P. 251 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25001-25100   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-pin 25001*	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 25002*	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.) (Proof shortened by AV, 2-Mar-2024.)
⊢ (*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 25003*	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 25004	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 25005	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 25006	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 25007	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 25008	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 25009	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 25010	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 25011*	Lemma for pcohtpy 25012. (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 25012	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 25013	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 25014	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 25015	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 25016*	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 25017*	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 25018*	Lemma for pcorev 25019. 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 25019*	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 25020*	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 25021*	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 25022*	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 25023*	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 25024	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 25025	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 25026	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 25027	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 25028	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 25029	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 25030	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 25031	Lemma for pi1buni 25032. (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 25032	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 25033	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 25034	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 25035	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 25036	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 25037*	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 25038	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 25039	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 25040	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 25041	Lemma for pi1grp 25042. (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 25042	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 25043	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 25044*	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 25045*	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 25046*	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 25047*	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 25048*	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 25049*	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 25050*	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 25051*	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 25052*	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 25053*	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 25054	Syntax for the class of subcomplex modules.
class ℂMod
Definition	df-clm 25055*	Define the class of subcomplex modules, which are left modules over a subring of the field of complex numbers ℂfld, which allows 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 20554), left modules over such subrings are the same as right modules, see rmodislmod 20927. 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 25056	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 25057	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 25058	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 25059	A subcomplex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
Theorem	clmgrp 25060	A subcomplex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Grp)
Theorem	clmabl 25061	A subcomplex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Abel)
Theorem	clmring 25062	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 25063	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 25064	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 25065	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 25066	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 25067	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 25068	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 25069	Reverse direction of isclm 25056. (Contributed by Mario Carneiro, 30-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)
Theorem	clmzss 25070	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 25071	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 25072	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 25073	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 25074	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 25075	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 25076	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 25077	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 25078	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 25079	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 25080	Closure of scalar product for a subcomplex module. Analogue of lmodvscl 20875. (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 25081	Associative law for scalar product. Analogue of lmodvsass 20884. (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)))
Theorem	clmvscom 25082	Commutative law for the scalar product. (Contributed by NM, 14-Feb-2008.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝑄 · (𝑅 · 𝑋)) = (𝑅 · (𝑄 · 𝑋)))
Theorem	clmvsdir 25083	Distributive law for scalar product (right-distributivity). (lmodvsdir 20883 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 + 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
Theorem	clmvsdi 25084	Distributive law for scalar product (left-distributivity). (lmodvsdi 20882 analog.) (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌)))
Theorem	clmvs1 25085	Scalar product with ring unity. (lmodvs1 20887 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (1 · 𝑋) = 𝑋)
Theorem	clmvs2 25086	A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (Revised by AV, 21-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + 𝐴) = (2 · 𝐴))
Theorem	clm0vs 25087	Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 20892 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (0 · 𝑋) = 0 )
Theorem	clmopfne 25088	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.)
⊢ · = ( ·sf ‘𝑊)    &   ⊢ + = (+𝑓‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → + ≠ · )
Theorem	isclmp 25089*	The predicate "is a subcomplex module". (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑊 ∈ ℂMod ↔ ((𝑊 ∈ Grp ∧ 𝑆 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥 ∈ 𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))))
Theorem	isclmi0 25090*	Properties that determine a subcomplex module. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑆 = (ℂfld ↾s 𝐾)    &   ⊢ 𝑊 ∈ Grp    &   ⊢ 𝐾 ∈ (SubRing‘ℂfld)    &   ⊢ (𝑥 ∈ 𝑉 → (1 · 𝑥) = 𝑥)    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑦 · 𝑥) ∈ 𝑉)    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)))    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))    ⇒   ⊢ 𝑊 ∈ ℂMod
Theorem	clmvneg1 25091	Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 20902 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (invg‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (-1 · 𝑋) = (𝑁‘𝑋))
Theorem	clmvsneg 25092	Multiplication of a vector by a negated scalar. (lmodvsneg 20903 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (invg‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ ℂMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝑁‘(𝑅 · 𝑋)) = (-𝑅 · 𝑋))
Theorem	clmmulg 25093	The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ∙ = (.g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ 𝑉) → (𝐴 ∙ 𝐵) = (𝐴 · 𝐵))
Theorem	clmsubdir 25094	Scalar multiplication distributive law for subtraction. (lmodsubdir 20917 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ − = (-g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ ℂMod)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) · 𝑋) = ((𝐴 · 𝑋) − (𝐵 · 𝑋)))
Theorem	clmpm1dir 25095	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)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐾 = (Base‘(Scalar‘𝑊))    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (-1 · (𝐵 · 𝐶))))
Theorem	clmnegneg 25096	Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 21-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (-1 · (-1 · 𝐴)) = 𝐴)
Theorem	clmnegsubdi2 25097	Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 29-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴)))
Theorem	clmsub4 25098	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)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 + 𝐵) + (-1 · (𝐶 + 𝐷))) = ((𝐴 + (-1 · 𝐶)) + (𝐵 + (-1 · 𝐷))))
Theorem	clmvsrinv 25099	A vector minus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + (-1 · 𝐴)) = 0 )
Theorem	clmvslinv 25100	Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ((-1 · 𝐴) + 𝐴) = 0 )