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Theorem List for Metamath Proof Explorer - 22901-23000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempi1xfr 22901* 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 22902* 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 22903* 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 22904* 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 22905* 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 22906* 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 22907* 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 22908 Syntax for the class of subcomplex modules.
class ℂMod

Definitiondf-clm 22909* 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 18832), left modules over such subrings are the same as right modules, see rmodislmod 18979. 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 22910 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 22911 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 22912 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 22913 A subcomplex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ LMod)

Theoremclmgrp 22914 A subcomplex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ Grp)

Theoremclmabl 22915 A subcomplex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ Abel)

Theoremclmring 22916 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 22917 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 22918 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 22919 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 22920 The addition of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → + = (+g𝐹))

Theoremclmmul 22921 The multiplication of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → · = (.r𝐹))

Theoremclmcj 22922 The conjugation of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → ∗ = (*𝑟𝐹))

Theoremisclmi 22923 Reverse direction of isclm 22910. (Contributed by Mario Carneiro, 30-Oct-2015.)
𝐹 = (Scalar‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)

Theoremclmzss 22924 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 22925 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 22926 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 22927 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 22928 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 22929 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 22930 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 22931 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 22932 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 22933 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 22934 Closure of scalar product for a subcomplex module. Analogue of lmodvscl 18928. (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 22935 Associative law for scalar product. Analogue of lmodvsass 18936. (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 22936 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 22937 Distributive law for scalar product (right-distributivity). (lmodvsdir 18935 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 22938 Distributive law for scalar product (left-distributivity). (lmodvsdi 18934 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 22939 Scalar product with ring unit. (lmodvs1 18939 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (1 · 𝑋) = 𝑋)

Theoremclmvs2 22940 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 22941 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 18944 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (0 · 𝑋) = 0 )

Theoremclmopfne 22942 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 22943* 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 22944* 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 22945 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 18954 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (invg𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (-1 · 𝑋) = (𝑁𝑋))

Theoremclmvsneg 22946 Multiplication of a vector by a negated scalar. (lmodvsneg 18955 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 22947 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 22948 Scalar multiplication distributive law for subtraction. (lmodsubdir 18969 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 22949 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 22950 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 22951 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 22952 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))))

Theoremclmvsrinv 22953 A vector minus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → (𝐴 + (-1 · 𝐴)) = 0 )

Theoremclmvslinv 22954 Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → ((-1 · 𝐴) + 𝐴) = 0 )

Theoremclmvsubval 22955 Value of vector subtraction in terms of addition in a subcomplex module. Analogue of lmodvsubval2 18966. (Contributed by NM, 31-Mar-2014.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉𝐵𝑉) → (𝐴 𝐵) = (𝐴 + (-1 · 𝐵)))

Theoremclmvsubval2 22956 Value of vector subtraction on a subcomplex module. (Contributed by Mario Carneiro, 19-Nov-2013.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉𝐵𝑉) → (𝐴 𝐵) = ((-1 · 𝐵) + 𝐴))

Theoremclmvz 22957 Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → ( 0 𝐴) = (-1 · 𝐴))

Theoremzlmclm 22958 The -module operation turns an arbitrary abelian group into a subcomplex module. (Contributed by Mario Carneiro, 30-Oct-2015.)
𝑊 = (ℤMod‘𝐺)       (𝐺 ∈ Abel ↔ 𝑊 ∈ ℂMod)

Theoremclmzlmvsca 22959 The scalar product of a subcomplex module matches the scalar product of the derived -module, which implies, together with zlmbas 19914 and zlmplusg 19915, that any module over is structure-equivalent to the canonical -module ℤMod‘𝐺. (Contributed by Mario Carneiro, 30-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &   𝑋 = (Base‘𝐺)       ((𝐺 ∈ ℂMod ∧ (𝐴 ∈ ℤ ∧ 𝐵𝑋)) → (𝐴( ·𝑠𝐺)𝐵) = (𝐴( ·𝑠𝑊)𝐵))

