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Theorem List for Metamath Proof Explorer - 20901-21000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrngosn 20901 The trivial or zero ring defined on a singleton set (see http://en.wikipedia.org/wiki/Trivial_ring). (Contributed by Steve Rodriguez, 10-Feb-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

15.2.3  Division Rings

Syntaxcdrng 20902 Extend class notation with the class of all division rings.

Definitiondf-drngo 20903* Define the class of all division rings (sometimes called skew fields). A division ring is a unital ring where every element except the additive identity has a multiplicative inverse. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)
GId GId

Theoremdrngoi 20904 The properties of a division ring. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)
GId

15.2.4  Star Fields

Syntaxcsfld 20905 Extend class notation with the class of all star fields.

Definitiondf-sfld 20906* Define the class of all star fields, which are all division rings with involutions. (Contributed by NM, 29-Aug-2010.) (New usage is discouraged.)

15.2.5  Fields and Rings

Syntaxccm2 20907 Extend class notation with a class that adds commutativity to various flavors of rings.

Definitiondf-com2 20908* A device to add commutativity to various sorts of rings. I use because I suppose has a neutral element and therefore is onto. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)

Theoremiscom2 20909* A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)

Syntaxcfld 20910 Extend class notation with the class of all fields.

Definitiondf-fld 20911 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 20912 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 20913 The base set of a ring is not empty. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)

Theoremrngmgmbs4 20914* 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 20915 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 20916 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 20917 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 20918 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 20919 In a unital ring the addition is an abelian group. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)

Theoremrngoidmlem 20920 The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
GId

Theoremrngolidm 20921 The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)
GId

Theoremrngoridm 20922 The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)
GId

Theoremrngosn3 20923 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 20924 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 20925 The only unital ring with one element is the zero ring. (Contributed by FL, 15-Feb-2010.) (New usage is discouraged.)
GId

Theoremrngo1cl 20926 The unit of a ring belongs to the base set. (Contributed by FL, 12-Feb-2010.) (New usage is discouraged.)
GId

Theoremrngoueqz 20927 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 20928 The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
GId GId

Theoremzrdivrng 20929 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 20930 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

15.3  Complex vector spaces

15.3.1  Definition and basic properties

Syntaxcvc 20931 Extend class notation with the class of all complex vector spaces.

Definitiondf-vc 20932* Define the class of all complex vector spaces. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcrel 20933 The class of all complex vector spaces is a relation. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)

Theoremvci 20934* 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 20935 Functionality of th scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvccl 20936 Closure of the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcid 20937 Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcdi 20938 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcdir 20939 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcass 20940 Associative law for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvc2 20941 A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

Theoremvcsubdir 20942 Subtractive distributive law for the scalar product of a complex vector space. (Contributed by NM, 31-Jul-2007.) (New usage is discouraged.)

Theoremvcablo 20943 Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Theoremvcgrp 20944 Vector addition is a group operation. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvcgcl 20945 Closure law for the vector addition (group) operation of a complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

Theoremvccom 20946 Vector addition is commutative. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvcaass 20947 Vector addition is associative. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvca32 20948 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 20949 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)

Theoremvcrcan 20950 Right cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvclcan 20951 Left cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvczcl 20952 The zero vector is a vector. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
GId

Theoremvc0rid 20953 The zero vector is a right identity element. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
GId

Theoremvc0lid 20954 The zero vector is a left identity element. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
GId

Theoremvc0 20955 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 20956 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 20957 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 20958 A vector minus itself. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
GId

Theoremvclinv 20959 Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
GId

Theoremvcnegneg 20960 Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (New usage is discouraged.)

Theoremvcnegsubdi2 20961 Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (New usage is discouraged.)

Theoremvcsub4 20962 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)

Theoremisvclem 20963* Lemma for isvc 20967. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcoprnelem 20964 Lemma for vcoprne 20965. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcoprne 20965 The operations of a complex vector space cannot be identical. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcex 20966 The components of a complex vector space are sets. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremisvc 20967* The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremisvci 20968* Properties that determine a complex vector space. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

15.3.2  Examples of complex vector spaces

Theoremcncvc 20969 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.)

15.4  Normed complex vector spaces

15.4.1  Definition and basic properties

Syntaxcnv 20970 Extend class notation with the class of all normed complex vector spaces.

Syntaxcpv 20971 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 8620.

Syntaxcba 20972 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 20973 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 20974 Extend class notation with zero vector in a normed complex vector space.

Syntaxcnsb 20975 Extend class notation with vector subtraction in a normed complex vector space.

Syntaxcnmcv 20976 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 20977 Extend class notation with the class of the induced metrics on normed complex vector spaces.

Definitiondf-nv 20978* Define the class of all normed complex vector spaces. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
GId

Theoremnvss 20979 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 20980 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 20981 Define vector addition on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

Definitiondf-ba 20982 Define the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

Definitiondf-sm 20983 Define scalar multiplication on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)

Definitiondf-0v 20984 Define the zero vector in a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
GId

Definitiondf-vs 20985 Define vector subtraction on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Definitiondf-nmcv 20986 Define the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
CV

Definitiondf-ims 20987 Define the induced metric on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
CV

Theoremnvrel 20988 The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)

Theoremvafval 20989 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 20990 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 20991 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 20992 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 20993 Value of the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
CV

Theoremnvop2 20994 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 20995 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 20996* Lemma for isnv 20998. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
GId

Theoremnvex 20997 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 20998* The predicate "is a normed complex vector space." (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
GId

Theoremisnvi 20999* Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
GId

Theoremnvi 21000* 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

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