P. 289 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28801-28900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	grpodivf 28801	Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋)
Theorem	grpodivcl 28802	Closure of group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)
Theorem	grpodivdiv 28803	Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = (𝐴𝐺(𝐶𝐷𝐵)))
Theorem	grpomuldivass 28804	Associative-type law for multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = (𝐴𝐺(𝐵𝐷𝐶)))
Theorem	grpodivid 28805	Division of a group member by itself. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    &   ⊢ 𝑈 = (GId‘𝐺)    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 𝑈)
Theorem	grponpcan 28806	Cancellation law for group division. (npcan 11160 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)
18.1.2  Abelian groups
Syntax	cablo 28807	Extend class notation with the class of all Abelian group operations.
class AbelOp
Definition	df-ablo 28808*	Define the class of all Abelian group operations. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
⊢ AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}
Theorem	isablo 28809*	The predicate "is an Abelian (commutative) group operation." (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))
Theorem	ablogrpo 28810	An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
Theorem	ablocom 28811	An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))
Theorem	ablo32 28812	Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))
Theorem	ablo4 28813	Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))
Theorem	isabloi 28814*	Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
⊢ 𝐺 ∈ GrpOp    &   ⊢ dom 𝐺 = (𝑋 × 𝑋)    &   ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))    ⇒   ⊢ 𝐺 ∈ AbelOp
Theorem	ablomuldiv 28815	Law for group multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))
Theorem	ablodivdiv 28816	Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = ((𝐴𝐷𝐵)𝐺𝐶))
Theorem	ablodivdiv4 28817	Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = (𝐴𝐷(𝐵𝐺𝐶)))
Theorem	ablodiv32 28818	Swap the second and third terms in a double division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐷𝐵))
Theorem	ablonncan 28819	Cancellation law for group division. (nncan 11180 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = 𝐵)
Theorem	ablonnncan1 28820	Cancellation law for group division. (nnncan1 11187 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
⊢ 𝑋 = ran 𝐺    &   ⊢ 𝐷 = ( /𝑔 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐷(𝐴𝐷𝐶)) = (𝐶𝐷𝐵))
18.2  Complex vector spaces
18.2.1  Definition and basic properties
Syntax	cvc 28821	Extend class notation with the class of all complex vector spaces.
class CVecOLD
Definition	df-vc 28822*	Define the class of all complex vector spaces. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
⊢ CVecOLD = {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ AbelOp ∧ 𝑠:(ℂ × ran 𝑔)⟶ran 𝑔 ∧ ∀𝑥 ∈ ran 𝑔((1𝑠𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ ran 𝑔(𝑦𝑠(𝑥𝑔𝑧)) = ((𝑦𝑠𝑥)𝑔(𝑦𝑠𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑠𝑥) = ((𝑦𝑠𝑥)𝑔(𝑧𝑠𝑥)) ∧ ((𝑦 · 𝑧)𝑠𝑥) = (𝑦𝑠(𝑧𝑠𝑥))))))}
Theorem	vcrel 28823	The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
⊢ Rel CVecOLD
Theorem	vciOLD 28824*	Obsolete version of cvsi 24199. The properties of a complex 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. The variable 𝑊 was chosen because V is already used for the universal class. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑊)    &   ⊢ 𝑆 = (2nd ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑊 ∈ 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	vcsm 28825	Functionality of th scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑊)    &   ⊢ 𝑆 = (2nd ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ (𝑊 ∈ CVecOLD → 𝑆:(ℂ × 𝑋)⟶𝑋)
Theorem	vccl 28826	Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑊)    &   ⊢ 𝑆 = (2nd ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)
Theorem	vcidOLD 28827	Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) Obsolete theorem, use clmvs1 24162 together with cvsclm 24195 instead. (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑊)    &   ⊢ 𝑆 = (2nd ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴)
Theorem	vcdi 28828	Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑊)    &   ⊢ 𝑆 = (2nd ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))
Theorem	vcdir 28829	Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑊)    &   ⊢ 𝑆 = (2nd ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))
Theorem	vcass 28830	Associative law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑊)    &   ⊢ 𝑆 = (2nd ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))
Theorem	vc2OLD 28831	A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) Obsolete theorem, use clmvs2 24163 together with cvsclm 24195 instead. (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐺 = (1st ‘𝑊)    &   ⊢ 𝑆 = (2nd ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))
Theorem	vcablo 28832	Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑊)    ⇒   ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp)
Theorem	vcgrp 28833	Vector addition is a group operation. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑊)    ⇒   ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)
Theorem	vclcan 28834	Left cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    ⇒   ⊢ ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))
Theorem	vczcl 28835	The zero vector is a vector. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ (𝑊 ∈ CVecOLD → 𝑍 ∈ 𝑋)
Theorem	vc0rid 28836	The zero vector is a right identity element. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
⊢ 𝐺 = (1st ‘𝑊)    &   ⊢ 𝑋 = ran 𝐺    &   ⊢ 𝑍 = (GId‘𝐺)    ⇒   ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)
Theorem	vc0 28837	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 28838	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 28839	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 28840*	Lemma for isvcOLD 28842. (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 28841	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 28842*	The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) Obsolete version of iscvsp 24197. (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 28843*	Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) Obsolete version of iscvsi 24198. (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 28844	Obsolete version of cnaddabl 19385. 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 28845	Obsolete version of cnaddid 19386. 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 28846	Obsolete version of cncvs 24214. 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 28847	Extend class notation with the class of all normed complex vector spaces.
class NrmCVec
Syntax	cpv 28848	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 10805.
class +𝑣
Syntax	cba 28849	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 28850	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 28851	Extend class notation with zero vector in a normed complex vector space.
class 0vec
Syntax	cnsb 28852	Extend class notation with vector subtraction in a normed complex vector space.
class −𝑣
Syntax	cnmcv 28853	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 28854	Extend class notation with the class of the induced metrics on normed complex vector spaces.
class IndMet
Definition	df-nv 28855*	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 28856	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 28857	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 28858	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 28859	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 28860	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 28861	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 28862	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 28863	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 28864	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 28865	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 28866	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 28867	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 28868	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 28869	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 28870	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 28871	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 28872	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 28873*	Lemma for isnv 28875. (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 28874	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 28875*	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 28876*	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 28877*	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 28878	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 28879	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 28880	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 28881	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 28882	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 28883	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 28884	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 28885	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 28886	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 28887	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 28888	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 28889	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 28890	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 28891	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 28892	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 28893	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 28894	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 28895	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 28896	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 28897	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 28898	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 28899	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 28900	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)