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Theorem List for Metamath Proof Explorer - 16001-16100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlmodsubdi 16001 Scalar multiplication distributive law for subtraction. (hvsubdistr1 22551 analog, with longer proof since our scalar multiplication is not commutative.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlmodsubdir 16002 Scalar multiplication distributive law for subtraction. (hvsubdistr2 22552 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlmodsubeq0 16003 If the difference between two vectors is zero, they are equal. (hvsubeq0 22570 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodsubid 16004 Subtraction of a vector from itself. (hvsubid 22528 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvsghm 16005* Scalar multiplication of the vector space by a fixed scalar is an automorphism of the addiive group of vectors. (Contributed by Mario Carneiro, 5-May-2015.)
Scalar

Theoremlmodprop2d 16006* If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 16007 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
Scalar       Scalar

Theoremlmodpropd 16007* If two structures have the same components (properties), one is a left module iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 27-Jun-2015.)
Scalar       Scalar

10.6.2  Subspaces and spans in a left module

Syntaxclss 16008 Extend class notation with linear subspaces of a left module or left vector space.

Definitiondf-lss 16009* Define the set of linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
Scalar

Theoremlssset 16010* The set of all (not necessarily closed) linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 15-Jul-2014.)
Scalar

Theoremislss 16011* The predicate "is a subspace" (of a left module or left vector space). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
Scalar

Theoremislssd 16012* Properties that determine a subspace of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
Scalar

Theoremlssss 16013 A subspace is a set of vectors. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)

Theoremlssel 16014 A subspace member is a vector. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)

Theoremlss1 16015 The set of vectors in a left module is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssuni 16016 The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.)

Theoremlssn0 16017 A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)

Theorem00lss 16018 The empty structure has no subspaces (for use with fvco4i 5801). (Contributed by Stefan O'Rear, 31-Mar-2015.)

Theoremlsscl 16019 Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
Scalar

Theoremlssvsubcl 16020 Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssvancl1 16021 Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. TODO: notice similarity to lspindp3 16208. Can it be used along with lspsnne1 16189, lspsnne2 16190 to shorten this proof? (Contributed by NM, 14-May-2015.)

Theoremlssvancl2 16022 Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. (Contributed by NM, 20-May-2015.)

Theoremlss0cl 16023 The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)

Theoremlsssn0 16024 The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlss0ss 16025 The zero subspace is included in every subspace. (sh0le 22942 analog.) (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssle0 16026 No subspace is smaller than the zero subspace. (shle0 22944 analog.) (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssne0 16027* A nonzero subspace has a nonzero vector. (shne0i 22950 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)

Theoremlssneln0 16028 A vector which doesn't belong to a subspace is nonzero. (Contributed by NM, 14-May-2015.)

Theoremlssssr 16029* Conclude subspace ordering from nonzero vector membership. (ssrdv 3354 analog.) (Contributed by NM, 17-Aug-2014.)

Theoremlssvacl 16030 Closure of vector addition in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssvscl 16031 Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlssvnegcl 16032 Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.)

Theoremlsssubg 16033 All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
SubGrp

Theoremlsssssubg 16034 All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
SubGrp

Theoremislss3 16035 A linear subspace of a module is a subset which is a module in its own right. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
s

Theoremlsslmod 16036 A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
s

Theoremlsslss 16037 The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
s

Theoremislss4 16038* A linear subspace is a subgroup which respects scalar multiplication. (Contributed by Stefan O'Rear, 11-Dec-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Scalar                                   SubGrp

Theoremlss1d 16039* One-dimensional subspace (or zero-dimensional if is the zero vector). (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlssintcl 16040 The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssincl 16041 The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssmre 16042 The subspaces of a module comprise a Moore system on the vectors of the module. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Moore

Theoremlssacs 16043 Submodules are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
ACS

Theoremprdsvscacl 16044* Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s                                                               Scalar

Theoremprdslmodd 16045* The product of a family of left modules is a left module. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s                            Scalar

Theorempwslmod 16046 The product of a family of left modules is a left module. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Syntaxclspn 16047 Extend class notation with span of a set of vectors.

Definitiondf-lsp 16048* Define span of a set of vectors of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)

Theoremlspfval 16049* The span function for a left vector space (or a left module). (df-span 22811 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspf 16050 The span operator on a left module maps subsets to subsets. (Contributed by Stefan O'Rear, 12-Dec-2014.)

