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

Theoremvc0 22001 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 22002 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 22003 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 22004 A vector minus itself. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
GId

Theoremvclinv 22005 Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
GId

Theoremvcnegneg 22006 Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (New usage is discouraged.)

Theoremvcnegsubdi2 22007 Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (New usage is discouraged.)

Theoremvcsub4 22008 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 5-Aug-2007.) (New usage is discouraged.)

Theoremisvclem 22009* Lemma for isvc 22013. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcoprnelem 22010 Lemma for vcoprne 22011. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcoprne 22011 The operations of a complex vector space cannot be identical. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremvcex 22012 The components of a complex vector space are sets. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremisvc 22013* The predicate "is a complex vector space." (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremisvci 22014* 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 22015 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 22016 Extend class notation with the class of all normed complex vector spaces.

Syntaxcpv 22017 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 8949.

Syntaxcba 22018 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 22019 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 22020 Extend class notation with zero vector in a normed complex vector space.

Syntaxcnsb 22021 Extend class notation with vector subtraction in a normed complex vector space.

Syntaxcnmcv 22022 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 22023 Extend class notation with the class of the induced metrics on normed complex vector spaces.

Definitiondf-nv 22024* Define the class of all normed complex vector spaces. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
GId

Theoremnvss 22025 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 22026 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 22027 Define vector addition on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

Definitiondf-ba 22028 Define the base set of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)

Definitiondf-sm 22029 Define scalar multiplication on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)

Definitiondf-0v 22030 Define the zero vector in a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
GId

Definitiondf-vs 22031 Define vector subtraction on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Definitiondf-nmcv 22032 Define the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
CV

Definitiondf-ims 22033 Define the induced metric on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
CV

Theoremnvrel 22034 The class of all normed complex vectors spaces is a relation. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)

Theoremvafval 22035 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 22036 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 22037 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 22038 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 22039 Value of the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
CV

Theoremnvop2 22040 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 22041 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 22042* Lemma for isnv 22044. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
GId

Theoremnvex 22043 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 22044* The predicate "is a normed complex vector space." (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
GId

Theoremisnvi 22045* Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
GId

Theoremnvi 22046* 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 22047 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 22048 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 22049 The vector addition operation of a normed complex vector space is a group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvgf 22050 Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvsf 22051 Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvgcl 22052 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 22053 The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvass 22054 The vector addition (group) operation is associative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvadd12 22055 Commutative/associative law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvadd32 22056 Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)

Theoremnvrcan 22057 Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvlcan 22058 Left cancellation law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvadd4 22059 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)

Theoremnvscl 22060 Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

Theoremnvsid 22061 Identity element for the scalar product of a normed complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvsass 22062 Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

Theoremnvscom 22063 Commutative law for the scalar product of a normed complex vector space. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)

Theoremnvdi 22064 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvdir 22065 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnv2 22066 A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008.) (New usage is discouraged.)

Theoremvsfval 22067 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 22068 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 22069 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 22070 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 22071 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 22072 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 22073 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 22074 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 22075 Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvmval 22076 Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)

Theoremnvmval2 22077 Value of vector subtraction on a normed complex vector space. (Contributed by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremnvmfval 22078* 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 22079 Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremnvmf 22080 Mapping for the vector subtraction operation. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremnvmcl 22081 Closure law for the vector subtraction operation of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)

Theoremnvnnncan1 22082 Cancellation law for vector subtraction. (nnncan1 9293 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)

Theoremnvnnncan2 22083 Cancellation law for vector subtraction. (nnncan2 9294 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvmdi 22084 Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)

Theoremnvnegneg 22085 Double negative of a vector. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvmul0or 22086 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)

Theoremnvrinv 22087 A vector minus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnvlinv 22088 Minus a vector plus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnvsubadd 22089 Relationship between vector subtraction and addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvpncan2 22090 Cancellation law for vector subtraction. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)

Theoremnvpncan 22091 Cancellation law for vector subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvaddsubass 22092 Associative-type law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvaddsub 22093 Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvnpcan 22094 Cancellation law for a normed complex vector space. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvaddsub4 22095 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)

Theoremnvsubsub23 22096 Swap subtrahend and result of vector subtraction. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvnncan 22097 Cancellation law for a normed complex vector space. (Contributed by NM, 17-Dec-2007.) (New usage is discouraged.)

Theoremnvmeq0 22098 The difference between two vectors is zero iff they are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvmid 22099 A vector minus itself is the zero vector. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvf 22100 Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
CV

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