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Theorem List for Metamath Proof Explorer - 29701-29800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlsatssn0 29701 A subspace (or any class) including an atom is nonzero. (Contributed by NM, 3-Feb-2015.)
LSAtoms

Theoremlsatcmp 29702 If two atoms are comparable, they are equal. (atsseq 23840 analog.) TODO: can lspsncmp 16178 shorten this? (Contributed by NM, 25-Aug-2014.)
LSAtoms

Theoremlsatcmp2 29703 If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 29702. TODO: can lspsncmp 16178 shorten this? (Contributed by NM, 3-Feb-2015.)
LSAtoms

Theoremlsatel 29704 A nonzero vector in an atom determines the atom. (Contributed by NM, 25-Aug-2014.)
LSAtoms

TheoremlsatelbN 29705 A nonzero vector in an atom determines the atom. (Contributed by NM, 3-Feb-2015.) (New usage is discouraged.)
LSAtoms

Theoremlsat2el 29706 Two atoms sharing a nonzero vector are equal. (Contributed by NM, 8-Mar-2015.)
LSAtoms

Theoremlsmsat 29707* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 30503 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.)
LSAtoms

TheoremlsatfixedN 29708* Show equality with the span of the sum of two vectors, one of which () is fixed in advance. Compare lspfixed 16190. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
LSAtoms

Theoremlsmsatcv 29709 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 23144 analog.) Explicit atom version of lsmcv 16203. (Contributed by NM, 29-Oct-2014.)
LSAtoms

Theoremlssatomic 29710* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 23851 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlssats 29711* 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 23854 analog.) (Contributed by NM, 9-Apr-2014.)
LSAtoms

Theoremlpssat 29712* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 23856 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremlrelat 29713* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 23857 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremlssatle 29714* The ordering of two subspaces is determined by the atoms under them. (chrelat3 23864 analog.) (Contributed by NM, 29-Oct-2014.)
LSAtoms

Theoremlssat 29715* 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 23856 analog.) (Contributed by NM, 9-Apr-2014.)
LSAtoms

Theoremislshpat 29716* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 29679. (Contributed by NM, 11-Jan-2015.)
LSHyp       LSAtoms

Syntaxclcv 29717 Extend class notation with the covering relation for a left module or left vector space.
L

Definitiondf-lcv 29718* 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 29720 for binary relation. (df-cv 23772 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvfbr 29719* The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvbr 29720* The covers relation for a left vector space (or a left module). (cvbr 23775 analog.) (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvbr2 29721* The covers relation for a left vector space (or a left module). (cvbr2 23776 analog.) (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvbr3 29722* The covers relation for a left vector space (or a left module). (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvpss 29723 The covers relation implies proper subset. (cvpss 23778 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvnbtwn 29724 The covers relation implies no in-betweenness. (cvnbtwn 23779 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvntr 29725 The covers relation is not transitive. (cvntr 23785 analog.) (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvnbtwn2 29726 The covers relation implies no in-betweenness. (cvnbtwn2 23780 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvnbtwn3 29727 The covers relation implies no in-betweenness. (cvnbtwn3 23781 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlsmcv2 29728 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 23786 analog.) (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvat 29729* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 23859 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoremlsatcv0 29730 An atom covers the zero subspace. (atcv0 23835 analog.) (Contributed by NM, 7-Jan-2015.)
LSAtoms       L

Theoremlsatcveq0 29731 A subspace covered by an atom must be the zero subspace. (atcveq0 23841 analog.) (Contributed by NM, 7-Jan-2015.)
LSAtoms       L

Theoremlsat0cv 29732 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 29733 Lemma for lcvexch 29738. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem2 29734 Lemma for lcvexch 29738. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem3 29735 Lemma for lcvexch 29738. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem4 29736 Lemma for lcvexch 29738. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem5 29737 Lemma for lcvexch 29738. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexch 29738 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 23862 analog.) TODO: combine some lemmas. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvp 29739 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 23868 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlcv1 29740 Covering property of a subspace plus an atom. (chcv1 23848 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlcv2 29741 Covering property of a subspace plus an atom. (chcv2 23849 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatexch 29742 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 23874 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatnle 29743 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 23869 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatnem0 29744 The meet of distinct atoms is the zero subspace. (atnemeq0 23870 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatexch1 29745 The atom exch1ange property. (hlatexch1 30093 analog.) (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremlsatcv0eq 29746 If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 23872 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcv1 29747 Two atoms covering the zero subspace are equal. (atcv1 23873 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcvatlem 29748 Lemma for lsatcvat 29749. (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatcvat 29749 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 23879 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatcvat2 29750 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 23880 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcvat3 29751 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 23889 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremislshpcv 29752 Hyperplane properties expressed with covers relation. (Contributed by NM, 11-Jan-2015.)
LSHyp       L

