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Theorem List for Metamath Proof Explorer - 27401-27500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvcsm 27401 Functionality of th scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       (𝑊 ∈ CVecOLD𝑆:(ℂ × 𝑋)⟶𝑋)

Theoremvccl 27402 Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

TheoremvcidOLD 27403 Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) Obsolete as of 21-Sep-2021. Use clmvs1 22887 together with cvsclm 22920 instead. (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)

Theoremvcdi 27404 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))

Theoremvcdir 27405 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

Theoremvcass 27406 Associative law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

Theoremvc2OLD 27407 A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) Obsolete as of 21-Sep-2021. Use clmvs2 22888 together with cvsclm 22920 instead. (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))

Theoremvcablo 27408 Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)       (𝑊 ∈ CVecOLD𝐺 ∈ AbelOp)

Theoremvcgrp 27409 Vector addition is a group operation. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)       (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)

Theoremvclcan 27410 Left cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))

Theoremvczcl 27411 The zero vector is a vector. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝑊 ∈ CVecOLD𝑍𝑋)

Theoremvc0rid 27412 The zero vector is a right identity element. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)

Theoremvc0 27413 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       ((𝑊 ∈ CVecOLD𝐴𝑋) → (0𝑆𝐴) = 𝑍)

Theoremvcz 27414 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

Theoremvcm 27415 Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 25-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺    &   𝑀 = (inv‘𝐺)       ((𝑊 ∈ CVecOLD𝐴𝑋) → (-1𝑆𝐴) = (𝑀𝐴))

Theoremisvclem 27416* Lemma for isvcOLD 27418. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (⟨𝐺, 𝑆⟩ ∈ CVecOLD ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))))

Theoremvcex 27417 The components of a complex vector space are sets. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
(⟨𝐺, 𝑆⟩ ∈ CVecOLD → (𝐺 ∈ V ∧ 𝑆 ∈ V))

TheoremisvcOLD 27418* The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) Obsolete as of 4-Oct-2021. Use iscvsp 22922 instead. (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = ran 𝐺       (⟨𝐺, 𝑆⟩ ∈ CVecOLD ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))

TheoremisvciOLD 27419* Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) Obsolete as of 4-Oct-2021. Use iscvsi 22923 instead. (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 ∈ AbelOp    &   dom 𝐺 = (𝑋 × 𝑋)    &   𝑆:(ℂ × 𝑋)⟶𝑋    &   (𝑥𝑋 → (1𝑆𝑥) = 𝑥)    &   ((𝑦 ∈ ℂ ∧ 𝑥𝑋𝑧𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))    &   ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥𝑋) → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))    &   ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥𝑋) → ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))    &   𝑊 = ⟨𝐺, 𝑆       𝑊 ∈ CVecOLD

18.2.2  Examples of complex vector spaces

TheoremcnaddabloOLD 27420 Obsolete as of 23-Jan-2020. Use cnaddabl 18266 instead. Complex number addition is an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
+ ∈ AbelOp

TheoremcnidOLD 27421 Obsolete as of 23-Jan-2020. Use cnaddid 18267 instead. The group identity element of complex number addition is zero. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
0 = (GId‘ + )

TheoremcncvcOLD 27422 Obsolete version of cncvs 22939 as of 20-Sep-2021. The set of complex numbers is a complex vector space. The vector operation is +, and the scalar product is ·. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
⟨ + , · ⟩ ∈ CVecOLD

18.3  Normed complex vector spaces

18.3.1  Definition and basic properties

Syntaxcnv 27423 Extend class notation with the class of all normed complex vector spaces.
class NrmCVec

Syntaxcpv 27424 Extend class notation with vector addition in a normed complex vector space. In the literature, the subscript "v" is omitted, but we need it to avoid ambiguity with complex number addition + caddc 9936.
class +𝑣

Syntaxcba 27425 Extend class notation with the base set of a normed complex vector space. (Note that BaseSet is capitalized because, once it is fixed for a particular vector space 𝑈, it is not a function, unlike e.g. normCV. This is our typical convention.) (New usage is discouraged.)
class BaseSet

Syntaxcns 27426 Extend class notation with scalar multiplication in a normed complex vector space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity.
class ·𝑠OLD

Syntaxcn0v 27427 Extend class notation with zero vector in a normed complex vector space.
class 0vec

