)
Theorem	pi1id 23262	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) ∧ 𝑌 ∈ 𝑋) → [ 0 ]( ≃ph‘𝐽) = (0g‘𝐺))
Theorem	pi1inv 23263*	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 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘0) = 𝑌)    &   ⊢ (𝜑 → (𝐹‘1) = 𝑌)    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    ⇒   ⊢ (𝜑 → (𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽))
Theorem	pi1xfrf 23264*	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]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘1) = (𝐼‘0))    &   ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))    ⇒   ⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄))
Theorem	pi1xfrval 23265*	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]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘1) = (𝐼‘0))    &   ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))    &   ⊢ (𝜑 → 𝐴 ∈ ∪ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘[𝐴]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝐴(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))
Theorem	pi1xfr 23266*	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]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))
Theorem	pi1xfrcnvlem 23267*	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]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    &   ⊢ 𝐻 = ran (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)    ⇒   ⊢ (𝜑 → ◡𝐺 ⊆ 𝐻)
Theorem	pi1xfrcnv 23268*	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]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    &   ⊢ 𝐻 = ran (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)    ⇒   ⊢ (𝜑 → (◡𝐺 = 𝐻 ∧ ◡𝐺 ∈ (𝑄 GrpHom 𝑃)))
Theorem	pi1xfrgim 23269*	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]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpIso 𝑄))
Theorem	pi1cof 23270*	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]( ≃ph‘𝐽), [(𝐹 ∘ 𝑔)]( ≃ph‘𝐾)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐵)    ⇒   ⊢ (𝜑 → 𝐺:𝑉⟶(Base‘𝑄))
Theorem	pi1coval 23271*	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]( ≃ph‘𝐽), [(𝐹 ∘ 𝑔)]( ≃ph‘𝐾)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝑇 ∈ ∪ 𝑉) → (𝐺‘[𝑇]( ≃ph‘𝐽)) = [(𝐹 ∘ 𝑇)]( ≃ph‘𝐾))
Theorem	pi1coghm 23272*	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]( ≃ph‘𝐽), [(𝐹 ∘ 𝑔)]( ≃ph‘𝐾)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐵)    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))
12.5  Metric subcomplex vector spaces
12.5.1  Subcomplex modules
Syntax	cclm 23273	Syntax for the class of subcomplex modules.
class ℂMod
Definition	df-clm 23274*	Define the class of subcomplex modules, which are left modules over a subring of the field of complex numbers ℂfld, which allows us to use the complex addition, multiplication, etc. in theorems about subcomplex modules. Since the field of complex numbers is commutative and so are its subrings (see subrgcrng 19180), left modules over such subrings are the same as right modules, see rmodislmod 19327. Therefore, we drop the word "left" from "subcomplex left module". (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
Theorem	isclm 23275	A subcomplex module is a left module over a subring of the field of complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
Theorem	clmsca 23276	The ring of scalars 𝐹 of a subcomplex module is the restriction of the field of complex numbers to the base set of 𝐹. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾))
Theorem	clmsubrg 23277	The base set of the ring of scalars of a subcomplex module is the base set of a subring of the field of complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld))
Theorem	clmlmod 23278	A subcomplex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
Theorem	clmgrp 23279	A subcomplex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Grp)
Theorem	clmabl 23280	A subcomplex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Abel)
Theorem	clmring 23281	The scalar ring of a subcomplex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Ring)
Theorem	clmfgrp 23282	The scalar ring of a subcomplex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)
Theorem	clm0 23283	The zero of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → 0 = (0g‘𝐹))
Theorem	clm1 23284	The identity of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → 1 = (1r‘𝐹))
Theorem	clmadd 23285	The addition of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → + = (+g‘𝐹))
Theorem	clmmul 23286	The multiplication of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → · = (.r‘𝐹))
Theorem	clmcj 23287	The conjugation of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → ∗ = (*𝑟‘𝐹))
Theorem	isclmi 23288	Reverse direction of isclm 23275. (Contributed by Mario Carneiro, 30-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)
Theorem	clmzss 23289	The scalar ring of a subcomplex module contains the integers. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → ℤ ⊆ 𝐾)
Theorem	clmsscn 23290	The scalar ring of a subcomplex 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)
Theorem	clmsub 23291	Subtraction in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐴 − 𝐵) = (𝐴(-g‘𝐹)𝐵))
Theorem	clmneg 23292	Negation in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → -𝐴 = ((invg‘𝐹)‘𝐴))
Theorem	clmneg1 23293	Minus one is in the scalar ring of a subcomplex module. (Contributed by AV, 28-Sep-2021.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → -1 ∈ 𝐾)
Theorem	clmabs 23294	Norm in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → (abs‘𝐴) = ((norm‘𝐹)‘𝐴))
Theorem	clmacl 23295	Closure of ring addition for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
Theorem	clmmcl 23296	Closure of ring multiplication for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 · 𝑌) ∈ 𝐾)
Theorem	clmsubcl 23297	Closure of ring subtraction for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 − 𝑌) ∈ 𝐾)
Theorem	lmhmclm 23298	The domain of a linear operator is a subcomplex module iff the range is. (Contributed by Mario Carneiro, 21-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ ℂMod ↔ 𝑇 ∈ ℂMod))
Theorem	clmvscl 23299	Closure of scalar product for a subcomplex module. Analogue of lmodvscl 19276. (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉)
Theorem	clmvsass 23300	Associative law for scalar product. Analogue of lmodvsass 19284. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 · 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))