Theoremnmoleub2lem 22960* Lemma for nmoleub2a 22963 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ ∀𝑥𝑉 (𝜓 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)) → 0 ≤ 𝐴)    &   ((((𝜑 ∧ ∀𝑥𝑉 (𝜓 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦𝑉𝑦 ≠ (0g𝑆))) → (𝑀‘(𝐹𝑦)) ≤ (𝐴 · (𝐿𝑦)))    &   ((𝜑𝑥𝑉) → (𝜓 → (𝐿𝑥) ≤ 𝑅))       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 (𝜓 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmoleub2lem3 22961* Lemma for nmoleub2a 22963 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.) (Proof shortened by AV, 29-Sep-2021.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)    &    · = ( ·𝑠𝑆)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵𝑉)    &   (𝜑𝐵 ≠ (0g𝑆))    &   (𝜑 → ((𝑟 · 𝐵) ∈ 𝑉 → ((𝐿‘(𝑟 · 𝐵)) < 𝑅 → ((𝑀‘(𝐹‘(𝑟 · 𝐵))) / 𝑅) ≤ 𝐴)))    &   (𝜑 → ¬ (𝑀‘(𝐹𝐵)) ≤ (𝐴 · (𝐿𝐵)))        ¬ 𝜑

Theoremnmoleub2lem2 22962* Lemma for nmoleub2a 22963 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)    &   (((𝐿𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿𝑥)𝑂𝑅 → (𝐿𝑥) ≤ 𝑅))    &   (((𝐿𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿𝑥) < 𝑅 → (𝐿𝑥)𝑂𝑅))       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥)𝑂𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmoleub2a 22963* The operator norm is the supremum of the value of a linear operator in the closed unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥) ≤ 𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmoleub2b 22964* The operator norm is the supremum of the value of a linear operator in the open unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥) < 𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmoleub3 22965* The operator norm is the supremum of the value of a linear operator on the closed unit sphere. (Contributed by Mario Carneiro, 19-Oct-2015.) (Proof shortened by AV, 29-Sep-2021.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → ℝ ⊆ 𝐾)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥) = 𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmhmcn 22966 A linear operator over a normed subcomplex module is bounded iff it is continuous. (Contributed by Mario Carneiro, 22-Oct-2015.)
𝐽 = (TopOpen‘𝑆)    &   𝐾 = (TopOpen‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐵 = (Base‘𝐺)       ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))

Theoremcmodscexp 22967 The powers of i belong to the scalar subring of a subcomplex module if i belongs to the scalar subring . (Contributed by AV, 18-Oct-2021.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (((𝑊 ∈ ℂMod ∧ i ∈ 𝐾) ∧ 𝑁 ∈ ℕ) → (i↑𝑁) ∈ 𝐾)

Theoremcmodscmulexp 22968 The scalar product of a vector with powers of i belongs to the base set of a subcomplex module if the scalar subring of th subcomplex module contains i. (Contributed by AV, 18-Oct-2021.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑋 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾𝐵𝑋𝑁 ∈ ℕ)) → ((i↑𝑁) · 𝐵) ∈ 𝑋)

12.5.2  Subcomplex vector spaces

Usually, "complex vector spaces" are vector spaces over the field of the complex numbers, see for example the definition in [Roman] p. 36.

In the setting of set.mm, it is convenient to consider collectively vector spaces on subfields of the field of complex numbers. We call these, "subcomplex vector spaces" and collect them in the class ℂVec defined in df-cvs 22970 and characterized in iscvs 22973. These include rational vector spaces (qcvs 22993), real vector spaces (recvs 22992) and complex vector spaces (cncvs 22991).

This definition is analogous to the definition of subcomplex modules (and their class ℂMod), which are modules over subrings of the field of complex numbers. Note that ZZ-modules (that are roughly the same thing as Abelian groups, see zlmclm 22958) are subcomplex modules but are not subcomplex vector spaces (see zclmncvs 22994), because the ring ZZ is not a division ring (see zringndrg 19886).

