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Theorem List for Metamath Proof Explorer - 29501-29600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxclcv 29501 Extend class notation with the covering relation for a left module or left vector space.
L

Definitiondf-lcv 29502* 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 29504 for binary relation. (df-cv 23735 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvfbr 29503* The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvbr 29504* The covers relation for a left vector space (or a left module). (cvbr 23738 analog.) (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvbr2 29505* The covers relation for a left vector space (or a left module). (cvbr2 23739 analog.) (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvbr3 29506* The covers relation for a left vector space (or a left module). (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvpss 29507 The covers relation implies proper subset. (cvpss 23741 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvnbtwn 29508 The covers relation implies no in-betweenness. (cvnbtwn 23742 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvntr 29509 The covers relation is not transitive. (cvntr 23748 analog.) (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvnbtwn2 29510 The covers relation implies no in-betweenness. (cvnbtwn2 23743 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvnbtwn3 29511 The covers relation implies no in-betweenness. (cvnbtwn3 23744 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlsmcv2 29512 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 23749 analog.) (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvat 29513* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 23822 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoremlsatcv0 29514 An atom covers the zero subspace. (atcv0 23798 analog.) (Contributed by NM, 7-Jan-2015.)
LSAtoms       L

Theoremlsatcveq0 29515 A subspace covered by an atom must be the zero subspace. (atcveq0 23804 analog.) (Contributed by NM, 7-Jan-2015.)
LSAtoms       L

Theoremlsat0cv 29516 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 29517 Lemma for lcvexch 29522. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem2 29518 Lemma for lcvexch 29522. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem3 29519 Lemma for lcvexch 29522. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem4 29520 Lemma for lcvexch 29522. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem5 29521 Lemma for lcvexch 29522. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexch 29522 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 23825 analog.) TODO: combine some lemmas. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvp 29523 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 23831 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlcv1 29524 Covering property of a subspace plus an atom. (chcv1 23811 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlcv2 29525 Covering property of a subspace plus an atom. (chcv2 23812 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatexch 29526 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 23837 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatnle 29527 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 23832 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatnem0 29528 The meet of distinct atoms is the zero subspace. (atnemeq0 23833 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatexch1 29529 The atom exch1ange property. (hlatexch1 29877 analog.) (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremlsatcv0eq 29530 If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 23835 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcv1 29531 Two atoms covering the zero subspace are equal. (atcv1 23836 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcvatlem 29532 Lemma for lsatcvat 29533. (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatcvat 29533 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 23842 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatcvat2 29534 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 23843 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcvat3 29535 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 23852 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremislshpcv 29536 Hyperplane properties expressed with covers relation. (Contributed by NM, 11-Jan-2015.)
LSHyp       L

Theoreml1cvpat 29537 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 29957 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoreml1cvat 29538 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 29958 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoremlshpat 29539 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 30525 analog.) TODO: This changes in l1cvpat 29537 and l1cvat 29538 to , which in turn change in islshpcv 29536 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 29540 Extend class notation with all linear functionals of a left module or left vector space.
LFnl

Definitiondf-lfl 29541* 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 29542* 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 29543* The predicate "is a linear functional". (Contributed by NM, 15-Apr-2014.)
Scalar                                   LFnl

Theoremlfli 29544 Property of a linear functional. (lnfnli 23496 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar                                   LFnl

Theoremislfld 29545* Properties that determine a linear functional. TODO: use this in place of islfl 29543 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)
Scalar                                   LFnl

Theoremlflf 29546 A linear functional is a function from vectors to scalars. (lnfnfi 23497 analog.) (Contributed by NM, 15-Apr-2014.)
Scalar                     LFnl

Theoremlflcl 29547 A linear functional value is a scalar. (Contributed by NM, 15-Apr-2014.)
Scalar                     LFnl

Theoremlfl0 29548 A linear functional is zero at the zero vector. (lnfn0i 23498 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar                     LFnl

Theoremlfladd 29549 Property of a linear functional. (lnfnaddi 23499 analog.) (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl

Theoremlflsub 29550 Property of a linear functional. (lnfnaddi 23499 analog.) (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl

Theoremlflmul 29551 Property of a linear functional. (lnfnmuli 23500 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar                                   LFnl

Theoremlfl0f 29552 The zero function is a functional. (Contributed by NM, 16-Apr-2014.)
Scalar                     LFnl

Theoremlfl1 29553* A non-zero functional has a value of 1 at some argument. (Contributed by NM, 16-Apr-2014.)
Scalar                            LFnl

Theoremlfladdcl 29554 Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)
Scalar              LFnl

