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Theorem List for Metamath Proof Explorer - 22001-22100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcfld 22001 Extend class notation with the class of all fields.

Definitiondf-fld 22002 Definition of a field. A field is a commutative division ring. (Contributed by FL, 6-Sep-2009.) (Revised by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.)

Theoremflddivrng 22003 A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremrngon0 22004 The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)

Theoremrngmgmbs4 22005* The range of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Theoremrngodm1dm2 22006 In a unital ring the domain of the first variable of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)

Theoremrngorn1 22007 In a unital ring the range of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)

Theoremrngorn1eq 22008 In a unital ring the range of the addition equals the range of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)

Theoremrngomndo 22009 In a unital ring the multiplication is a monoid. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
MndOp

Theoremrngoablo2 22010 In a unital ring the addition is an abelian group. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)

Theoremrngoidmlem 22011 The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
GId

Theoremrngolidm 22012 The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)
GId

Theoremrngoridm 22013 The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)
GId

Theoremrngosn3 22014 The only unital ring with a base set consisting in one element is the zero ring. (Contributed by FL, 13-Feb-2010.) (Proof shortened by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.)

Theoremrngosn4 22015 The only unital ring with one element is the zero ring. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.)

Theoremrngosn6 22016 The only unital ring with one element is the zero ring. (Contributed by FL, 15-Feb-2010.) (New usage is discouraged.)
GId

Theoremrngo1cl 22017 The unit of a ring belongs to the base set. (Contributed by FL, 12-Feb-2010.) (New usage is discouraged.)
GId

Theoremrngoueqz 22018 In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (New usage is discouraged.)
GId       GId

Theoremisdivrngo 22019 The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
GId GId

Theoremzrdivrng 22020 The zero ring is not a division ring. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremdvrunz 22021 In a division ring the unit is different from the zero. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
GId       GId

Theoremzerdivemp1 22022* In a unitary ring a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.)
GId              GId

Theoremrngoridfz 22023* In a unitary ring a left invertible element is different from zero iff . (Contributed by FL, 18-Apr-2010.)
GId              GId

PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)

17.1  Complex vector spaces

17.1.1  Definition and basic properties

Syntaxcvc 22024 Extend class notation with the class of all complex vector spaces.

Definitiondf-vc 22025* Define the class of all complex vector spaces. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcrel 22026 The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theoremvci 22027* 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 was chosen because is already used for the universal class. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcsm 22028 Functionality of th scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvccl 22029 Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcid 22030 Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcdi 22031 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcdir 22032 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcass 22033 Associative law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvc2 22034 A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

Theoremvcsubdir 22035 Subtractive distributive law for the scalar product of a complex vector space. (Contributed by NM, 31-Jul-2007.) (New usage is discouraged.)

Theoremvcablo 22036 Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcgrp 22037 Vector addition is a group operation. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvcgcl 22038 Closure law for the vector addition (group) operation of a complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

Theoremvccom 22039 Vector addition is commutative. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvcaass 22040 Vector addition is associative. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvca32 22041 Commutative/associative law that swaps the last two terms in a triple vector sum. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)

Theoremvca4 22042 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)

Theoremvcrcan 22043 Right cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvclcan 22044 Left cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvczcl 22045 The zero vector is a vector. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
GId

Theoremvc0rid 22046 The zero vector is a right identity element. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
GId

Theoremvc0lid 22047 The zero vector is a left identity element. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
GId

Theoremvc0 22048 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
GId

Theoremvcz 22049 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.)
GId

Theoremvcm 22050 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.)

Theoremvcrinv 22051 A vector minus itself. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
GId

Theoremvclinv 22052 Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
GId

Theoremvcnegneg 22053 Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (New usage is discouraged.)

Theoremvcnegsubdi2 22054 Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (New usage is discouraged.)

Theoremvcsub4 22055 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)

Theoremisvclem 22056* Lemma for isvc 22060. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcoprnelem 22057 Lemma for vcoprne 22058. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcoprne 22058 The operations of a complex vector space cannot be identical. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcex 22059 The components of a complex vector space are sets. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremisvc 22060* The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremisvci 22061* Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

17.1.2  Examples of complex vector spaces

Theoremcncvc 22062 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.)

17.2  Normed complex vector spaces

17.2.1  Definition and basic properties

Syntaxcnv 22063 Extend class notation with the class of all normed complex vector spaces.

Syntaxcpv 22064 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 8993.

Syntaxcba 22065 Extend class notation with the base set of a normed complex vector space. (Note that is capitalized because, once it is fixed for a particular vector space , it is not a function, unlike e.g. CV. This is our typical convention.) (New usage is discouraged.)

Syntaxcns 22066 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.

Syntaxcn0v 22067 Extend class notation with zero vector in a normed complex vector space.

Syntaxcnsb 22068 Extend class notation with vector subtraction in a normed complex vector space.

Syntaxcnmcv 22069 Extend class notation with the norm function in a normed complex vector space. In the literature, the norm of is usually written "|| ||", but we use function notation to take advantage of our existing theorems about functions.
CV

Syntaxcims 22070 Extend class notation with the class of the induced metrics on normed complex vector spaces.

Definitiondf-nv 22071* Define the class of all normed complex vector spaces. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
GId

Theoremnvss 22072 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.)

Theoremnvvcop 22073 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.)

Definitiondf-va 22074 Define vector addition on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

Definitiondf-ba 22075 Define the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

Definitiondf-sm 22076 Define scalar multiplication on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)

Definitiondf-0v 22077 Define the zero vector in a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
GId

Definitiondf-vs 22078 Define vector subtraction on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Definitiondf-nmcv 22079 Define the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
CV

Definitiondf-ims 22080 Define the induced metric on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
CV

Theoremnvrel 22081 The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)

Theoremvafval 22082 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.)

Theorembafval 22083 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.)

Theoremsmfval 22084 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.)

Theorem0vfval 22085 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.)
GId

Theoremnmcvfval 22086 Value of the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
CV

Theoremnvop2 22087 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.)
CV

Theoremnvvop 22088 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.)

Theoremisnvlem 22089* Lemma for isnv 22091. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
GId

Theoremnvex 22090 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.)

Theoremisnv 22091* The predicate "is a normed complex vector space." (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
GId

Theoremisnvi 22092* Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
GId

Theoremnvi 22093* 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.)
CV

Theoremnvvc 22094 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.)

Theoremnvablo 22095 The vector addition operation of a normed complex vector space is an Abelian group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvgrp 22096 The vector addition operation of a normed complex vector space is a group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvgf 22097 Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvsf 22098 Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvgcl 22099 Closure law for the vector addition (group) operation of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

Theoremnvcom 22100 The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

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