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

Theorempwslmod 16001 The product of a family of left modules is a left module. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Syntaxclspn 16002 Extend class notation with span of a set of vectors.

Definitiondf-lsp 16003* Define span of a set of vectors of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)

Theoremlspfval 16004* The span function for a left vector space (or a left module). (df-span 22764 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspf 16005 The span operator on a left module maps subsets to subsets. (Contributed by Stefan O'Rear, 12-Dec-2014.)

Theoremlspval 16006* The span of a set of vectors (in a left module). (spanval 22788 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspcl 16007 The span of a set of vectors is a subspace. (spancl 22791 analog.) (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspsncl 16008 The span of a singleton is a subspace (frequently used special case of lspcl 16007). (Contributed by NM, 17-Jul-2014.)

Theoremlspprcl 16009 The span of a pair is a subspace (frequently used special case of lspcl 16007). (Contributed by NM, 11-Apr-2015.)

Theoremlsptpcl 16010 The span of an unordered triple is a subspace (frequently used special case of lspcl 16007). (Contributed by NM, 22-May-2015.)

Theoremlspsnsubg 16011 The span of a singleton is an additive subgroup (frequently used special case of lspcl 16007). (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp

Theorem00lsp 16012 fvco4i 5760 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)

Theoremlspid 16013 The span of a subspace is itself. (spanid 22802 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspssv 16014 A span is a set of vectors. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspss 16015 Span preserves subset ordering. (spanss 22803 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspssid 16016 A set of vectors is a subset of its span. (spanss2 22800 analog.) (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspidm 16017 The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspun 16018 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 16019 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 16020 Moore closure generalizes module span. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls

Theoremlspsnss 16021 The span of the singleton of a subspace member is included in the subspace. (spansnss 23026 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)

Theoremlspsnel3 16022 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 23027 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspprss 16023 The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.)

Theoremlspsnid 16024 A vector belongs to the span of its singleton. (spansnid 23018 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspsnel6 16025 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 16026 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.)

Theoremlspsnel5a 16027 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)

Theoremlspprid1 16028 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)

Theoremlspprid2 16029 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)

Theoremlspprvacl 16030 The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.)

Theoremlssats2 16031* A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.)

Theoremlspsneli 16032 A scalar product with a vector belongs to the span of its singleton. (spansnmul 23019 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlspsn 16033* Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlspsnel 16034* Member of span of the singleton of a vector. (elspansn 23021 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlspsnvsi 16035 Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.)
Scalar

Theoremlspsnss2 16036* Comparable spans of singletons must have proportional vectors. See lspsneq 16149 for equal span version. (Contributed by NM, 7-Jun-2015.)
Scalar

Theoremlspsnneg 16037 Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)

Theoremlspsnsub 16038 Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.)

Theoremlspsn0 16039 Span of the singleton of the zero vector. (spansn0 22996 analog.) (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)

Theoremlsp0 16040 Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)

Theoremlspuni0 16041 Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.)

Theoremlspun0 16042 The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.)

Theoremlspsneq0 16043 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 16044 Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.)

Theoremlmodindp1 16045 Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.)

Theoremlsslsp 16046 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 16047 The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.)
s

Theoremlsspropd 16048* 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 16049* 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 16050 Extend class notation with the generator of left module hom-sets.
LMHom

Syntaxclmim 16051 The class of left module isomorphism sets.
LMIso

Syntaxclmic 16052 The class of the left module isomorphism relation.
𝑚

Definitiondf-lmhm 16053* 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 16054* 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 16055 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

Theoremreldmlmhm 16056 Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
LMHom

Theoremlmimfn 16057 Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.)
LMIso

Theoremislmhm 16058* Property of being a homomorphism of left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
Scalar       Scalar                                   LMHom

Theoremislmhm3 16059* Property of a module homomorphism, similar to ismhm 14695. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Scalar       Scalar                                   LMHom

Theoremlmhmlem 16060 Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Scalar       Scalar       LMHom

Theoremlmhmsca 16061 A homomorphism of left modules constrains both modules to the same ring of scalars. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Scalar       Scalar       LMHom

Theoremlmghm 16062 A homomorphism of left modules is a homomorphism of groups. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmlmod2 16063 A homomorphism of left modules has a left module as codomain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmlmod1 16064 A homomorphism of left modules has a left module as domain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmf 16065 A homomorphism of left modules is a function. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmlin 16066 A homomorphism of left modules is -linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Scalar                                   LMHom

Theoremlmodvsinv 16067 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Scalar

Theoremlmodvsinv2 16068 Multiplying a negated vector by a scalar. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Scalar

Theoremislmhm2 16069* A one-equation proof of linearity of a left module homomorphism, similar to df-lss 15964. (Contributed by Mario Carneiro, 7-Oct-2015.)
Scalar       Scalar                                          LMHom

Theoremislmhmd 16070* Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
Scalar       Scalar                                                 LMHom

Theorem0lmhm 16071 The constant zero linear function between two modules. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Scalar       Scalar       LMHom

Theoremidlmhm 16072 The identity function on a module is linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
LMHom

Theoreminvlmhm 16073 The negative function on a module is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
LMHom

Theoremlmhmco 16074 The composition of two module-linear functions is module-linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
LMHom LMHom LMHom

Theoremlmhmplusg 16075 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
LMHom LMHom LMHom

Theoremlmhmvsca 16076 The pointwise scalar product of a linear function and a constant is linear, over a commutative ring. (Contributed by Mario Carneiro, 22-Sep-2015.)
Scalar              LMHom LMHom

Theoremlmhmf1o 16077 A bijective module homomorphism is also converse homomorphic. (Contributed by Stefan O'Rear, 25-Jan-2015.)
LMHom LMHom

Theoremlmhmima 16078 The image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmpreima 16079 The inverse image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmlsp 16080 Homomorphisms preserve spans. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmrnlss 16081 The range of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmkerlss 16082 The kernel of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremreslmhm 16083 Restriction of a homomorphism to a subspace. (Contributed by Stefan O'Rear, 1-Jan-2015.)
s        LMHom LMHom

Theoremreslmhm2 16084 Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
s               LMHom LMHom

Theoremreslmhm2b 16085 Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
s               LMHom LMHom

Theoremlmhmeql 16086 The equalizer of two module homomorphisms is a subspace. (Contributed by Stefan O'Rear, 7-Mar-2015.)
LMHom LMHom

Theoremlspextmo 16087* A linear function is completely determined (or overdetermined) by its values on a spanning subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by NM, 17-Jun-2017.)
LMHom

Theorempwsdiaglmhm 16088* Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s                      LMHom

Theoremislmim 16089 An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
LMIso LMHom

Theoremlmimf1o 16090 An isomorphism of left modules is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.)
LMIso

Theoremlmimlmhm 16091 An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
LMIso LMHom

Theoremlmimgim 16092 An isomorphism of modules is an isomorphism of groups. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
LMIso GrpIso

Theoremislmim2 16093 An isomorphism of left modules is a homomorphism whose converse is a homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
LMIso LMHom LMHom

Theoremlmimcnv 16094 The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
LMIso LMIso

Theorembrlmic 16095 The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑚 LMIso

Theorembrlmici 16096 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
LMIso 𝑚

Theoremlmiclcl 16097 Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑚

Theoremlmicrcl 16098 Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.)
𝑚

Theoremlmicsym 16099 Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015.)
𝑚 𝑚

Theoremlmhmpropd 16100* Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar       Scalar       Scalar       Scalar                                                 LMHom LMHom

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