P. 281 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28001-28100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	vc0 28001	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 = (1st ‘𝑊)    &   ⊢ 𝑆 = (2nd ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍)
Theorem	vcz 28002	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 = (1st ‘𝑊)    &   ⊢ 𝑆 = (2nd ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)
Theorem	vcm 28003	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 = (1st ‘𝑊)    &   ⊢ 𝑆 = (2nd ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑀 = (inv‘𝐺)    ⇒   ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴))
Theorem	isvclem 28004*	Lemma for isvcOLD 28006. (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 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))))
Theorem	vcex 28005	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))
Theorem	isvcOLD 28006*	The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) Obsolete version of iscvsp 23335. (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (⟨𝐺, 𝑆⟩ ∈ CVecOLD ↔ (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))
Theorem	isvciOLD 28007*	Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) Obsolete version of iscvsi 23336. (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 ∈ AbelOp    &   ⊢ dom 𝐺 = (𝑋 × 𝑋)    &   ⊢ 𝑆:(ℂ × 𝑋)⟶𝑋    &   ⊢ (𝑥 ∈ 𝑋 → (1𝑆𝑥) = 𝑥)    &   ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)))    &   ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)))    &   ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))    &   ⊢ 𝑊 = ⟨𝐺, 𝑆⟩    ⇒   ⊢ 𝑊 ∈ CVecOLD
18.2.2  Examples of complex vector spaces
Theorem	cnaddabloOLD 28008	Obsolete version of cnaddabl 18658. Complex number addition is an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ + ∈ AbelOp
Theorem	cnidOLD 28009	Obsolete version of cnaddid 18659. 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‘ + )
Theorem	cncvcOLD 28010	Obsolete version of cncvs 23352. 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
Syntax	cnv 28011	Extend class notation with the class of all normed complex vector spaces.
class NrmCVec
Syntax	cpv 28012	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 10275.
class +𝑣
Syntax	cba 28013	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
Syntax	cns 28014	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
Syntax	cn0v 28015	Extend class notation with zero vector in a normed complex vector space.
class 0vec
Syntax	cnsb 28016	Extend class notation with vector subtraction in a normed complex vector space.
class −𝑣
Syntax	cnmcv 28017	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
Syntax	cims 28018	Extend class notation with the class of the induced metrics on normed complex vector spaces.
class IndMet
Definition	df-nv 28019*	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⟩ ∈ CVecOLD ∧ 𝑛:ran 𝑔⟶ℝ ∧ ∀𝑥 ∈ ran 𝑔(((𝑛‘𝑥) = 0 → 𝑥 = (GId‘𝑔)) ∧ ∀𝑦 ∈ ℂ (𝑛‘(𝑦𝑠𝑥)) = ((abs‘𝑦) · (𝑛‘𝑥)) ∧ ∀𝑦 ∈ ran 𝑔(𝑛‘(𝑥𝑔𝑦)) ≤ ((𝑛‘𝑥) + (𝑛‘𝑦))))}
Theorem	nvss 28020	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)
Theorem	nvvcop 28021	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)
Definition	df-va 28022	Define vector addition on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
⊢ +𝑣 = (1st ∘ 1st )
Definition	df-ba 28023	Define the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
⊢ BaseSet = (𝑥 ∈ V ↦ ran ( +𝑣 ‘𝑥))
Definition	df-sm 28024	Define scalar multiplication on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
⊢ ·𝑠OLD = (2nd ∘ 1st )
Definition	df-0v 28025	Define the zero vector in a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
⊢ 0vec = (GId ∘ +𝑣 )
Definition	df-vs 28026	Define vector subtraction on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
⊢ −𝑣 = ( /𝑔 ∘ +𝑣 )
Definition	df-nmcv 28027	Define the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
⊢ normCV = 2nd
Definition	df-ims 28028	Define the induced metric on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)))
Theorem	nvrel 28029	The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)
⊢ Rel NrmCVec
Theorem	vafval 28030	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 = ( +𝑣 ‘𝑈)    ⇒   ⊢ 𝐺 = (1st ‘(1st ‘𝑈))
Theorem	bafval 28031	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 = ( +𝑣 ‘𝑈)    ⇒   ⊢ 𝑋 = ran 𝐺
Theorem	smfval 28032	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 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ 𝑆 = (2nd ‘(1st ‘𝑈))
Theorem	0vfval 28033	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝑍 = (GId‘𝐺))
Theorem	nmcvfval 28034	Value of the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ 𝑁 = (2nd ‘𝑈)
Theorem	nvop2 28035	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 = (1st ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)
Theorem	nvvop 28036	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 = (1st ‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)
Theorem	isnvlem 28037*	Lemma for isnv 28039. (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⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))))
Theorem	nvex 28038	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))
Theorem	isnv 28039*	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⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
Theorem	isnvi 28040*	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    &   ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑁‘𝑥) = 0) → 𝑥 = 𝑍)    &   ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)))    &   ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))    &   ⊢ 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩    ⇒   ⊢ 𝑈 ∈ NrmCVec
Theorem	nvi 28041*	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → (⟨𝐺, 𝑆⟩ ∈ CVecOLD ∧ 𝑁:𝑋⟶ℝ ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁‘𝑥)) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
Theorem	nvvc 28042	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 = (1st ‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)
Theorem	nvablo 28043	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 = ( +𝑣 ‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)
Theorem	nvgrp 28044	The vector addition operation of a normed complex vector space is a group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
⊢ 𝐺 = ( +𝑣 ‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
Theorem	nvgf 28045	Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝐺:(𝑋 × 𝑋)⟶𝑋)
Theorem	nvsf 28046	Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × 𝑋)⟶𝑋)
Theorem	nvgcl 28047	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 = ( +𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
