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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremgrpoinveu 27501* The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)

Theoremgrpoid 27502 Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴))

Theoremgrporn 27503 The range of a group operation. Useful for satisfying group base set hypotheses of the form 𝑋 = ran 𝐺. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
𝐺 ∈ GrpOp    &   dom 𝐺 = (𝑋 × 𝑋)       𝑋 = ran 𝐺

Theoremgrpoinvfval 27504* The inverse function of a group. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))

Theoremgrpoinvval 27505* The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))

Theoremgrpoinvcl 27506 A group element's inverse is a group element. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)

Theoremgrpoinv 27507 The properties of a group element's inverse. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈))

Theoremgrpolinv 27508 The left inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴)𝐺𝐴) = 𝑈)

Theoremgrporinv 27509 The right inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺(𝑁𝐴)) = 𝑈)

Theoremgrpoinvid1 27510 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐴𝐺𝐵) = 𝑈))

Theoremgrpoinvid2 27511 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) = 𝐵 ↔ (𝐵𝐺𝐴) = 𝑈))

Theoremgrpolcan 27512 Left cancellation law for groups. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))

Theoremgrpo2inv 27513 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁‘(𝑁𝐴)) = 𝐴)

Theoremgrpoinvf 27514 Mapping of the inverse function of a group. (Contributed by NM, 29-Mar-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)       (𝐺 ∈ GrpOp → 𝑁:𝑋1-1-onto𝑋)

Theoremgrpoinvop 27515 The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺𝐵)) = ((𝑁𝐵)𝐺(𝑁𝐴)))

Theoremgrpodivfval 27516* Group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)    &   𝐷 = ( /𝑔𝐺)       (𝐺 ∈ GrpOp → 𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(𝑁𝑦))))

Theoremgrpodivval 27517 Group division (or subtraction) operation value. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺(𝑁𝐵)))

Theoremgrpodivinv 27518 Group division by an inverse. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝑁𝐵)) = (𝐴𝐺𝐵))

Theoremgrpoinvdiv 27519 Inverse of a group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐷𝐵)) = (𝐵𝐷𝐴))

Theoremgrpodivf 27520 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 = ( /𝑔𝐺)       (𝐺 ∈ GrpOp → 𝐷:(𝑋 × 𝑋)⟶𝑋)

Theoremgrpodivcl 27521 Closure of group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ 𝑋)

Theoremgrpodivdiv 27522 Double group division. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = (𝐴𝐺(𝐶𝐷𝐵)))

Theoremgrpomuldivass 27523 Associative-type law for multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = (𝐴𝐺(𝐵𝐷𝐶)))

Theoremgrpodivid 27524 Division of a group member by itself. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐷𝐴) = 𝑈)

Theoremgrponpcan 27525 Cancellation law for group division. (npcan 10328 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ GrpOp ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴)

18.1.2  Abelian groups

Syntaxcablo 27526 Extend class notation with the class of all Abelian group operations.
class AbelOp

Definitiondf-ablo 27527* Define the class of all Abelian group operations. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)}

Theoremisablo 27528* 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 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)))

Theoremablogrpo 27529 An Abelian group operation is a group operation. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
(𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)

Theoremablocom 27530 An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Theoremablo32 27531 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))

Theoremablo4 27532 Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
𝑋 = ran 𝐺       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))

Theoremisabloi 27533* Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
𝐺 ∈ GrpOp    &   dom 𝐺 = (𝑋 × 𝑋)    &   ((𝑥𝑋𝑦𝑋) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))       𝐺 ∈ AbelOp

Theoremablomuldiv 27534 Law for group multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐺𝐵))

Theoremablodivdiv 27535 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷(𝐵𝐷𝐶)) = ((𝐴𝐷𝐵)𝐺𝐶))

Theoremablodivdiv4 27536 Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = (𝐴𝐷(𝐵𝐺𝐶)))

Theoremablodiv32 27537 Swap the second and third terms in a double division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷𝐶) = ((𝐴𝐷𝐶)𝐷𝐵))

Theoremablonnncan 27538 Cancellation law for group division. (nnncan 10354 analog.) (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷(𝐵𝐷𝐶))𝐷𝐶) = (𝐴𝐷𝐵))

Theoremablonncan 27539 Cancellation law for group division. (nncan 10348 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷(𝐴𝐷𝐵)) = 𝐵)

Theoremablonnncan1 27540 Cancellation law for group division. (nnncan1 10355 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝐺 ∈ AbelOp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐷(𝐴𝐷𝐶)) = (𝐶𝐷𝐵))

18.2  Complex vector spaces

18.2.1  Definition and basic properties

Syntaxcvc 27541 Extend class notation with the class of all complex vector spaces.
class CVecOLD

Definitiondf-vc 27542* 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))))))}

Theoremvcrel 27543 The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Rel CVecOLD

TheoremvciOLD 27544* Obsolete version of cvsi 22976 as of 21-Sep-2021. 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 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       (𝑊 ∈ 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)))))))

Theoremvcsm 27545 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 27546 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 27547 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 22939 together with cvsclm 22972 instead. (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)

Theoremvcdi 27548 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 27549 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 27550 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 27551 A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) Obsolete as of 21-Sep-2021. Use clmvs2 22940 together with cvsclm 22972 instead. (New usage is discouraged.) (Proof modification is discouraged.)
𝐺 = (1st𝑊)    &   𝑆 = (2nd𝑊)    &   𝑋 = ran 𝐺       ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))

Theoremvcablo 27552 Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)       (𝑊 ∈ CVecOLD𝐺 ∈ AbelOp)

Theoremvcgrp 27553 Vector addition is a group operation. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
𝐺 = (1st𝑊)       (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)

Theoremvclcan 27554 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 27555 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 27556 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 27557 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 27558 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 27559 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 27560* Lemma for isvcOLD 27562. (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 27561 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 27562* The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) Obsolete as of 4-Oct-2021. Use iscvsp 22974 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 27563* Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) Obsolete as of 4-Oct-2021. Use iscvsi 22975 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 27564 Obsolete as of 23-Jan-2020. Use cnaddabl 18318 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 27565 Obsolete as of 23-Jan-2020. Use cnaddid 18319 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 27566 Obsolete version of cncvs 22991 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 27567 Extend class notation with the class of all normed complex vector spaces.
class NrmCVec

Syntaxcpv 27568 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 9977.
class +𝑣

Syntaxcba 27569 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 27570 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 27571 Extend class notation with zero vector in a normed complex vector space.
class 0vec

Syntaxcnsb 27572 Extend class notation with vector subtraction in a normed complex vector space.
class 𝑣

Syntaxcnmcv 27573 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 27574 Extend class notation with the class of the induced metrics on normed complex vector spaces.
class IndMet

Definitiondf-nv 27575* 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 27576 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 27577 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 27578 Define vector addition on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
+𝑣 = (1st ∘ 1st )

Definitiondf-ba 27579 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 27580 Define scalar multiplication on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
·𝑠OLD = (2nd ∘ 1st )

Definitiondf-0v 27581 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 27582 Define vector subtraction on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑣 = ( /𝑔 ∘ +𝑣 )

Definitiondf-nmcv 27583 Define the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
normCV = 2nd

Definitiondf-ims 27584 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 27585 The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)
Rel NrmCVec

Theoremvafval 27586 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 27587 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 27588 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 27589 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 27590 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 27591 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 27592 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 27593* Lemma for isnv 27595. (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 27594 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 27595* 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 27596* 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 27597* 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 27598 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 27599 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 27600 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)

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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 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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