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Theorem List for Metamath Proof Explorer - 21901-22000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvccom 21901 Vector addition is commutative. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvcaass 21902 Vector addition is associative. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvca32 21903 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 21904 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)

Theoremvcrcan 21905 Right cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvclcan 21906 Left cancellation law for vector addition. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Theoremvczcl 21907 The zero vector is a vector. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
GId

Theoremvc0rid 21908 The zero vector is a right identity element. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
GId

Theoremvc0lid 21909 The zero vector is a left identity element. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
GId

Theoremvc0 21910 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 21911 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 21912 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 21913 A vector minus itself. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
GId

Theoremvclinv 21914 Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
GId

Theoremvcnegneg 21915 Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (New usage is discouraged.)

Theoremvcnegsubdi2 21916 Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (New usage is discouraged.)

Theoremvcsub4 21917 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)

Theoremisvclem 21918* Lemma for isvc 21922. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcoprnelem 21919 Lemma for vcoprne 21920. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcoprne 21920 The operations of a complex vector space cannot be identical. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcex 21921 The components of a complex vector space are sets. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremisvc 21922* The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremisvci 21923* 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 21924 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 21925 Extend class notation with the class of all normed complex vector spaces.

Syntaxcpv 21926 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 8940.

Syntaxcba 21927 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 21928 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 21929 Extend class notation with zero vector in a normed complex vector space.

Syntaxcnsb 21930 Extend class notation with vector subtraction in a normed complex vector space.

Syntaxcnmcv 21931 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 21932 Extend class notation with the class of the induced metrics on normed complex vector spaces.

Definitiondf-nv 21933* Define the class of all normed complex vector spaces. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
GId

Theoremnvss 21934 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 21935 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 21936 Define vector addition on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

Definitiondf-ba 21937 Define the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

Definitiondf-sm 21938 Define scalar multiplication on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)

Definitiondf-0v 21939 Define the zero vector in a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
GId

Definitiondf-vs 21940 Define vector subtraction on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Definitiondf-nmcv 21941 Define the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
CV

Definitiondf-ims 21942 Define the induced metric on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
CV

Theoremnvrel 21943 The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)

Theoremvafval 21944 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 21945 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 21946 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 21947 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 21948 Value of the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
CV

Theoremnvop2 21949 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 21950 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 21951* Lemma for isnv 21953. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
GId

Theoremnvex 21952 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 21953* The predicate "is a normed complex vector space." (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
GId

Theoremisnvi 21954* Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
GId

Theoremnvi 21955* 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 21956 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 21957 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 21958 The vector addition operation of a normed complex vector space is a group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvgf 21959 Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvsf 21960 Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvgcl 21961 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 21962 The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvass 21963 The vector addition (group) operation is associative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvadd12 21964 Commutative/associative law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvadd32 21965 Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)

Theoremnvrcan 21966 Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvlcan 21967 Left cancellation law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvadd4 21968 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)

Theoremnvscl 21969 Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

Theoremnvsid 21970 Identity element for the scalar product of a normed complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvsass 21971 Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

Theoremnvscom 21972 Commutative law for the scalar product of a normed complex vector space. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)

Theoremnvdi 21973 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvdir 21974 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnv2 21975 A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008.) (New usage is discouraged.)

Theoremvsfval 21976 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.)

Theoremnvzcl 21977 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.)

Theoremnv0rid 21978 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.)

Theoremnv0lid 21979 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.)

Theoremnv0 21980 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.)

Theoremnvsz 21981 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.)

Theoremnvinv 21982 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.)

Theoremnvinvfval 21983 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.)

Theoremnvm 21984 Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvmval 21985 Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)

Theoremnvmval2 21986 Value of vector subtraction on a normed complex vector space. (Contributed by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremnvmfval 21987* 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.)

Theoremnvzs 21988 Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremnvmf 21989 Mapping for the vector subtraction operation. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremnvmcl 21990 Closure law for the vector subtraction operation of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)

Theoremnvnnncan1 21991 Cancellation law for vector subtraction. (nnncan1 9283 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)

Theoremnvnnncan2 21992 Cancellation law for vector subtraction. (nnncan2 9284 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvmdi 21993 Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)

Theoremnvnegneg 21994 Double negative of a vector. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvmul0or 21995 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)

Theoremnvrinv 21996 A vector minus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnvlinv 21997 Minus a vector plus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnvsubadd 21998 Relationship between vector subtraction and addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvpncan2 21999 Cancellation law for vector subtraction. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)

Theoremnvpncan 22000 Cancellation law for vector subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

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