Since the field of complex numbers is commutative, so are its subrings, so there is no need to explicitly state "left module" or "left vector space" for subcomplex modules or vector spaces.

Syntaxccvs 22969 Syntax for the class of subcomplex vector spaces.
class ℂVec

Definitiondf-cvs 22970 Define the class of subcomplex vector spaces, which are the subcomplex modules which are also vector spaces. (Contributed by Thierry Arnoux, 22-May-2019.)
ℂVec = (ℂMod ∩ LVec)

Theoremcvslvec 22971 A subcomplex vector space is a (left) vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
(𝜑𝑊 ∈ ℂVec)       (𝜑𝑊 ∈ LVec)

Theoremcvsclm 22972 A subcomplex vector space is a subcomplex module. (Contributed by Thierry Arnoux, 22-May-2019.)
(𝜑𝑊 ∈ ℂVec)       (𝜑𝑊 ∈ ℂMod)

Theoremiscvs 22973 A subcomplex vector space is a subcomplex module over a division ring. For example, the subcomplex modules over the rational or real or complex numbers are subcomplex vector spaces. (Contributed by AV, 4-Oct-2021.)
(𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))

Theoremiscvsp 22974* The predicate "is a subcomplex vector space." (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.)
· = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝑊 ∈ ℂVec ↔ ((𝑊 ∈ Grp ∧ (𝑆 ∈ DivRing ∧ 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)))))))

Theoremiscvsi 22975* Properties that determine a subcomplex vector space. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.)
· = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &   𝑊 ∈ Grp    &   𝑆 = (ℂflds 𝐾)    &   𝑆 ∈ DivRing    &   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 ∈ ℂVec

Theoremcvsi 22976* The properties of a subcomplex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. (Contributed by NM, 3-Nov-2006.) (Revised by AV, 21-Sep-2021.)
𝑋 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑆 = (Base‘(Scalar‘𝑊))    &    = ( ·sf𝑊)    &    · = ( ·𝑠𝑊)       (𝑊 ∈ ℂVec → (𝑊 ∈ Abel ∧ (𝑆 ⊆ ℂ ∧ :(𝑆 × 𝑋)⟶𝑋) ∧ ∀𝑥𝑋 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦𝑆 (∀𝑧𝑋 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧𝑆 (((𝑦 + 𝑧) · 𝑥) = ((𝑦 · 𝑥) + (𝑧 · 𝑥)) ∧ ((𝑦 · 𝑧) · 𝑥) = (𝑦 · (𝑧 · 𝑥)))))))

Theoremcvsunit 22977 Unit group of the scalar ring of a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂVec → (𝐾 ∖ {0}) = (Unit‘𝐹))

Theoremcvsdiv 22978 Division of the scalar ring of a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂVec ∧ (𝐴𝐾𝐵𝐾𝐵 ≠ 0)) → (𝐴 / 𝐵) = (𝐴(/r𝐹)𝐵))

Theoremcvsdivcl 22979 The scalar field of a subcomplex vector space is closed under division. (Contributed by Thierry Arnoux, 22-May-2019.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂVec ∧ (𝐴𝐾𝐵𝐾𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾)

Theoremcvsmuleqdivd 22980 An equality involving ratios in a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐴 ≠ 0)    &   (𝜑 → (𝐴 · 𝑋) = (𝐵 · 𝑌))       (𝜑𝑋 = ((𝐵 / 𝐴) · 𝑌))

Theoremcvsdiveqd 22981 An equality involving ratios in a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝑋 = ((𝐴 / 𝐵) · 𝑌))       (𝜑 → ((𝐵 / 𝐴) · 𝑋) = 𝑌)

Theoremcnlmodlem1 22982 Lemma 1 for cnlmod 22986. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (Base‘𝑊) = ℂ

Theoremcnlmodlem2 22983 Lemma 2 for cnlmod 22986. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (+g𝑊) = +

