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Theorem List for Metamath Proof Explorer - 15701-15800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlssvsubcl 15701 Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssvancl1 15702 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 15889. Can it be used along with lspsnne1 15870, lspsnne2 15871 to shorten this proof? (Contributed by NM, 14-May-2015.)

Theoremlssvancl2 15703 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 15704 The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)

Theoremlsssn0 15705 The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlss0ss 15706 The zero subspace is included in every subspace. (sh0le 22019 analog.) (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssle0 15707 No subspace is smaller than the zero subspace. (shle0 22021 analog.) (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssne0 15708* A nonzero subspace has a nonzero vector. (shne0i 22027 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)

Theoremlssneln0 15709 A vector which doesn't belong to a subspace is nonzero. (Contributed by NM, 14-May-2015.)

Theoremlssssr 15710* Conclude subspace ordering from nonzero vector membership. (ssrdv 3185 analog.) (Contributed by NM, 17-Aug-2014.)

Theoremlssvacl 15711 Closure of vector addition in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssvscl 15712 Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlssvnegcl 15713 Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.)

Theoremlsssubg 15714 All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
SubGrp

Theoremlsssssubg 15715 All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
SubGrp

Theoremislss3 15716 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 15717 A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
s

Theoremlsslss 15718 The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
s

Theoremislss4 15719* 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 15720* 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 15721 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 15722 The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssmre 15723 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 15724 Submodules are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
ACS

Theoremprdsvscacl 15725* Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s                                                               Scalar

Theoremprdslmodd 15726* The product of a family of left modules is a left module. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s                            Scalar

Theorempwslmod 15727 The product of a family of left modules is a left module. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Syntaxclspn 15728 Extend class notation with span of a set of vectors.

Definitiondf-lsp 15729* Define span of a set of vectors of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)

Theoremlspfval 15730* The span function for a left vector space (or a left module). (df-span 21888 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspf 15731 The span operator on a left module maps subsets to subsets. (Contributed by Stefan O'Rear, 12-Dec-2014.)

Theoremlspval 15732* The span of a set of vectors (in a left module). (spanval 21912 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspcl 15733 The span of a set of vectors is a subspace. (spancl 21915 analog.) (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspsncl 15734 The span of a singleton is a subspace (frequently used special case of lspcl 15733). (Contributed by NM, 17-Jul-2014.)

Theoremlspprcl 15735 The span of a pair is a subspace (frequently used special case of lspcl 15733). (Contributed by NM, 11-Apr-2015.)

Theoremlsptpcl 15736 The span of an unordered triple is a subspace (frequently used special case of lspcl 15733). (Contributed by NM, 22-May-2015.)

Theoremlspsnsubg 15737 The span of a singleton is an additive subgroup (frequently used special case of lspcl 15733). (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp

Theorem00lsp 15738 fvco4i 5597 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)

Theoremlspid 15739 The span of a subspace is itself. (spanid 21926 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspssv 15740 A span is a set of vectors. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspss 15741 Span preserves subset ordering. (spanss 21927 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspssid 15742 A set of vectors is a subset of its span. (spanss2 21924 analog.) (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspidm 15743 The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspun 15744 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 15745 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 15746 Moore closure generalizes module span. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls

Theoremlspsnss 15747 The span of the singleton of a subspace member is included in the subspace. (spansnss 22150 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)

Theoremlspsnel3 15748 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 22151 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspprss 15749 The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.)

Theoremlspsnid 15750 A vector belongs to the span of its singleton. (spansnid 22142 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspsnel6 15751 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 15752 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.)

Theoremlspsnel5a 15753 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)

Theoremlspprid1 15754 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)

Theoremlspprid2 15755 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)

Theoremlspprvacl 15756 The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.)

Theoremlssats2 15757* A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.)

Theoremlspsneli 15758 A scalar product with a vector belongs to the span of its singleton. (spansnmul 22143 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlspsn 15759* Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlspsnel 15760* Member of span of the singleton of a vector. (elspansn 22145 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlspsnvsi 15761 Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.)
Scalar

Theoremlspsnss2 15762* Comparable spans of singletons must have proportional vectors. See lspsneq 15875 for equal span version. (Contributed by NM, 7-Jun-2015.)
Scalar

Theoremlspsnneg 15763 Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)

Theoremlspsnsub 15764 Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.)

Theoremlspsn0 15765 Span of the singleton of the zero vector. (spansn0 22120 analog.) (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)

Theoremlsp0 15766 Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)

Theoremlspuni0 15767 Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.)

Theoremlspun0 15768 The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.)

Theoremlspsneq0 15769 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 15770 Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.)

Theoremlmodindp1 15771 Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.)

Theoremlsslsp 15772 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 15773 The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.)
s

Theoremlsspropd 15774* 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 15775* 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 15776 Extend class notation with the generator of left module hom-sets.
LMHom

Syntaxclmim 15777 The class of left module isomorphism sets.
LMIso

Syntaxclmic 15778 The class of the left module isomorphism relation.
𝑚

Definitiondf-lmhm 15779* 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 15780* 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 15781 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 15782 Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
LMHom

Theoremlmimfn 15783 Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.)
LMIso

Theoremislmhm 15784* 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 15785* Property of a module homomorphism, similar to ismhm 14417. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Scalar       Scalar                                   LMHom

Theoremlmhmlem 15786 Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Scalar       Scalar       LMHom

Theoremlmhmsca 15787 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 15788 A homomorphism of left modules is a homomorphism of groups. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmlmod2 15789 A homomorphism of left modules has a left module as codomain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmlmod1 15790 A homomorphism of left modules has a left module as domain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmf 15791 A homomorphism of left modules is a function. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmlin 15792 A homomorphism of left modules is -linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Scalar                                   LMHom

Theoremlmodvsinv 15793 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Scalar

Theoremlmodvsinv2 15794 Multiplying a negated vector by a scalar. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Scalar

Theoremislmhm2 15795* A one-equation proof of linearity of a left module homomorphism, similar to df-lss 15690. (Contributed by Mario Carneiro, 7-Oct-2015.)
Scalar       Scalar                                          LMHom

Theoremislmhmd 15796* Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
Scalar       Scalar                                                 LMHom

Theorem0lmhm 15797 The constant zero linear function between two modules. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Scalar       Scalar       LMHom

Theoremidlmhm 15798 The identity function on a module is linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
LMHom

Theoreminvlmhm 15799 The negative function on a module is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
LMHom

Theoremlmhmco 15800 The composition of two module-linear functions is module-linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
LMHom LMHom LMHom

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