Theorem	nvcom 28048	The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
Theorem	nvass 28049	The vector addition (group) operation is associative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))
Theorem	nvadd32 28050	Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))
Theorem	nvrcan 28051	Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))
Theorem	nvadd4 28052	Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
Theorem	nvscl 28053	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 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
Theorem	nvsid 28054	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 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
Theorem	nvsass 28055	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 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))
Theorem	nvscom 28056	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 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝑆𝐶)) = (𝐵𝑆(𝐴𝑆𝐶)))
Theorem	nvdi 28057	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))
Theorem	nvdir 28058	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
Theorem	nv2 28059	A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))
Theorem	vsfval 28060	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    ⇒   ⊢ 𝑀 = ( /𝑔 ‘𝐺)
Theorem	nvzcl 28061	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 = (0vec‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋)
Theorem	nv0rid 28062	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)
Theorem	nv0lid 28063	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝐴) = 𝐴)
Theorem	nv0 28064	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 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍)
Theorem	nvsz 28065	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 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)
Theorem	nvinv 28066	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑀 = (inv‘𝐺)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴))
Theorem	nvinvfval 28067	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑁 = (𝑆 ∘ ◡(2nd ↾ ({-1} × V)))    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺))
Theorem	nvm 28068	Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    &   ⊢ 𝑁 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵))
Theorem	nvmval 28069	Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝐺(-1𝑆𝐵)))
Theorem	nvmval2 28070	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = ((-1𝑆𝐵)𝐺𝐴))
Theorem	nvmfval 28071*	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(-1𝑆𝑦))))
Theorem	nvmf 28072	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 = ( −𝑣 ‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋)
Theorem	nvmcl 28073	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 = ( −𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) ∈ 𝑋)
Theorem	nvnnncan1 28074	Cancellation law for vector subtraction. (nnncan1 10659 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑀𝐵)𝑀(𝐴𝑀𝐶)) = (𝐶𝑀𝐵))
Theorem	nvmdi 28075	Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝑀𝐶)) = ((𝐴𝑆𝐵)𝑀(𝐴𝑆𝐶)))
Theorem	nvnegneg 28076	Double negative of a vector. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆(-1𝑆𝐴)) = 𝐴)
Theorem	nvmul0or 28077	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 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑆𝐵) = 𝑍 ↔ (𝐴 = 0 ∨ 𝐵 = 𝑍)))
Theorem	nvrinv 28078	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = 𝑍)
Theorem	nvlinv 28079	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = 𝑍)
Theorem	nvpncan2 28080	Cancellation law for vector subtraction. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀𝐴) = 𝐵)
Theorem	nvpncan 28081	Cancellation law for vector subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀𝐵) = 𝐴)
Theorem	nvaddsub 28082	Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑀𝐶) = ((𝐴𝑀𝐶)𝐺𝐵))
Theorem	nvnpcan 28083	Cancellation law for a normed complex vector space. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑀𝐵)𝐺𝐵) = 𝐴)
Theorem	nvaddsub4 28084	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 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)))
Theorem	nvmeq0 28085	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 = ( −𝑣 ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑀𝐵) = 𝑍 ↔ 𝐴 = 𝐵))
Theorem	nvmid 28086	A vector minus itself is the zero vector. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑀𝐴) = 𝑍)
Theorem	nvf 28087	Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → 𝑁:𝑋⟶ℝ)
Theorem	nvcl 28088	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 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ)
Theorem	nvcli 28089	The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    &   ⊢ 𝑈 ∈ NrmCVec    &   ⊢ 𝐴 ∈ 𝑋    ⇒   ⊢ (𝑁‘𝐴) ∈ ℝ
Theorem	nvs 28090	Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵)))
Theorem	nvsge0 28091	The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑆𝐵)) = (𝐴 · (𝑁‘𝐵)))
Theorem	nvm1 28092	The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(-1𝑆𝐴)) = (𝑁‘𝐴))
Theorem	nvdif 28093	The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺(-1𝑆𝐵))) = (𝑁‘(𝐵𝐺(-1𝑆𝐴))))
Theorem	nvpi 28094	The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺(i𝑆𝐵))) = (𝑁‘(𝐵𝐺(-i𝑆𝐴))))
Theorem	nvz0 28095	The norm of a zero vector is zero. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
⊢ 𝑍 = (0vec‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ (𝑈 ∈ NrmCVec → (𝑁‘𝑍) = 0)
Theorem	nvz 28096	The norm of a vector is zero iff the vector is zero. First part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑍 = (0vec‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = 0 ↔ 𝐴 = 𝑍))
Theorem	nvtri 28097	Triangle inequality for the norm of a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵)))
Theorem	nvmtri 28098	Triangle inequality for the norm of a vector difference. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑀 = ( −𝑣 ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑀𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵)))
Theorem	nvabs 28099	Norm difference property of a normed complex vector space. Problem 3 of [Kreyszig] p. 64. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝐺 = ( +𝑣 ‘𝑈)    &   ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (abs‘((𝑁‘𝐴) − (𝑁‘𝐵))) ≤ (𝑁‘(𝐴𝐺(-1𝑆𝐵))))
Theorem	nvge0 28100	The norm of a normed complex vector space is nonnegative. Second part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 28-Nov-2006.) (Proof shortened by AV, 10-Jul-2022.) (New usage is discouraged.)
⊢ 𝑋 = (BaseSet‘𝑈)    &   ⊢ 𝑁 = (normCV‘𝑈)    ⇒   ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝑁‘𝐴))