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Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlshpne0 29101 The member of the span in the hyperplane definition does not belong to the hyperplane. (Contributed by NM, 14-Jul-2014.)
LSHyp

Theoremlshpdisj 29102 A hyperplane and the span in the hyperplane definition are disjoint. (Contributed by NM, 3-Jul-2014.)
LSHyp

Theoremlshpcmp 29103 If two hyperplanes are comparable, they are equal. (Contributed by NM, 9-Oct-2014.)
LSHyp

TheoremlshpinN 29104 The intersection of two different hyperplanes is not a hyperplane. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
LSHyp

Theoremlsatset 29105* 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 29106* 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 29107 The span of a non-zero singleton is an atom. TODO: make this obsolete and use lsatlspsn 29108 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
LSAtoms

Theoremlsatlspsn 29108 The span of a non-zero singleton is an atom. (Contributed by NM, 16-Jan-2015.)
LSAtoms

Theoremislsati 29109* 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 29110* 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 29111 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 29112 An atom is a subspace. (Contributed by NM, 25-Aug-2014.)
LSAtoms

Theoremlsatssv 29113 An atom is a set of vectors. (Contributed by NM, 27-Feb-2015.)
LSAtoms

Theoremlsatn0 29114 A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 23696 analog.) (Contributed by NM, 25-Aug-2014.)
LSAtoms

Theoremlsatspn0 29115 The span of a vector is an atom iff the vector is nonzero. (Contributed by NM, 4-Feb-2015.)
LSAtoms

Theoremlsator0sp 29116 The span of a vector is either an atom or the zero subspace. (Contributed by NM, 15-Mar-2015.)
LSAtoms

Theoremlsatssn0 29117 A subspace (or any class) including an atom is nonzero. (Contributed by NM, 3-Feb-2015.)
LSAtoms

Theoremlsatcmp 29118 If two atoms are comparable, they are equal. (atsseq 23698 analog.) TODO: can lspsncmp 16115 shorten this? (Contributed by NM, 25-Aug-2014.)
LSAtoms

Theoremlsatcmp2 29119 If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 29118. TODO: can lspsncmp 16115 shorten this? (Contributed by NM, 3-Feb-2015.)
LSAtoms

Theoremlsatel 29120 A nonzero vector in an atom determines the atom. (Contributed by NM, 25-Aug-2014.)
LSAtoms

TheoremlsatelbN 29121 A nonzero vector in an atom determines the atom. (Contributed by NM, 3-Feb-2015.) (New usage is discouraged.)
LSAtoms

Theoremlsat2el 29122 Two atoms sharing a nonzero vector are equal. (Contributed by NM, 8-Mar-2015.)
LSAtoms

Theoremlsmsat 29123* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 29919 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.)
LSAtoms

TheoremlsatfixedN 29124* Show equality with the span of the sum of two vectors, one of which () is fixed in advance. Compare lspfixed 16127. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
LSAtoms

Theoremlsmsatcv 29125 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 23002 analog.) Explicit atom version of lsmcv 16140. (Contributed by NM, 29-Oct-2014.)
LSAtoms

Theoremlssatomic 29126* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 23709 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlssats 29127* 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 23712 analog.) (Contributed by NM, 9-Apr-2014.)
LSAtoms

Theoremlpssat 29128* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 23714 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremlrelat 29129* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 23715 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremlssatle 29130* The ordering of two subspaces is determined by the atoms under them. (chrelat3 23722 analog.) (Contributed by NM, 29-Oct-2014.)
LSAtoms

Theoremlssat 29131* 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 23714 analog.) (Contributed by NM, 9-Apr-2014.)
LSAtoms

Theoremislshpat 29132* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 29095. (Contributed by NM, 11-Jan-2015.)
LSHyp       LSAtoms

Syntaxclcv 29133 Extend class notation with the covering relation for a left module or left vector space.
L

Definitiondf-lcv 29134* 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 29136 for binary relation. (df-cv 23630 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvfbr 29135* The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvbr 29136* The covers relation for a left vector space (or a left module). (cvbr 23633 analog.) (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvbr2 29137* The covers relation for a left vector space (or a left module). (cvbr2 23634 analog.) (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvbr3 29138* The covers relation for a left vector space (or a left module). (Contributed by NM, 9-Jan-2015.)
L

Theoremlcvpss 29139 The covers relation implies proper subset. (cvpss 23636 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvnbtwn 29140 The covers relation implies no in-betweenness. (cvnbtwn 23637 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvntr 29141 The covers relation is not transitive. (cvntr 23643 analog.) (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvnbtwn2 29142 The covers relation implies no in-betweenness. (cvnbtwn2 23638 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlcvnbtwn3 29143 The covers relation implies no in-betweenness. (cvnbtwn3 23639 analog.) (Contributed by NM, 7-Jan-2015.)
L

