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Theorem List for Metamath Proof Explorer - 22801-22900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-omn 22801* 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 22802* 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 22803* 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 22804* 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 22805* 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 22806 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 22807 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 22808 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 22809 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 22810 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 22811 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 22812 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 22813* Lemma for pcohtpy 22814. (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 22814 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 22815 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 22816 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 22817 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 22818* 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 22819* 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 22820* Lemma for pcorev 22821. 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 22821* 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 22822* 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 22823* 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 22824* 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 22825* 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 22826 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 22827 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 22828 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 22829 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 22830 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 22831 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 22832 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 22833 Lemma for pi1buni 22834. (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 22834 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 22835 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 22836 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 22837 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 22838 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 22839* 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 22840 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 22841 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 22842 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 22843 Lemma for pi1grp 22844. (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 22844 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 22845 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 22846* 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 22847* 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 22848* 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 22849* 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 22850* 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 22851* 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 22852* 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 22853* 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 22854* 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 22855* 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  Metric subcomplex vector spaces
 
12.5.1  Subcomplex modules
 
Syntaxcclm 22856 Syntax for the class of subcomplex modules.
class ℂMod
 
Definitiondf-clm 22857* Define the class of subcomplex modules, which are left modules over a subring of the field of complex numbers fld, 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 18778), left modules over such subrings are the same as right modules, see rmodislmod 18925. 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 = (ℂflds 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
 
Theoremisclm 22858 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 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
 
Theoremclmsca 22859 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 = (ℂflds 𝐾))
 
Theoremclmsubrg 22860 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))
 
Theoremclmlmod 22861 A subcomplex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
 
Theoremclmgrp 22862 A subcomplex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ Grp)
 
Theoremclmabl 22863 A subcomplex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ Abel)
 
Theoremclmring 22864 The scalar ring of a subcomplex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 𝐹 ∈ Ring)
 
Theoremclmfgrp 22865 The scalar ring of a subcomplex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)
 
Theoremclm0 22866 The zero of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 0 = (0g𝐹))
 
Theoremclm1 22867 The identity of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 1 = (1r𝐹))
 
Theoremclmadd 22868 The addition of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → + = (+g𝐹))
 
Theoremclmmul 22869 The multiplication of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → · = (.r𝐹))
 
Theoremclmcj 22870 The conjugation of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → ∗ = (*𝑟𝐹))
 
Theoremisclmi 22871 Reverse direction of isclm 22858. (Contributed by Mario Carneiro, 30-Oct-2015.)
𝐹 = (Scalar‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)
 
Theoremclmzss 22872 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)
 
Theoremclmsscn 22873 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)
 
Theoremclmsub 22874 Subtraction in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝐴𝐾𝐵𝐾) → (𝐴𝐵) = (𝐴(-g𝐹)𝐵))
 
Theoremclmneg 22875 Negation in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝐴𝐾) → -𝐴 = ((invg𝐹)‘𝐴))
 
Theoremclmneg1 22876 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)
 
Theoremclmabs 22877 Norm in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝐴𝐾) → (abs‘𝐴) = ((norm‘𝐹)‘𝐴))
 
Theoremclmacl 22878 Closure of ring addition for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
 
Theoremclmmcl 22879 Closure of ring multiplication for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 · 𝑌) ∈ 𝐾)
 
Theoremclmsubcl 22880 Closure of ring subtraction for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋𝑌) ∈ 𝐾)
 
Theoremlmhmclm 22881 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))
 
Theoremclmvscl 22882 Closure of scalar product for a subcomplex module. Analogue of lmodvscl 18874. (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 22883 Associative law for scalar product. Analogue of lmodvsass 18882. (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 22884 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 22885 Distributive law for scalar product (right-distributivity). (lmodvsdir 18881 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 22886 Distributive law for scalar product (left-distributivity). (lmodvsdi 18880 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)))
 
Theoremclmvs1 22887 Scalar product with ring unit. (lmodvs1 18885 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (1 · 𝑋) = 𝑋)
 
Theoremclmvs2 22888 A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (Revised by AV, 21-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → (𝐴 + 𝐴) = (2 · 𝐴))
 
Theoremclm0vs 22889 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 18890 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (0 · 𝑋) = 0 )
 
Theoremclmopfne 22890 The (functionalized) operations of addition and multiplication by a scalar of a subcomplex module cannot be identical. (Contributed by NM, 31-May-2008.) (Revised by AV, 3-Oct-2021.)
· = ( ·sf𝑊)    &    + = (+𝑓𝑊)       (𝑊 ∈ ℂMod → +· )
 
Theoremisclmp 22891* The predicate "is a subcomplex module." (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.)
· = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝑊 ∈ ℂMod ↔ ((𝑊 ∈ Grp ∧ 𝑆 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))))
 
Theoremisclmi0 22892* Properties that determine a subcomplex module. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.)
· = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &   𝑆 = (ℂflds 𝐾)    &   𝑊 ∈ Grp    &   𝐾 ∈ (SubRing‘ℂfld)    &   (𝑥𝑉 → (1 · 𝑥) = 𝑥)    &   ((𝑦𝐾𝑥𝑉) → (𝑦 · 𝑥) ∈ 𝑉)    &   ((𝑦𝐾𝑥𝑉𝑧𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))    &   ((𝑦𝐾𝑧𝐾𝑥𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)))    &   ((𝑦𝐾𝑧𝐾𝑥𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))       𝑊 ∈ ℂMod
 
Theoremclmvneg1 22893 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 18900 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (invg𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (-1 · 𝑋) = (𝑁𝑋))
 
Theoremclmvsneg 22894 Multiplication of a vector by a negated scalar. (lmodvsneg 18901 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (invg𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂMod)    &   (𝜑𝑋𝐵)    &   (𝜑𝑅𝐾)       (𝜑 → (𝑁‘(𝑅 · 𝑋)) = (-𝑅 · 𝑋))
 
Theoremclmmulg 22895 The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (.g𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ ℤ ∧ 𝐵𝑉) → (𝐴 𝐵) = (𝐴 · 𝐵))
 
Theoremclmsubdir 22896 Scalar multiplication distributive law for subtraction. (lmodsubdir 18915 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (-g𝑊)    &   (𝜑𝑊 ∈ ℂMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐴𝐵) · 𝑋) = ((𝐴 · 𝑋) (𝐵 · 𝑋)))
 
Theoremclmpm1dir 22897 Subtractive distributive law for the scalar product of a subcomplex module. (Contributed by NM, 31-Jul-2007.) (Revised by AV, 21-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝐾 = (Base‘(Scalar‘𝑊))       ((𝑊 ∈ ℂMod ∧ (𝐴𝐾𝐵𝐾𝐶𝑉)) → ((𝐴𝐵) · 𝐶) = ((𝐴 · 𝐶) + (-1 · (𝐵 · 𝐶))))
 
Theoremclmnegneg 22898 Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 21-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → (-1 · (-1 · 𝐴)) = 𝐴)
 
Theoremclmnegsubdi2 22899 Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 29-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉𝐵𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴)))
 
Theoremclmsub4 22900 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 5-Aug-2007.) (Revised by AV, 29-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐴 + 𝐵) + (-1 · (𝐶 + 𝐷))) = ((𝐴 + (-1 · 𝐶)) + (𝐵 + (-1 · 𝐷))))
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