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Theorem List for Metamath Proof Explorer - 29901-30000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlcv1 29901 Covering property of a subspace plus an atom. (chcv1 23860 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlcv2 29902 Covering property of a subspace plus an atom. (chcv2 23861 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatexch 29903 The atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem was originally proved by Hermann Grassmann in 1862. (atexch 23886 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatnle 29904 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 23881 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatnem0 29905 The meet of distinct atoms is the zero subspace. (atnemeq0 23882 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatexch1 29906 The atom exch1ange property. (hlatexch1 30254 analog.) (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremlsatcv0eq 29907 If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 23884 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcv1 29908 Two atoms covering the zero subspace are equal. (atcv1 23885 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcvatlem 29909 Lemma for lsatcvat 29910. (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatcvat 29910 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 23891 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatcvat2 29911 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 23892 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcvat3 29912 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 23901 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremislshpcv 29913 Hyperplane properties expressed with covers relation. (Contributed by NM, 11-Jan-2015.)
LSHyp       L

Theoreml1cvpat 29914 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 30334 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoreml1cvat 29915 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 30335 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoremlshpat 29916 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 30902 analog.) TODO: This changes in l1cvpat 29914 and l1cvat 29915 to , which in turn change in islshpcv 29913 to , with a couple of conversions of span to atom. Seems convoluted. Would a direct proof be better? (Contributed by NM, 11-Jan-2015.)
LSHyp       LSAtoms

19.26.4  Functionals and kernels of a left vector space (or module)

Syntaxclfn 29917 Extend class notation with all linear functionals of a left module or left vector space.
LFnl

Definitiondf-lfl 29918* Define the set of all linear functionals (maps from vectors to the ring) of a left module or left vector space. (Contributed by NM, 15-Apr-2014.)
LFnl Scalar Scalar Scalar Scalar

Theoremlflset 29919* The set of linear functionals in a left module or left vector space. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar                                   LFnl

Theoremislfl 29920* The predicate "is a linear functional". (Contributed by NM, 15-Apr-2014.)
Scalar                                   LFnl

Theoremlfli 29921 Property of a linear functional. (lnfnli 23545 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar                                   LFnl

Theoremislfld 29922* Properties that determine a linear functional. TODO: use this in place of islfl 29920 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)
Scalar                                   LFnl

Theoremlflf 29923 A linear functional is a function from vectors to scalars. (lnfnfi 23546 analog.) (Contributed by NM, 15-Apr-2014.)
Scalar                     LFnl

Theoremlflcl 29924 A linear functional value is a scalar. (Contributed by NM, 15-Apr-2014.)
Scalar                     LFnl

Theoremlfl0 29925 A linear functional is zero at the zero vector. (lnfn0i 23547 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar                     LFnl

Theoremlfladd 29926 Property of a linear functional. (lnfnaddi 23548 analog.) (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl

Theoremlflsub 29927 Property of a linear functional. (lnfnaddi 23548 analog.) (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl

Theoremlflmul 29928 Property of a linear functional. (lnfnmuli 23549 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar                                   LFnl

Theoremlfl0f 29929 The zero function is a functional. (Contributed by NM, 16-Apr-2014.)
Scalar                     LFnl

Theoremlfl1 29930* A non-zero functional has a value of 1 at some argument. (Contributed by NM, 16-Apr-2014.)
Scalar                            LFnl

Theoremlfladdcl 29931 Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)
Scalar              LFnl

Theoremlfladdcom 29932 Commutativity of functional addition. (Contributed by NM, 19-Oct-2014.)
Scalar              LFnl

Theoremlfladdass 29933 Associativity of functional addition. (Contributed by NM, 19-Oct-2014.)
Scalar              LFnl

Theoremlfladd0l 29934 Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.)
Scalar                     LFnl

Theoremlflnegcl 29935* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 30006, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Scalar                     LFnl

Theoremlflnegl 29936* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 30006, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Scalar                     LFnl

Theoremlflvscl 29937 Closure of a scalar product with a functional. Note that this is the scalar product for a right vector space with the scalar after the vector; reversing these fails closure. (Contributed by NM, 9-Oct-2014.) (Revised by Mario Carneiro, 22-Apr-2015.)
Scalar                     LFnl

Theoremlflvsdi1 29938 Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                            LFnl

Theoremlflvsdi2 29939 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                            LFnl

Theoremlflvsdi2a 29940 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-Oct-2014.)
Scalar                            LFnl

Theoremlflvsass 29941 Associative law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                     LFnl

Theoremlfl0sc 29942 The (right vector space) scalar product of a functional with zero is the zero functional. Note that the first occurrence of represents the zero scalar, and the second is the zero functional. (Contributed by NM, 7-Oct-2014.)
Scalar       LFnl