Theoremcnlmodlem3 22984 Lemma 3 for cnlmod 22986. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (Scalar‘𝑊) = ℂfld

Theoremcnlmod4 22985 Lemma 4 for cnlmod 22986. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       ( ·𝑠𝑊) = ·

Theoremcnlmod 22986 The set of complex numbers is a left module over itself. The vector operation is +, and the scalar product is ·. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       𝑊 ∈ LMod

Theoremcnstrcvs 22987 The set of complex numbers is a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       𝑊 ∈ ℂVec

Theoremcnrbas 22988 The set of complex numbers is the base set of the complex left module of complex numbers. (Contributed by AV, 21-Sep-2021.)
𝐶 = (ringLMod‘ℂfld)       (Base‘𝐶) = ℂ

Theoremcnrlmod 22989 The complex left module of complex numbers is a left module. The vector operation is +, and the scalar product is ·. (Contributed by AV, 21-Sep-2021.)
𝐶 = (ringLMod‘ℂfld)       𝐶 ∈ LMod

Theoremcnrlvec 22990 The complex left module of complex numbers is a left vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 21-Sep-2021.)
𝐶 = (ringLMod‘ℂfld)       𝐶 ∈ LVec

Theoremcncvs 22991 The complex left module of complex numbers is a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 21-Sep-2021.)
𝐶 = (ringLMod‘ℂfld)       𝐶 ∈ ℂVec

Theoremrecvs 22992 The field of the real numbers as left module over itself is a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.)
𝑅 = (ringLMod‘ℝfld)       𝑅 ∈ ℂVec

Theoremqcvs 22993 The field of rational numbers as left module over itself is a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.)
𝑄 = (ringLMod‘(ℂflds ℚ))       𝑄 ∈ ℂVec

Theoremzclmncvs 22994 The ring of integers as left module over itself is a subcomplex module, but not a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.)
𝑍 = (ringLMod‘ℤring)       (𝑍 ∈ ℂMod ∧ 𝑍 ∉ ℂVec)

12.5.3  Normed subcomplex vector spaces

This section characterizes normed subcomplex vector spaces as subcomplex vector spaces which are also normed vector spaces (that is, normed groups with a positively homogeneous norm). For the moment, there is no need of a special token to represent their class, so we only use the characterization isncvsngp 22995. Most theorems for normed subcomplex vector spaces have a label containing "ncvs". The idiom 𝑊 ∈ (NrmVec ∩ ℂVec) is used in the following to say that 𝑊 is a normed subcomplex vector space, i.e. a subcomplex vector space which is also a normed vector space.

Theoremisncvsngp 22995* A normed subcomplex vector space is a subcomplex vector space which is a normed group with a positively homogeneous norm. (Contributed by NM, 5-Jun-2008.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥𝑉𝑘𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁𝑥))))

Theoremisncvsngpd 22996* Properties that determine a normed subcomplex vector space. (Contributed by NM, 15-Apr-2007.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂVec)    &   (𝜑𝑊 ∈ NrmGrp)    &   ((𝜑 ∧ (𝑥𝑉𝑘𝐾)) → (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁𝑥)))       (𝜑𝑊 ∈ (NrmVec ∩ ℂVec))

Theoremncvsi 22997* The properties of a normed subcomplex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (-g𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ (NrmVec ∩ ℂVec) → (𝑊 ∈ ℂVec ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥𝑉 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑉 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ∧ ∀𝑘𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁𝑥)))))

Theoremncvsprp 22998 Proportionality property of the norm of a scalar product in a normed subcomplex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴𝐾𝐵𝑉) → (𝑁‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))

Theoremncvsge0 22999 The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ (𝐾 ∩ ℝ) ∧ 0 ≤ 𝐴) ∧ 𝐵𝑉) → (𝑁‘(𝐴 · 𝐵)) = (𝐴 · (𝑁𝐵)))

Theoremncvsm1 23000 The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴𝑉) → (𝑁‘(-1 · 𝐴)) = (𝑁𝐴))

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