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Theorem List for Metamath Proof Explorer - 15501-15600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlmodvnegid 15501 Addition of a vector with its negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvneg1 15502 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

TheoremlmodvsnegOLD 15503 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Scalar

Theoremlmodvsneg 15504 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Scalar

Theoremlmodvsubcl 15505 Closure of vector subtraction. (hvsubcl 21427 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodcom 15506 Left module vector sum is commutative. (Contributed by Gérard Lang, 25-Jun-2014.)

Theoremlmodabl 15507 A left module is an abelian group (of vectors, under addition). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)

Theoremlmodcmn 15508 A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
CMnd

Theoremlmodnegadd 15509 Distribute negation through addition of scalar products. (Contributed by NM, 9-Apr-2015.)
Scalar

Theoremlmod4 15510 Commutative/associative law for left module vector sum. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvsubadd 15511 Relationship between vector subtraction and addition. (hvsubadd 21486 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvaddsub4 15512 Vector addition/subtraction law. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvpncan 15513 Addition/subtraction cancellation law for vectors. (hvpncan 21448 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvnpcan 15514 Cancellation law for vector subtraction (npcan 8940 analog). (Contributed by NM, 19-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvsubval2 15515 Value of vector subtraction in terms of addition. (hvsubval 21426 analog.) (Contributed by NM, 31-Mar-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlmodsubvs 15516 Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.)
Scalar

Theoremlmodsubdi 15517 Scalar multiplication distributive law for subtraction. (hvsubdistr1 21458 analog, with longer proof since our scalar multiplication is not commutative.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlmodsubdir 15518 Scalar multiplication distributive law for subtraction. (hvsubdistr2 21459 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlmodsubeq0 15519 If the difference between two vectors is zero, they are equal. (hvsubeq0 21477 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodsubid 15520 Subtraction of a vector from itself. (hvsubid 21435 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlmodvsghm 15521* 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 15522* If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 15523 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
Scalar       Scalar

Theoremlmodpropd 15523* 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 15524 Extend class notation with linear subspaces of a left module or left vector space.

Definitiondf-lss 15525* Define the set of linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
Scalar

Theoremlssset 15526* 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 15527* 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 15528* 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 15529 A subspace is a set of vectors. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)

Theoremlssel 15530 A subspace member is a vector. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)

Theoremlss1 15531 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 15532 The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.)

Theoremlssn0 15533 A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)

Theorem00lss 15534 The empty structure has no subspaces (for use with fvco4i 5449). (Contributed by Stefan O'Rear, 31-Mar-2015.)

Theoremlsscl 15535 Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
Scalar

Theoremlssvsubcl 15536 Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssvancl1 15537 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 15724. Can it be used along with lspsnne1 15705, lspsnne2 15706 to shorten this proof? (Contributed by NM, 14-May-2015.)

Theoremlssvancl2 15538 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 15539 The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)

Theoremlsssn0 15540 The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlss0ss 15541 The zero subspace is included in every subspace. (sh0le 21849 analog.) (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssle0 15542 No subspace is smaller than the zero subspace. (shle0 21851 analog.) (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssne0 15543* A nonzero subspace has a nonzero vector. (shne0i 21857 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)

Theoremlssneln0 15544 A vector which doesn't belong to a subspace is nonzero. (Contributed by NM, 14-May-2015.)

Theoremlssssr 15545* Conclude subspace ordering from nonzero vector membership. (ssrdv 3106 analog.) (Contributed by NM, 17-Aug-2014.)

Theoremlssvacl 15546 Closure of vector addition in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssvscl 15547 Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlssvnegcl 15548 Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.)

Theoremlsssubg 15549 All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
SubGrp

Theoremlsssssubg 15550 All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
SubGrp

Theoremislss3 15551 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 15552 A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
s

Theoremlsslss 15553 The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
s

Theoremislss4 15554* 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 15555* 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 15556 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 15557 The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlssmre 15558 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 15559 Submodules are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
ACS

Theoremprdsvscacl 15560* Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s                                                               Scalar

Theoremprdslmodd 15561* The product of a family of left modules is a left module. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s                            Scalar

Theorempwslmod 15562 The product of a family of left modules is a left module. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Syntaxclspn 15563 Extend class notation with span of a set of vectors.

Definitiondf-lsp 15564* Define span of a set of vectors of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)

Theoremlspfval 15565* The span function for a left vector space (or a left module). (df-span 21718 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspf 15566 The span operator on a left module maps subsets to subsets. (Contributed by Stefan O'Rear, 12-Dec-2014.)

Theoremlspval 15567* The span of a set of vectors (in a left module). (spanval 21742 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspcl 15568 The span of a set of vectors is a subspace. (spancl 21745 analog.) (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspsncl 15569 The span of a singleton is a subspace (frequently used special case of lspcl 15568). (Contributed by NM, 17-Jul-2014.)

Theoremlspprcl 15570 The span of a pair is a subspace (frequently used special case of lspcl 15568). (Contributed by NM, 11-Apr-2015.)

Theoremlsptpcl 15571 The span of an unordered triple is a subspace (frequently used special case of lspcl 15568). (Contributed by NM, 22-May-2015.)

Theoremlspsnsubg 15572 The span of a singleton is an additive subgroup (frequently used special case of lspcl 15568). (Contributed by Mario Carneiro, 21-Apr-2016.)
SubGrp

Theorem00lsp 15573 fvco4i 5449 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)

Theoremlspid 15574 The span of a subspace is itself. (spanid 21756 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspssv 15575 A span is a set of vectors. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspss 15576 Span preserves subset ordering. (spanss 21757 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspssid 15577 A set of vectors is a subset of its span. (spanss2 21754 analog.) (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspidm 15578 The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspun 15579 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 15580 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 15581 Moore closure generalizes module span. (Contributed by Stefan O'Rear, 31-Jan-2015.)
mrCls

Theoremlspsnss 15582 The span of the singleton of a subspace member is included in the subspace. (spansnss 21980 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)

Theoremlspsnel3 15583 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 21981 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspprss 15584 The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.)

Theoremlspsnid 15585 A vector belongs to the span of its singleton. (spansnid 21972 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)

Theoremlspsnel6 15586 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 15587 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.)

Theoremlspsnel5a 15588 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)

Theoremlspprid1 15589 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)

Theoremlspprid2 15590 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)

Theoremlspprvacl 15591 The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.)

Theoremlssats2 15592* A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.)

Theoremlspsneli 15593 A scalar product with a vector belongs to the span of its singleton. (spansnmul 21973 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlspsn 15594* Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlspsnel 15595* Member of span of the singleton of a vector. (elspansn 21975 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Scalar

Theoremlspsnvsi 15596 Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.)
Scalar

Theoremlspsnss2 15597* Comparable spans of singletons must have proportional vectors. See lspsneq 15710 for equal span version. (Contributed by NM, 7-Jun-2015.)
Scalar

Theoremlspsnneg 15598 Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)

Theoremlspsnsub 15599 Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.)

Theoremlspsn0 15600 Span of the singleton of the zero vector. (spansn0 21950 analog.) (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)

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