Syntaxcnsb 27428 Extend class notation with vector subtraction in a normed complex vector space.
class 𝑣

Syntaxcnmcv 27429 Extend class notation with the norm function in a normed complex vector space. In the literature, the norm of 𝐴 is usually written "|| 𝐴 ||", but we use function notation to take advantage of our existing theorems about functions.
class normCV

Syntaxcims 27430 Extend class notation with the class of the induced metrics on normed complex vector spaces.
class IndMet

Definitiondf-nv 27431* Define the class of all normed complex vector spaces. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
NrmCVec = {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ (⟨𝑔, 𝑠⟩ ∈ CVecOLD𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛𝑥) + (𝑛𝑦))))}

Theoremnvss 27432 Structure of the class of all normed complex vectors spaces. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
NrmCVec ⊆ (CVecOLD × V)

Theoremnvvcop 27433 A normed complex vector space is a vector space. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
(⟨𝑊, 𝑁⟩ ∈ NrmCVec → 𝑊 ∈ CVecOLD)

Definitiondf-va 27434 Define vector addition on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
+𝑣 = (1st ∘ 1st )

Definitiondf-ba 27435 Define the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
BaseSet = (𝑥 ∈ V ↦ ran ( +𝑣𝑥))

Definitiondf-sm 27436 Define scalar multiplication on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
·𝑠OLD = (2nd ∘ 1st )

Definitiondf-0v 27437 Define the zero vector in a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
0vec = (GId ∘ +𝑣 )

Definitiondf-vs 27438 Define vector subtraction on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑣 = ( /𝑔 ∘ +𝑣 )

Definitiondf-nmcv 27439 Define the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
normCV = 2nd

Definitiondf-ims 27440 Define the induced metric on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV𝑢) ∘ ( −𝑣𝑢)))

Theoremnvrel 27441 The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)
Rel NrmCVec

Theoremvafval 27442 Value of the function for the vector addition (group) operation on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)       𝐺 = (1st ‘(1st𝑈))

Theorembafval 27443 Value of the function for the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       𝑋 = ran 𝐺

Theoremsmfval 27444 Value of the function for the scalar multiplication operation on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
𝑆 = ( ·𝑠OLD𝑈)       𝑆 = (2nd ‘(1st𝑈))

Theorem0vfval 27445 Value of the function for the zero vector on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)    &   𝑍 = (0vec𝑈)       (𝑈𝑉𝑍 = (GId‘𝐺))

Theoremnmcvfval 27446 Value of the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
𝑁 = (normCV𝑈)       𝑁 = (2nd𝑈)

Theoremnvop2 27447 A normed complex vector space is an ordered pair of a vector space and a norm operation. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
𝑊 = (1st𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)

Theoremnvvop 27448 The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
𝑊 = (1st𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)

Theoremisnvlem 27449* Lemma for isnv 27451. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))

Theoremnvex 27450 The components of a normed complex vector space are sets. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
(⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))

Theoremisnv 27451* The predicate "is a normed complex vector space." (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))

Theoremisnvi 27452* Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)    &   𝐺, 𝑆⟩ ∈ CVecOLD    &   𝑁:𝑋⟶ℝ    &   ((𝑥𝑋 ∧ (𝑁𝑥) = 0) → 𝑥 = 𝑍)    &   ((𝑦 ∈ ℂ ∧ 𝑥𝑋) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))    &   ((𝑥𝑋𝑦𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))    &   𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁       𝑈 ∈ NrmCVec

Theoremnvi 27453* The properties of a normed complex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ NrmCVec → (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))

Theoremnvvc 27454 The vector space component of a normed complex vector space. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑊 = (1st𝑈)       (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)

Theoremnvablo 27455 The vector addition operation of a normed complex vector space is an Abelian group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)       (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)

Theoremnvgrp 27456 The vector addition operation of a normed complex vector space is a group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)       (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)

Theoremnvgf 27457 Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       (𝑈 ∈ NrmCVec → 𝐺:(𝑋 × 𝑋)⟶𝑋)

Theoremnvsf 27458 Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       (𝑈 ∈ NrmCVec → 𝑆:(ℂ × 𝑋)⟶𝑋)

Theoremnvgcl 27459 Closure law for the vector addition (group) operation of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Theoremnvcom 27460 The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Theoremnvass 27461 The vector addition (group) operation is associative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