Theoreml1cvpat 29753 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 30173 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoreml1cvat 29754 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 30174 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoremlshpat 29755 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 30741 analog.) TODO: This changes in l1cvpat 29753 and l1cvat 29754 to , which in turn change in islshpcv 29752 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 29756 Extend class notation with all linear functionals of a left module or left vector space.
LFnl

Definitiondf-lfl 29757* 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 29758* 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 29759* The predicate "is a linear functional". (Contributed by NM, 15-Apr-2014.)
Scalar                                   LFnl

Theoremlfli 29760 Property of a linear functional. (lnfnli 23533 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar                                   LFnl

Theoremislfld 29761* Properties that determine a linear functional. TODO: use this in place of islfl 29759 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)
Scalar                                   LFnl

Theoremlflf 29762 A linear functional is a function from vectors to scalars. (lnfnfi 23534 analog.) (Contributed by NM, 15-Apr-2014.)
Scalar                     LFnl

Theoremlflcl 29763 A linear functional value is a scalar. (Contributed by NM, 15-Apr-2014.)
Scalar                     LFnl

Theoremlfl0 29764 A linear functional is zero at the zero vector. (lnfn0i 23535 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar                     LFnl

Theoremlfladd 29765 Property of a linear functional. (lnfnaddi 23536 analog.) (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl

Theoremlflsub 29766 Property of a linear functional. (lnfnaddi 23536 analog.) (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl

Theoremlflmul 29767 Property of a linear functional. (lnfnmuli 23537 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar                                   LFnl

Theoremlfl0f 29768 The zero function is a functional. (Contributed by NM, 16-Apr-2014.)
Scalar                     LFnl

Theoremlfl1 29769* A non-zero functional has a value of 1 at some argument. (Contributed by NM, 16-Apr-2014.)
Scalar                            LFnl

Theoremlfladdcl 29770 Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)
Scalar              LFnl

Scalar              LFnl

Scalar              LFnl

Theoremlfladd0l 29773 Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.)
Scalar                     LFnl

Theoremlflnegcl 29774* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 29845, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Scalar                     LFnl

Theoremlflnegl 29775* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 29845, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Scalar                     LFnl

Theoremlflvscl 29776 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 29777 Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                            LFnl

Theoremlflvsdi2 29778 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                            LFnl

Theoremlflvsdi2a 29779 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-Oct-2014.)
Scalar                            LFnl

Theoremlflvsass 29780 Associative law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                     LFnl

Theoremlfl0sc 29781 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 29782 The scalar product with the zero functional is the zero functional. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
Scalar

Theoremlfl1sc 29783 The (right vector space) scalar product of a functional with one is the functional. (Contributed by NM, 21-Oct-2014.)
Scalar       LFnl

Syntaxclk 29784 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 29785* 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 29786* The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar              LFnl       LKer

Theoremlkrval 29787 Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
Scalar              LFnl       LKer

Theoremellkr 29788 Membership in the kernel of a functional. (elnlfn 23421 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar              LFnl       LKer

Theoremlkrval2 29789* Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
Scalar              LFnl       LKer

Theoremellkr2 29790 Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.)
Scalar              LFnl       LKer

Theoremlkrcl 29791 A member of the kernel of a functional is a vector. (Contributed by NM, 16-Apr-2014.)
LFnl       LKer

Theoremlkrf0 29792 The value of a functional at a member of its kernel is zero. (Contributed by NM, 16-Apr-2014.)
Scalar              LFnl       LKer

Theoremlkr0f 29793 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 17-Apr-2014.)
Scalar                     LFnl       LKer

Theoremlkrlss 29794 The kernel of a linear functional is a subspace. (nlelshi 23553 analog.) (Contributed by NM, 16-Apr-2014.)
LFnl       LKer

Theoremlkrssv 29795 The kernel of a linear functional is a set of vectors. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremlkrsc 29796 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 29797 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 29798* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl       LKer

Theoremeqlkr2 29799* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 10-Oct-2014.)
Scalar                            LFnl       LKer

Theoremeqlkr3 29800 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

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