Theoremlfladdcom 29555 Commutativity of functional addition. (Contributed by NM, 19-Oct-2014.)
Scalar              LFnl

Theoremlfladdass 29556 Associativity of functional addition. (Contributed by NM, 19-Oct-2014.)
Scalar              LFnl

Theoremlfladd0l 29557 Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.)
Scalar                     LFnl

Theoremlflnegcl 29558* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 29629, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Scalar                     LFnl

Theoremlflnegl 29559* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 29629, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Scalar                     LFnl

Theoremlflvscl 29560 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 29561 Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                            LFnl

Theoremlflvsdi2 29562 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                            LFnl

Theoremlflvsdi2a 29563 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-Oct-2014.)
Scalar                            LFnl

Theoremlflvsass 29564 Associative law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                     LFnl

Theoremlfl0sc 29565 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 29566 The scalar product with the zero functional is the zero functional. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
Scalar

Theoremlfl1sc 29567 The (right vector space) scalar product of a functional with one is the functional. (Contributed by NM, 21-Oct-2014.)
Scalar       LFnl

Syntaxclk 29568 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 29569* 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 29570* The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar              LFnl       LKer

Theoremlkrval 29571 Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
Scalar              LFnl       LKer

Theoremellkr 29572 Membership in the kernel of a functional. (elnlfn 23384 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar              LFnl       LKer

Theoremlkrval2 29573* Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
Scalar              LFnl       LKer

Theoremellkr2 29574 Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.)
Scalar              LFnl       LKer

Theoremlkrcl 29575 A member of the kernel of a functional is a vector. (Contributed by NM, 16-Apr-2014.)
LFnl       LKer

Theoremlkrf0 29576 The value of a functional at a member of its kernel is zero. (Contributed by NM, 16-Apr-2014.)
Scalar              LFnl       LKer

Theoremlkr0f 29577 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 17-Apr-2014.)
Scalar                     LFnl       LKer

Theoremlkrlss 29578 The kernel of a linear functional is a subspace. (nlelshi 23516 analog.) (Contributed by NM, 16-Apr-2014.)
LFnl       LKer

Theoremlkrssv 29579 The kernel of a linear functional is a set of vectors. (Contributed by NM, 1-Jan-2015.)
LFnl       LKer

Theoremlkrsc 29580 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 29581 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 29582* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl       LKer

Theoremeqlkr2 29583* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 10-Oct-2014.)
Scalar                            LFnl       LKer

Theoremeqlkr3 29584 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 29585 The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 29572) is the whole vector space. (Contributed by NM, 19-Apr-2014.)
Scalar                                   LFnl       LKer

Theoremlkrlsp2 29586 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 29587 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 29588 The kernel of a nonzero functional is a hyperplane. (Contributed by NM, 29-Jun-2014.)
Scalar              LSHyp       LFnl       LKer

Theoremlkrshp3 29589 The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 17-Jul-2014.)
Scalar              LSHyp       LFnl       LKer

Theoremlkrshpor 29590 The kernel of a functional is either a hyperplane or the full vector space. (Contributed by NM, 7-Oct-2014.)
LSHyp       LFnl       LKer

Theoremlkrshp4 29591 A kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.)
LSHyp       LFnl       LKer

Theoremlshpsmreu 29592* Lemma for lshpkrex 29601. 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 2893 for to ? (Contributed by NM, 4-Jan-2015.)
LSHyp                                          Scalar

Theoremlshpkrlem1 29593* Lemma for lshpkrex 29601. The value of tentative functional is zero iff its argument belongs to hyperplane . (Contributed by NM, 14-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem2 29594* Lemma for lshpkrex 29601. The value of tentative functional is a scalar. (Contributed by NM, 16-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem3 29595* Lemma for lshpkrex 29601. Defining property of . (Contributed by NM, 15-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem4 29596* Lemma for lshpkrex 29601. Part of showing linearity of . (Contributed by NM, 16-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem5 29597* Lemma for lshpkrex 29601. Part of showing linearity of . (Contributed by NM, 16-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrlem6 29598* Lemma for lshpkrex 29601. Show linearlity of . (Contributed by NM, 17-Jul-2014.)
LSHyp                                          Scalar

Theoremlshpkrcl 29599* The set defined by hyperplane is a linear functional. (Contributed by NM, 17-Jul-2014.)
LSHyp                                   Scalar                            LFnl

Theoremlshpkr 29600* The kernel of functional is the hyperplane defining it. (Contributed by NM, 17-Jul-2014.)
LSHyp                                   Scalar                            LKer

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