Theoremlspval 16051* The span of a set of vectors (in a left module). (spanval 22835 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspcl 16052 The span of a set of vectors is a subspace. (spancl 22838 analog.) (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspsncl 16053 The span of a singleton is a subspace (frequently used special case of lspcl 16052). (Contributed by NM, 17-Jul-2014.)

Theoremlspprcl 16054 The span of a pair is a subspace (frequently used special case of lspcl 16052). (Contributed by NM, 11-Apr-2015.)

Theoremlsptpcl 16055 The span of an unordered triple is a subspace (frequently used special case of lspcl 16052). (Contributed by NM, 22-May-2015.)

Theoremlspsnsubg 16056 The span of a singleton is an additive subgroup (frequently used special case of lspcl 16052). (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp

Theorem00lsp 16057 fvco4i 5801 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)

Theoremlspid 16058 The span of a subspace is itself. (spanid 22849 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspssv 16059 A span is a set of vectors. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspss 16060 Span preserves subset ordering. (spanss 22850 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspssid 16061 A set of vectors is a subset of its span. (spanss2 22847 analog.) (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspidm 16062 The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspun 16063 The span of union is the span of the union of spans. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspssp 16064 If a set of vectors is a subset of a subspace, then the span of those vectors is also contained in the subspace. (Contributed by Mario Carneiro, 4-Sep-2014.)

Theoremmrclsp 16065 Moore closure generalizes module span. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls

Theoremlspsnss 16066 The span of the singleton of a subspace member is included in the subspace. (spansnss 23073 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)

Theoremlspsnel3 16067 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 23074 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspprss 16068 The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.)

Theoremlspsnid 16069 A vector belongs to the span of its singleton. (spansnid 23065 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspsnel6 16070 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)

Theoremlspsnel5 16071 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.)

Theoremlspsnel5a 16072 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)

Theoremlspprid1 16073 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)

Theoremlspprid2 16074 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)

Theoremlspprvacl 16075 The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.)

Theoremlssats2 16076* A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.)

Theoremlspsneli 16077 A scalar product with a vector belongs to the span of its singleton. (spansnmul 23066 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlspsn 16078* Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlspsnel 16079* Member of span of the singleton of a vector. (elspansn 23068 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlspsnvsi 16080 Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.)
Scalar

Theoremlspsnss2 16081* Comparable spans of singletons must have proportional vectors. See lspsneq 16194 for equal span version. (Contributed by NM, 7-Jun-2015.)
Scalar

Theoremlspsnneg 16082 Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)

Theoremlspsnsub 16083 Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.)

Theoremlspsn0 16084 Span of the singleton of the zero vector. (spansn0 23043 analog.) (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)

Theoremlsp0 16085 Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)

Theoremlspuni0 16086 Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.)

Theoremlspun0 16087 The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.)

Theoremlspsneq0 16088 Span of the singleton is the zero subspace iff the vector is zero. (Contributed by NM, 27-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspsneq0b 16089 Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.)

Theoremlmodindp1 16090 Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.)

Theoremlsslsp 16091 Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) TODO: Shouldn't we swap and since we are computing a property of ? (Like we say sin 0 = 0 and not 0 = sin 0.) - NM 15-Mar-2015.
s

Theoremlss0v 16092 The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.)
s

Theoremlsspropd 16093* If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
Scalar       Scalar

Theoremlsppropd 16094* If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
Scalar       Scalar

10.6.3  Homomorphisms and isomorphisms of left modules

Syntaxclmhm 16095 Extend class notation with the generator of left module hom-sets.
LMHom

Syntaxclmim 16096 The class of left module isomorphism sets.
LMIso

Syntaxclmic 16097 The class of the left module isomorphism relation.
𝑚

Definitiondf-lmhm 16098* A homomorphism of left modules is a group homomorphism which additionally preserves the scalar product. This requires both structures to be left modules over the same ring. (Contributed by Stefan O'Rear, 31-Dec-2014.)
LMHom Scalar Scalar

Definitiondf-lmim 16099* An isomorphism of modules is a homomorphism which is also a bijection, i.e. it preserves equality as well as the group and scalar operations. (Contributed by Stefan O'Rear, 21-Jan-2015.)
LMIso LMHom

Definitiondf-lmic 16100 Two modules are said to be isomorphic iff they are connected by at least one isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑚 LMIso

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