Theoremlflsc0N 29943 The scalar product with the zero functional is the zero functional. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
Scalar

Theoremlfl1sc 29944 The (right vector space) scalar product of a functional with one is the functional. (Contributed by NM, 21-Oct-2014.)
Scalar       LFnl

Syntaxclk 29945 Extend class notation with the kernel of a functional (set of vectors whose functional value is zero) on a left module or left vector space.
LKer

Definitiondf-lkr 29946* Define the kernel of a functional (set of vectors whose functional value is zero) on a left module or left vector space. (Contributed by NM, 15-Apr-2014.)
LKer LFnl Scalar

Theoremlkrfval 29947* The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar              LFnl       LKer

Theoremlkrval 29948 Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
Scalar              LFnl       LKer

Theoremellkr 29949 Membership in the kernel of a functional. (elnlfn 23433 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar              LFnl       LKer

Theoremlkrval2 29950* Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
Scalar              LFnl       LKer

Theoremellkr2 29951 Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.)
Scalar              LFnl       LKer

Theoremlkrcl 29952 A member of the kernel of a functional is a vector. (Contributed by NM, 16-Apr-2014.)
LFnl       LKer

Theoremlkrf0 29953 The value of a functional at a member of its kernel is zero. (Contributed by NM, 16-Apr-2014.)
Scalar              LFnl       LKer

Theoremlkr0f 29954 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 17-Apr-2014.)
Scalar                     LFnl       LKer

Theoremlkrlss 29955 The kernel of a linear functional is a subspace. (nlelshi 23565 analog.) (Contributed by NM, 16-Apr-2014.)
LFnl       LKer

Theoremlkrssv 29956 The kernel of a linear functional is a set of vectors. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremlkrsc 29957 The kernel of a non-zero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 9-Oct-2014.)
Scalar                     LFnl       LKer

Theoremlkrscss 29958 The kernel of a scalar product of a functional includes the kernel of the functional. (The inclusion is proper for the zero product and equality otherwise.) (Contributed by NM, 9-Oct-2014.)
Scalar                     LFnl       LKer

Theoremeqlkr 29959* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl       LKer

Theoremeqlkr2 29960* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 10-Oct-2014.)
Scalar                            LFnl       LKer

Theoremeqlkr3 29961 Two functionals with the same kernel are equal if they are equal at any nonzero value. (Contributed by NM, 2-Jan-2015.)
Scalar                     LFnl       LKer

Theoremlkrlsp 29962 The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 29949) is the whole vector space. (Contributed by NM, 19-Apr-2014.)
Scalar                                   LFnl       LKer

Theoremlkrlsp2 29963 The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 12-May-2014.)
LFnl       LKer

Theoremlkrlsp3 29964 The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 29-Jun-2014.)
LFnl       LKer

Theoremlkrshp 29965 The kernel of a nonzero functional is a hyperplane. (Contributed by NM, 29-Jun-2014.)
Scalar              LSHyp       LFnl       LKer

Theoremlkrshp3 29966 The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 17-Jul-2014.)
Scalar              LSHyp       LFnl       LKer

Theoremlkrshpor 29967 The kernel of a functional is either a hyperplane or the full vector space. (Contributed by NM, 7-Oct-2014.)
LSHyp       LFnl       LKer

Theoremlkrshp4 29968 A kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.)
LSHyp       LFnl       LKer

Theoremlshpsmreu 29969* Lemma for lshpkrex 29978. Show uniqueness of ring multiplier when a vector is broken down into components, one in a hyperplane and the other outside of it . TODO: do we need the cbvrexv 2935 for to ? (Contributed by NM, 4-Jan-2015.)
LSHyp                                          Scalar

Theoremlshpkrlem1 29970* Lemma for lshpkrex 29978. The value of tentative functional is zero iff its argument belongs to hyperplane . (Contributed by NM, 14-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem2 29971* Lemma for lshpkrex 29978. The value of tentative functional is a scalar. (Contributed by NM, 16-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem3 29972* Lemma for lshpkrex 29978. Defining property of . (Contributed by NM, 15-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem4 29973* Lemma for lshpkrex 29978. Part of showing linearity of . (Contributed by NM, 16-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem5 29974* Lemma for lshpkrex 29978. Part of showing linearity of . (Contributed by NM, 16-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem6 29975* Lemma for lshpkrex 29978. Show linearlity of . (Contributed by NM, 17-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrcl 29976* The set defined by hyperplane is a linear functional. (Contributed by NM, 17-Jul-2014.)
LSHyp                                   Scalar                            LFnl