Theoremlsmcv2 29144 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 23644 analog.) (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvat 29145* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 23717 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoremlsatcv0 29146 An atom covers the zero subspace. (atcv0 23693 analog.) (Contributed by NM, 7-Jan-2015.)
LSAtoms       L

Theoremlsatcveq0 29147 A subspace covered by an atom must be the zero subspace. (atcveq0 23699 analog.) (Contributed by NM, 7-Jan-2015.)
LSAtoms       L

Theoremlsat0cv 29148 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 29149 Lemma for lcvexch 29154. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem2 29150 Lemma for lcvexch 29154. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem3 29151 Lemma for lcvexch 29154. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem4 29152 Lemma for lcvexch 29154. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexchlem5 29153 Lemma for lcvexch 29154. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvexch 29154 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 23720 analog.) TODO: combine some lemmas. (Contributed by NM, 10-Jan-2015.)
L

Theoremlcvp 29155 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 23726 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlcv1 29156 Covering property of a subspace plus an atom. (chcv1 23706 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlcv2 29157 Covering property of a subspace plus an atom. (chcv2 23707 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatexch 29158 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 23732 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatnle 29159 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 23727 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatnem0 29160 The meet of distinct atoms is the zero subspace. (atnemeq0 23728 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatexch1 29161 The atom exch1ange property. (hlatexch1 29509 analog.) (Contributed by NM, 14-Jan-2015.)
LSAtoms

Theoremlsatcv0eq 29162 If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 23730 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcv1 29163 Two atoms covering the zero subspace are equal. (atcv1 23731 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcvatlem 29164 Lemma for lsatcvat 29165. (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatcvat 29165 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 23737 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms

Theoremlsatcvat2 29166 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 23738 analog.) (Contributed by NM, 10-Jan-2015.)
LSAtoms       L

Theoremlsatcvat3 29167 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 23747 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms

Theoremislshpcv 29168 Hyperplane properties expressed with covers relation. (Contributed by NM, 11-Jan-2015.)
LSHyp       L

Theoreml1cvpat 29169 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 29589 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoreml1cvat 29170 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 29590 analog.) (Contributed by NM, 11-Jan-2015.)
LSAtoms       L

Theoremlshpat 29171 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 30157 analog.) TODO: This changes in l1cvpat 29169 and l1cvat 29170 to , which in turn change in islshpcv 29168 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 29172 Extend class notation with all linear functionals of a left module or left vector space.
LFnl

Definitiondf-lfl 29173* 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 29174* 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 29175* The predicate "is a linear functional". (Contributed by NM, 15-Apr-2014.)
Scalar                                   LFnl

Theoremlfli 29176 Property of a linear functional. (lnfnli 23391 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar                                   LFnl

Theoremislfld 29177* Properties that determine a linear functional. TODO: use this in place of islfl 29175 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)
Scalar                                   LFnl

Theoremlflf 29178 A linear functional is a function from vectors to scalars. (lnfnfi 23392 analog.) (Contributed by NM, 15-Apr-2014.)
Scalar                     LFnl

Theoremlflcl 29179 A linear functional value is a scalar. (Contributed by NM, 15-Apr-2014.)
Scalar                     LFnl

Theoremlfl0 29180 A linear functional is zero at the zero vector. (lnfn0i 23393 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Scalar                     LFnl

Theoremlfladd 29181 Property of a linear functional. (lnfnaddi 23394 analog.) (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl

Theoremlflsub 29182 Property of a linear functional. (lnfnaddi 23394 analog.) (Contributed by NM, 18-Apr-2014.)
Scalar                            LFnl

Theoremlflmul 29183 Property of a linear functional. (lnfnmuli 23395 analog.) (Contributed by NM, 16-Apr-2014.)
Scalar                                   LFnl

Theoremlfl0f 29184 The zero function is a functional. (Contributed by NM, 16-Apr-2014.)
Scalar                     LFnl

Theoremlfl1 29185* A non-zero functional has a value of 1 at some argument. (Contributed by NM, 16-Apr-2014.)
Scalar                            LFnl

Theoremlfladdcl 29186 Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)
Scalar              LFnl

Scalar              LFnl

Scalar              LFnl

Theoremlfladd0l 29189 Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.)
Scalar                     LFnl

Theoremlflnegcl 29190* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 29261, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Scalar                     LFnl

Theoremlflnegl 29191* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 29261, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
Scalar                     LFnl

Theoremlflvscl 29192 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 29193 Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                            LFnl

Theoremlflvsdi2 29194 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                            LFnl

Theoremlflvsdi2a 29195 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-Oct-2014.)
Scalar                            LFnl

Theoremlflvsass 29196 Associative law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
Scalar                     LFnl

Theoremlfl0sc 29197 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 29198 The scalar product with the zero functional is the zero functional. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
Scalar

Theoremlfl1sc 29199 The (right vector space) scalar product of a functional with one is the functional. (Contributed by NM, 21-Oct-2014.)
Scalar       LFnl

Syntaxclk 29200 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

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