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Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnvass 22101 The vector addition (group) operation is associative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvadd12 22102 Commutative/associative law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvadd32 22103 Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)

Theoremnvrcan 22104 Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvlcan 22105 Left cancellation law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvadd4 22106 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)

Theoremnvscl 22107 Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)

Theoremnvsid 22108 Identity element for the scalar product of a normed complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvsass 22109 Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

Theoremnvscom 22110 Commutative law for the scalar product of a normed complex vector space. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)

Theoremnvdi 22111 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvdir 22112 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnv2 22113 A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008.) (New usage is discouraged.)

Theoremvsfval 22114 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 22115 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 22116 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 22117 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 22118 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 22119 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 22120 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 22121 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 22122 Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvmval 22123 Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)

Theoremnvmval2 22124 Value of vector subtraction on a normed complex vector space. (Contributed by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremnvmfval 22125* 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 22126 Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)

Theoremnvmf 22127 Mapping for the vector subtraction operation. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremnvmcl 22128 Closure law for the vector subtraction operation of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)

Theoremnvnnncan1 22129 Cancellation law for vector subtraction. (nnncan1 9337 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)

Theoremnvnnncan2 22130 Cancellation law for vector subtraction. (nnncan2 9338 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Theoremnvmdi 22131 Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)

Theoremnvnegneg 22132 Double negative of a vector. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)

Theoremnvmul0or 22133 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)

Theoremnvrinv 22134 A vector minus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnvlinv 22135 Minus a vector plus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Theoremnvsubadd 22136 Relationship between vector subtraction and addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvpncan2 22137 Cancellation law for vector subtraction. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)

Theoremnvpncan 22138 Cancellation law for vector subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvaddsubass 22139 Associative-type law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvaddsub 22140 Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvnpcan 22141 Cancellation law for a normed complex vector space. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvaddsub4 22142 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)

Theoremnvsubsub23 22143 Swap subtrahend and result of vector subtraction. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)

Theoremnvnncan 22144 Cancellation law for a normed complex vector space. (Contributed by NM, 17-Dec-2007.) (New usage is discouraged.)

Theoremnvmeq0 22145 The difference between two vectors is zero iff they are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Theoremnvmid 22146 A vector minus itself is the zero vector. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Theoremnvf 22147 Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvcl 22148 The norm of a normed complex vector space is a real number. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvcli 22149 The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007.) (New usage is discouraged.)
CV

Theoremnvdm 22150 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 22151 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 22152 The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.)
CV

Theoremnvm1 22153 The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvdif 22154 The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (New usage is discouraged.)
CV

Theoremnvpi 22155 The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
CV

Theoremnvsub 22156 The norm of the difference between two vectors. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
CV

Theoremnvz0 22157 The norm of a zero vector is zero. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
CV

Theoremnvz 22158 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 22159 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 22160 Triangle inequality for the norm of a vector difference. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
CV

Theoremnvmtri2 22161 Triangle inequality for the norm of a vector difference. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
CV

Theoremnvabs 22162 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 22163 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 22164 A nonzero norm is positive. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
CV

Theoremnv1 22165 From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
CV

Theoremnvop 22166 A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
CV

Theoremnvoprne 22167 The vector addition and scalar product operations are not identical. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

17.2.2  Examples of normed complex vector spaces

Theoremcnnv 22168 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 22169 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 22170 The base set of the normed complex vector space of complex numbers. (Contributed by NM, 7-Nov-2007.) (New usage is discouraged.)

Theoremcnnvdemo 22171 Derive the associative law for complex number addition addass 9077 to demonstrate the use of cnnv 22168, cnnvg 22169, and cnnvba 22170. (Contributed by NM, 12-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcnnvs 22172 The scalar product operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)

Theoremcnnvnm 22173 The norm operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
CV

Theoremcnnvm 22174 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 22175 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)

Theoremelimnvu 22176 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)

17.2.3  Induced metric of a normed complex vector space

Theoremimsval 22177 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 22178 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 22179 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 22180 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 22181 Mapping for the induced metric distance function of a normed complex vector space. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)

Theoremimsmetlem 22182 Lemma for imsmet 22183. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)
CV

Theoremimsmet 22183 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 22184 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 22185 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 22186 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 22187 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 22188* 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 22189 The metric induced on the complex numbers. cnmet 18806 proves that it is a metric. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by NM, 15-Jan-2008.) (New usage is discouraged.)

Theoremvacn 22190 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 22191 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 22192 The norm of a normed complex vector space is a continuous function to . (For , see nmcvcn 22191.) (Contributed by NM, 12-Aug-2007.) (New usage is discouraged.)
CV                     fld

Theoremsmcnlem 22193* Lemma for smcn 22194. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
fld              CV

Theoremsmcn 22194 Scalar multiplication is jointly continuous in both arguments. (Contributed by NM, 16-Jun-2009.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
fld

Theoremvmcn 22195 Vector subtraction is jointly continuous in both arguments. (Contributed by Mario Carneiro, 6-May-2014.) (New usage is discouraged.)

17.2.4  Inner product

Syntaxcdip 22196 Extend class notation with the class inner product functions.

Definitiondf-dip 22197* Define a function that maps a complex normed vector space to its inner product operation in case its norm satisfies the parallelogram identity (otherwise the operation is still defined, but not meaningful). Based on Exercise 4(a) of [ReedSimon] p. 63 and Theorem 6.44 of [Ponnusamy] p. 361. Vector addition is , the scalar product is , and the norm is . (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.)
CV

Theoremdipfval 22198* The inner product function on a normed complex vector space. The definition is meaningful for vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law. (Contributed by NM, 10-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
CV

Theoremipval 22199* Value of the inner product. The definition is meaningful for normed complex vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law, although for convenience we define it for any normed complex vector space. The vector (group) addition operation is , the scalar product is , the norm is , and the set of vectors is . Equation 6.45 of [Ponnusamy] p. 361. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
CV

Theoremipval2lem2 22200 Lemma for ipval3 22205. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
CV

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