Theoremlshpkr 29977* The kernel of functional is the hyperplane defining it. (Contributed by NM, 17-Jul-2014.)
LSHyp                                   Scalar                            LKer

Theoremlshpkrex 29978* There exists a functional whose kernel equals a given hyperplane. Part of Th. 1.27 of Barbu and Precupanu, Convexity and Optimization in Banach Spaces. (Contributed by NM, 17-Jul-2014.)
LSHyp       LFnl       LKer

Theoremlshpset2N 29979* The set of all hyperplanes of a left module or left vector space equals the set of all kernels of nonzero functionals. (Contributed by NM, 17-Jul-2014.) (New usage is discouraged.)
Scalar              LSHyp       LFnl       LKer

TheoremislshpkrN 29980* The predicate "is a hyperplane" (of a left module or left vector space). TODO: should it be or as in lshpkrex 29978? Both standards seem to be used randomly throughout set.mm; we should decide on a preferred one. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
Scalar              LSHyp       LFnl       LKer

Theoremlfl1dim 29981* Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
Scalar       LFnl       LKer

Theoremlfl1dim2N 29982* Equivalent expressions for a 1-dim subspace (ray) of functionals. TODO: delete this if not useful; lfl1dim 29981 may be more compatible with lspsn 16080. (Contributed by NM, 24-Oct-2014.) (New usage is discouraged.)
Scalar       LFnl       LKer

19.26.5  Opposite rings and dual vector spaces

Syntaxcld 29983 Extend class notation with left dualvector space.
LDual

Definitiondf-ldual 29984* Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows us to reuse our existing collection of left vector space theorems. The restriction on allows it to be a set; see ofmres 6345. Note the operation reversal in the scalar product to allow us to use the original scalar ring instead of the oppr ring, for convenience. (Contributed by NM, 18-Oct-2014.)
LDual LFnl Scalar LFnl LFnl Scalar opprScalar Scalar LFnl Scalar

Theoremldualset 29985* Define the (left) dual of a left vector space (or module) in which the vectors are functionals. In many texts, this is defined as a right vector space, but by reversing the multiplication we achieve a left vector space, as is done in definition of dual vector space in [Holland95] p. 218. This allows us to reuse our existing collection of left vector space theorems. Note the operation reversal in the scalar product to allow us to use the original scalar ring instead of the oppr ring, for convenience. (Contributed by NM, 18-Oct-2014.)
LFnl       LDual       Scalar                     oppr                     Scalar

Theoremldualvbase 29986 The vectors of a dual space are functionals of the original space. (Contributed by NM, 18-Oct-2014.)
LFnl       LDual

Theoremldualelvbase 29987 Utility theorem for converting a functional to a vector of the dual space in order to use standard vector theorems. (Contributed by NM, 6-Jan-2015.)
LFnl       LDual

Theoremldualfvadd 29988 Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.)
LFnl       Scalar              LDual

Theoremldualvadd 29989 Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.)
LFnl       Scalar              LDual

Theoremldualvaddcl 29990 The value of vector addition in the dual of a vector space is a functional. (Contributed by NM, 21-Oct-2014.)
LFnl       LDual

Theoremldualvaddval 29991 The value of the value of vector addition in the dual of a vector space. (Contributed by NM, 7-Jan-2015.)
Scalar              LFnl       LDual

Theoremldualsca 29992 The ring of scalars of the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
Scalar       oppr       LDual       Scalar

Theoremldualsbase 29993 Base set of scalar ring for the dual of a vector space. (Contributed by NM, 24-Oct-2014.)
Scalar              LDual       Scalar

TheoremldualsaddN 29994 Scalar addition for the dual of a vector space. (Contributed by NM, 24-Oct-2014.) (New usage is discouraged.)
Scalar              LDual       Scalar

Theoremldualsmul 29995 Scalar multiplication for the dual of a vector space. (Contributed by NM, 19-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
Scalar                     LDual       Scalar

Theoremldualfvs 29996* Scalar product operation for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
LFnl              Scalar                     LDual

Theoremldualvs 29997 Scalar product operation value (which is a functional) for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
LFnl              Scalar                     LDual

Theoremldualvsval 29998 Value of scalar product operation value for the dual of a vector space. (Contributed by NM, 18-Oct-2014.)
LFnl              Scalar                     LDual

Theoremldualvscl 29999 The scalar product operation value is a functional. (Contributed by NM, 18-Oct-2014.)
LFnl       Scalar              LDual

Theoremldualvaddcom 30000 Commutative law for vector (functional) addition. (Contributed by NM, 17-Jan-2015.)
LFnl       LDual

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