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Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnvvc 21001 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 21002 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 21003 The vector addition operation of a normed complex vector space is a group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvgf 21004 Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvsf 21005 Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvgcl 21006 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 21007 The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvass 21008 The vector addition (group) operation is associative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvadd12 21009 Commutative/associative law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvadd32 21010 Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)

Theoremnvrcan 21011 Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvlcan 21012 Left cancellation law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvadd4 21013 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)

Theoremnvscl 21014 Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

Theoremnvsid 21015 Identity element for the scalar product of a normed complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvsass 21016 Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

Theoremnvscom 21017 Commutative law for the scalar product of a normed complex vector space. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)

Theoremnvdi 21018 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvdir 21019 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnv2 21020 A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008.) (New usage is discouraged.)

Theoremvsfval 21021 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 21022 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 21023 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 21024 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 21025 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 21026 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 21027 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 21028 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 21029 Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvmval 21030 Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)

Theoremnvmval2 21031 Value of vector subtraction on a normed complex vector space. (Contributed by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremnvmfval 21032* 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 21033 Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremnvmf 21034 Mapping for the vector subtraction operation. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremnvmcl 21035 Closure law for the vector subtraction operation of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)

Theoremnvnnncan1 21036 Cancellation law for vector subtraction. (nnncan1 8963 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)

Theoremnvnnncan2 21037 Cancellation law for vector subtraction. (nnncan2 8964 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvmdi 21038 Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)

Theoremnvnegneg 21039 Double negative of a vector. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvmul0or 21040 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)

Theoremnvrinv 21041 A vector minus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnvlinv 21042 Minus a vector plus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnvsubadd 21043 Relationship between vector subtraction and addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvpncan2 21044 Cancellation law for vector subtraction. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)

Theoremnvpncan 21045 Cancellation law for vector subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvaddsubass 21046 Associative-type law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvaddsub 21047 Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvnpcan 21048 Cancellation law for a normed complex vector space. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvaddsub4 21049 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)

Theoremnvsubsub23 21050 Swap subtrahend and result of vector subtraction. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvnncan 21051 Cancellation law for a normed complex vector space. (Contributed by NM, 17-Dec-2007.) (New usage is discouraged.)

Theoremnvmeq0 21052 The difference between two vectors is zero iff they are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvmid 21053 A vector minus itself is the zero vector. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvf 21054 Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvcl 21055 The norm of a normed complex vector space is a real number. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvcli 21056 The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007.) (New usage is discouraged.)
CV

Theoremnvdm 21057 Two ways to express the set of vectors in a normed complex vector space. (Contributed by NM, 31-Jan-2007.) (New usage is discouraged.)
CV

Theoremnvs 21058 Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvsge0 21059 The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.)
CV

Theoremnvm1 21060 The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvdif 21061 The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (New usage is discouraged.)
CV

Theoremnvpi 21062 The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
CV

Theoremnvsub 21063 The norm of the difference between two vectors. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
CV

Theoremnvz0 21064 The norm of a zero vector is zero. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvz 21065 The norm of a vector is zero iff the vector is zero. First part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvtri 21066 Triangle inequality for the norm of a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
CV

Theoremnvmtri 21067 Triangle inequality for the norm of a vector difference. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
CV

Theoremnvmtri2 21068 Triangle inequality for the norm of a vector difference. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
CV

Theoremnvabs 21069 Norm difference property of a normed complex vector space. Problem 3 of [Kreyszig] p. 64. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
CV

Theoremnvge0 21070 The norm of a normed complex vector space is nonnegative. Second part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvgt0 21071 A nonzero norm is positive. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
CV

Theoremnv1 21072 From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
CV

Theoremnvop 21073 A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
CV

Theoremnvoprne 21074 The vector addition and scalar product operations are not identical. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

15.4.2  Examples of normed complex vector spaces

Theoremcnnv 21075 The set of complex numbers is a normed complex vector space. The vector operation is , the scalar product is , and the norm function is . (Contributed by Steve Rodriguez, 3-Dec-2006.) (New usage is discouraged.)

Theoremcnnvg 21076 The vector addition (group) operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)

Theoremcnnvba 21077 The base set of the normed complex vector space of complex numbers. (Contributed by NM, 7-Nov-2007.) (New usage is discouraged.)

Theoremcnnvdemo 21078 Derive the associative law for complex number addition addass 8704 to demonstrate the use of cnnv 21075, cnnvg 21076, and cnnvba 21077. (Contributed by NM, 12-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcnnvs 21079 The scalar product operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)

Theoremcnnvnm 21080 The norm operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
CV

Theoremcnnvm 21081 The vector subtraction operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremelimnv 21082 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)

Theoremelimnvu 21083 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)

15.4.3  Induced metric of a normed complex vector space

Theoremimsval 21084 Value of the induced metric of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
CV

Theoremimsdval 21085 Value of the induced metric (distance function) of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)
CV

Theoremimsdval2 21086 Value of the distance function of the induced metric of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvnd 21087 The norm of a normed complex vector space expressed in terms of the distance function of its induced metric. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
CV

Theoremimsdf 21088 Mapping for the induced metric distance function of a normed complex vector space. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)

Theoremimsmetlem 21089 Lemma for imsmet 21090. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)
CV

Theoremimsmet 21090 The induced metric of a normed complex vector space is a metric space. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

Theoremimsxmet 21091 The induced metric of a normed complex vector space is an extended metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

Theoremnvelbl 21092 Membership of a vector in a ball. (Contributed by NM, 27-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
CV

Theoremnvelbl2 21093 Membership of an off-center vector in a ball. (Contributed by NM, 27-Dec-2007.) (Revised by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
CV

Theoremnvlmcl 21094 Closure of the limit of a converging vector sequence. (Contributed by NM, 26-Dec-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

Theoremnvlmle 21095* If the norm of each member of a converging sequence is less than or equal to a given amount, so is the norm of the convergence value. (Contributed by NM, 25-Dec-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
CV

Theoremcnims 21096 The metric induced on the complex numbers. cnmet 18113 proves that it is a metric. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by NM, 15-Jan-2008.) (New usage is discouraged.)

Theoremvacn 21097 Vector addition is jointly continuous in both arguments. (Contributed by Jeffrey Hankins, 16-Jun-2009.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

Theoremnmcvcn 21098 The norm of a normed complex vector space is a continuous function. (Contributed by NM, 16-May-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.)
CV

Theoremnmcnc 21099 The norm of a normed complex vector space is a continuous function to . (For , see nmcvcn 21098.) (Contributed by NM, 12-Aug-2007.) (New usage is discouraged.)
CV                     fld

Theoremsmcnlem 21100* Lemma for smcn 21101. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
fld              CV

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