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Theorem List for Metamath Proof Explorer - 29801-29900   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremlshpinN 29801 The intersection of two different hyperplanes is not a hyperplane. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
LSHyp

Theoremlsatset 29802* The set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
LSAtoms

Theoremislsat 29803* The predicate "is a 1-dim subspace (atom)" (of a left module or left vector space). (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
LSAtoms

Theoremlsatlspsn2 29804 The span of a non-zero singleton is an atom. TODO: make this obsolete and use lsatlspsn 29805 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
LSAtoms

Theoremlsatlspsn 29805 The span of a non-zero singleton is an atom. (Contributed by NM, 16-Jan-2015.)
LSAtoms

Theoremislsati 29806* A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014.)
LSAtoms

Theoremlsateln0 29807* A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.)
LSAtoms

Theoremlsatlss 29808 The set of 1-dim subspaces is a set of subspaces. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
LSAtoms

Theoremlsatlssel 29809 An atom is a subspace. (Contributed by NM, 25-Aug-2014.)
LSAtoms

Theoremlsatssv 29810 An atom is a set of vectors. (Contributed by NM, 27-Feb-2015.)
LSAtoms

Theoremlsatn0 29811 A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 22941 analog.) (Contributed by NM, 25-Aug-2014.)
LSAtoms

Theoremlsatspn0 29812 The span of a vector is an atom iff the vector is nonzero. (Contributed by NM, 4-Feb-2015.)
LSAtoms

Theoremlsator0sp 29813 The span of a vector is either an atom or the zero subspace. (Contributed by NM, 15-Mar-2015.)
LSAtoms

Theoremlsatssn0 29814 A subspace (or any class) including an atom is nonzero. (Contributed by NM, 3-Feb-2015.)
LSAtoms

Theoremlsatcmp 29815 If two atoms are comparable, they are equal. (atsseq 22943 analog.) TODO: can lspsncmp 15885 shorten this? (Contributed by NM, 25-Aug-2014.)
LSAtoms

Theoremlsatcmp2 29816 If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 29815. TODO: can lspsncmp 15885 shorten this? (Contributed by NM, 3-Feb-2015.)
LSAtoms

Theoremlsatel 29817 A nonzero vector in an atom determines the atom. (Contributed by NM, 25-Aug-2014.)
LSAtoms

TheoremlsatelbN 29818 A nonzero vector in an atom determines the atom. (Contributed by NM, 3-Feb-2015.) (New usage is discouraged.)
LSAtoms

Theoremlsat2el 29819 Two atoms sharing a nonzero vector are equal. (Contributed by NM, 8-Mar-2015.)
LSAtoms

Theoremlsmsat 29820* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 30616 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.)
LSAtoms

TheoremlsatfixedN 29821* Show equality with the span of the sum of two vectors, one of which () is fixed in advance. Compare lspfixed 15897. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
LSAtoms

Theoremlsmsatcv 29822 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 22247 analog.) Explicit atom version of lsmcv 15910. (Contributed by NM, 29-Oct-2014.)
LSAtoms

Theoremlssatomic 29823* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 22954 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlssats 29824* The lattice of subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. Hypothesis (shatomistici 22957 analog.) (Contributed by NM, 9-Apr-2014.)
LSAtoms

Theoremlpssat 29825* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 22959 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremlrelat 29826* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 22960 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremlssatle 29827* The ordering of two subspaces is determined by the atoms under them. (chrelat3 22967 analog.) (Contributed by NM, 29-Oct-2014.)
LSAtoms

Theoremlssat 29828* Two subspaces in a proper subset relationship imply the existence of a 1-dim subspace less than or equal to one but not the other. (chpssati 22959 analog.) (Contributed by NM, 9-Apr-2014.)
LSAtoms

Theoremislshpat 29829* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 29792. (Contributed by NM, 11-Jan-2015.)
LSHyp       LSAtoms

Syntaxclcv 29830 Extend class notation with the covering relation for a left module or left vector space.
L

Definitiondf-lcv 29831* Define the covering relation for subspaces of a left vector space. Similar to Definition 3.2.18 of [PtakPulmannova] p. 68. Ptak/Pulmannova's notation L is read " covers " or " is covered by " , and it means that is larger than and there is nothing in between. See lcvbr 29833 for binary relation. (df-cv 22875 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvfbr 29832* The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvbr 29833* The covers relation for a left vector space (or a left module). (cvbr 22878 analog.) (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvbr2 29834* The covers relation for a left vector space (or a left module). (cvbr2 22879 analog.) (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvbr3 29835* The covers relation for a left vector space (or a left module). (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvpss 29836 The covers relation implies proper subset. (cvpss 22881 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvnbtwn 29837 The covers relation implies no in-betweenness. (cvnbtwn 22882 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvntr 29838 The covers relation is not transitive. (cvntr 22888 analog.) (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvnbtwn2 29839 The covers relation implies no in-betweenness. (cvnbtwn2 22883 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvnbtwn3 29840 The covers relation implies no in-betweenness. (cvnbtwn3 22884 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlsmcv2 29841 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 22889 analog.) (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvat 29842* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 22962 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoremlsatcv0 29843 An atom covers the zero subspace. (atcv0 22938 analog.) (Contributed by NM, 7-Jan-2015.)
LSAtoms       L