Theoremnvadd32 27462 Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))

Theoremnvrcan 27463 Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))

Theoremnvadd4 27464 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))

Theoremnvscl 27465 Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

Theoremnvsid 27466 Identity element for the scalar product of a normed complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (1𝑆𝐴) = 𝐴)

Theoremnvsass 27467 Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

Theoremnvscom 27468 Commutative law for the scalar product of a normed complex vector space. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → (𝐴𝑆(𝐵𝑆𝐶)) = (𝐵𝑆(𝐴𝑆𝐶)))

Theoremnvdi 27469 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))

Theoremnvdir 27470 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

Theoremnv2 27471 A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))

Theoremvsfval 27472 Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       𝑀 = ( /𝑔𝐺)

Theoremnvzcl 27473 Closure law for the zero vector of a normed complex vector space. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)       (𝑈 ∈ NrmCVec → 𝑍𝑋)

Theoremnv0rid 27474 The zero vector is a right identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)

Theoremnv0lid 27475 The zero vector is a left identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑍𝐺𝐴) = 𝐴)

Theoremnv0 27476 Zero times a vector is the zero vector. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (0𝑆𝐴) = 𝑍)

Theoremnvsz 27477 Anything times the zero vector is the zero vector. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

Theoremnvinv 27478 Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑀 = (inv‘𝐺)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (-1𝑆𝐴) = (𝑀𝐴))

Theoremnvinvfval 27479 Function for the negative of a vector on a normed complex vector space, in terms of the underlying addition group inverse. (We currently do not have a separate notation for the negative of a vector.) (Contributed by NM, 27-Mar-2008.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (𝑆(2nd ↾ ({-1} × V)))       (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺))

Theoremnvm 27480 Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = ( /𝑔𝐺)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵))

Theoremnvmval 27481 Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) = (𝐴𝐺(-1𝑆𝐵)))

Theoremnvmval2 27482 Value of vector subtraction on a normed complex vector space. (Contributed by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) = ((-1𝑆𝐵)𝐺𝐴))

Theoremnvmfval 27483* Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑀 = ( −𝑣𝑈)       (𝑈 ∈ NrmCVec → 𝑀 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(-1𝑆𝑦))))

Theoremnvmf 27484 Mapping for the vector subtraction operation. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)       (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋)

Theoremnvmcl 27485 Closure law for the vector subtraction operation of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) ∈ 𝑋)

Theoremnvnnncan1 27486 Cancellation law for vector subtraction. (nnncan1 10314 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑀𝐵)𝑀(𝐴𝑀𝐶)) = (𝐶𝑀𝐵))

Theoremnvmdi 27487 Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝐴𝑆(𝐵𝑀𝐶)) = ((𝐴𝑆𝐵)𝑀(𝐴𝑆𝐶)))

Theoremnvnegneg 27488 Double negative of a vector. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (-1𝑆(-1𝑆𝐴)) = 𝐴)

Theoremnvmul0or 27489 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → ((𝐴𝑆𝐵) = 𝑍 ↔ (𝐴 = 0 ∨ 𝐵 = 𝑍)))

Theoremnvrinv 27490 A vector minus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺(-1𝑆𝐴)) = 𝑍)

Theoremnvlinv 27491 Minus a vector plus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → ((-1𝑆𝐴)𝐺𝐴) = 𝑍)

Theoremnvpncan2 27492 Cancellation law for vector subtraction. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝑀𝐴) = 𝐵)

Theoremnvpncan 27493 Cancellation law for vector subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝑀𝐵) = 𝐴)

Theoremnvaddsub 27494 Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝑀𝐶) = ((𝐴𝑀𝐶)𝐺𝐵))

Theoremnvnpcan 27495 Cancellation law for a normed complex vector space. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝑀𝐵)𝐺𝐵) = 𝐴)

Theoremnvaddsub4 27496 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)))

Theoremnvmeq0 27497 The difference between two vectors is zero iff they are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝑀𝐵) = 𝑍𝐴 = 𝐵))

Theoremnvmid 27498 A vector minus itself is the zero vector. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝑀𝐴) = 𝑍)

Theoremnvf 27499 Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ NrmCVec → 𝑁:𝑋⟶ℝ)

Theoremnvcl 27500 The norm of a normed complex vector space is a real number. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑁𝐴) ∈ ℝ)

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42322
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