Theoremlsatcveq0 29844 A subspace covered by an atom must be the zero subspace. (atcveq0 22944 analog.) (Contributed by NM, 7-Jan-2015.)
LSAtoms       L

Theoremlsat0cv 29845 A subspace is an atom iff it covers the zero subspace. This could serve as an alternate definition of an atom. TODO: this is a quick-and-dirty proof that could probably be more efficient. (Contributed by NM, 14-Mar-2015.)
LSAtoms       L

Theoremlcvexchlem1 29846 Lemma for lcvexch 29851. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem2 29847 Lemma for lcvexch 29851. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem3 29848 Lemma for lcvexch 29851. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem4 29849 Lemma for lcvexch 29851. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem5 29850 Lemma for lcvexch 29851. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexch 29851 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 22965 analog.) TODO: combine some lemmas. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvp 29852 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22971 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlcv1 29853 Covering property of a subspace plus an atom. (chcv1 22951 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlcv2 29854 Covering property of a subspace plus an atom. (chcv2 22952 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatexch 29855 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 22977 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatnle 29856 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 22972 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatnem0 29857 The meet of distinct atoms is the zero subspace. (atnemeq0 22973 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatexch1 29858 The atom exch1ange property. (hlatexch1 30206 analog.) (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremlsatcv0eq 29859 If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 22975 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcv1 29860 Two atoms covering the zero subspace are equal. (atcv1 22976 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcvatlem 29861 Lemma for lsatcvat 29862. (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatcvat 29862 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 22982 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatcvat2 29863 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 22983 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcvat3 29864 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22992 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremislshpcv 29865 Hyperplane properties expressed with covers relation. (Contributed by NM, 11-Jan-2015.)
LSHyp       L

Theoreml1cvpat 29866 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 30286 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoreml1cvat 29867 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 30287 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoremlshpat 29868 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 30854 analog.) TODO: This changes in l1cvpat 29866 and l1cvat 29867 to , which in turn change in islshpcv 29865 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

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

Syntaxclfn 29869 Extend class notation with all linear functionals of a left module or left vector space.
LFnl

Definitiondf-lfl 29870* 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 29871* 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 29872* The predicate "is a linear functional". (Contributed by NM, 15-Apr-2014.)
Scalar                                   LFnl

Theoremlfli 29873 Property of a linear functional. (lnfnli 22636 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar                                   LFnl

Theoremislfld 29874* Properties that determine a linear functional. TODO: use this in place of islfl 29872 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)
Scalar                                   LFnl

Theoremlflf 29875 A linear functional is a function from vectors to scalars. (lnfnfi 22637 analog.) (Contributed by NM, 15-Apr-2014.)
Scalar                     LFnl

Theoremlflcl 29876 A linear functional value is a scalar. (Contributed by NM, 15-Apr-2014.)
Scalar                     LFnl

Theoremlfl0 29877 A linear functional is zero at the zero vector. (lnfn0i 22638 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar                     LFnl

Theoremlfladd 29878 Property of a linear functional. (lnfnaddi 22639 analog.) (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl

Theoremlflsub 29879 Property of a linear functional. (lnfnaddi 22639 analog.) (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl

Theoremlflmul 29880 Property of a linear functional. (lnfnmuli 22640 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar                                   LFnl

Theoremlfl0f 29881 The zero function is a functional. (Contributed by NM, 16-Apr-2014.)
Scalar                     LFnl

Theoremlfl1 29882* A non-zero functional has a value of 1 at some argument. (Contributed by NM, 16-Apr-2014.)
Scalar                            LFnl

Theoremlfladdcl 29883 Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)
Scalar              LFnl

Theoremlfladdcom 29884 Commutativity of functional addition. (Contributed by NM, 19-Oct-2014.)
Scalar              LFnl

Theoremlfladdass 29885 Associativity of functional addition. (Contributed by NM, 19-Oct-2014.)
Scalar              LFnl

Theoremlfladd0l 29886 Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.)
Scalar                     LFnl

Theoremlflnegcl 29887* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 29958, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Scalar                     LFnl

Theoremlflnegl 29888* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 29958, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Scalar                     LFnl

Theoremlflvscl 29889 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 29890 Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                            LFnl

Theoremlflvsdi2 29891 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                            LFnl

Theoremlflvsdi2a 29892 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-Oct-2014.)
Scalar                            LFnl

Theoremlflvsass 29893 Associative law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                     LFnl

Theoremlfl0sc 29894 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 29895 The scalar product with the zero functional is the zero functional. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
Scalar

Theoremlfl1sc 29896 The (right vector space) scalar product of a functional with one is the functional. (Contributed by NM, 21-Oct-2014.)
Scalar       LFnl

Syntaxclk 29897 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 29898* 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 29899* The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar              LFnl       LKer

Theoremlkrval 29900 Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
Scalar              LFnl       LKer

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