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Theorem List for Metamath Proof Explorer - 15801-15900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlmhmplusg 15801 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
LMHom LMHom LMHom

Theoremlmhmvsca 15802 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 15803 A bijective module homomorphism is also converse homomorphic. (Contributed by Stefan O'Rear, 25-Jan-2015.)
LMHom LMHom

Theoremlmhmima 15804 The image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmpreima 15805 The inverse image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmlsp 15806 Homomorphisms preserve spans. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmrnlss 15807 The range of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremlmhmkerlss 15808 The kernel of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.)
LMHom

Theoremreslmhm 15809 Restriction of a homomorphism to a subspace. (Contributed by Stefan O'Rear, 1-Jan-2015.)
s        LMHom LMHom

Theoremreslmhm2 15810 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 15811 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 15812 The equalizer of two module homomorphisms is a subspace. (Contributed by Stefan O'Rear, 7-Mar-2015.)
LMHom LMHom

Theoremlspextmo 15813* 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 15814* Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
s                      LMHom

Theoremislmim 15815 An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
LMIso LMHom

Theoremlmimf1o 15816 An isomorphism of left modules is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.)
LMIso

Theoremlmimlmhm 15817 An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
LMIso LMHom

Theoremlmimgim 15818 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 15819 An isomorphism of left modules is a homomorphism whose converse is a homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
LMIso LMHom LMHom

Theoremlmimcnv 15820 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 15821 The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑚 LMIso

Theorembrlmici 15822 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
LMIso 𝑚

Theoremlmiclcl 15823 Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑚

Theoremlmicrcl 15824 Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.)
𝑚

Theoremlmicsym 15825 Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015.)
𝑚 𝑚

Theoremlmhmpropd 15826* Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar       Scalar       Scalar       Scalar                                                 LMHom LMHom

10.6.4  Subspace sum; bases for a left module

Syntaxclbs 15827 Extend class notation with the set of bases for a vector space.
LBasis

Definitiondf-lbs 15828* Define the set of bases to a left module or left vector space. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis Scalar

Theoremislbs 15829* The predicate " is a basis for the left module or vector space ". A subset of the base set is a basis if zero is not in the set, it spans the set, and no nonzero multiple of an element of the basis is in the span of the rest of the family. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 14-Jan-2015.)
Scalar                     LBasis

Theoremlbsss 15830 A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis

Theoremlbsel 15831 An element of a basis is a vector. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis

Theoremlbssp 15832 The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis

Theoremlbsind 15833 A basis is linearly independent; that is, every element has a span which trivially intersects the span of the remainder of the basis. (Contributed by Mario Carneiro, 12-Jan-2015.)
LBasis              Scalar

Theoremlbsind2 15834 A basis is linearly independent; that is, every element is not in the span of the remainder of the basis. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 12-Jan-2015.)
LBasis              Scalar

Theoremlbspss 15835 No proper subset of a basis spans the space. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis              Scalar

Theoremlsmcl 15836 The sum of two subspaces is a subspace. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)

Theoremlsmspsn 15837* Member of subspace sum of spans of singletons. (Contributed by NM, 8-Apr-2015.)
Scalar

Theoremlsmelval2 15838* Subspace sum membership in terms of a sum of 1-dim subspaces (atoms), which can be useful for treating subspaces as projective lattice elements. (Contributed by NM, 9-Aug-2014.)

Theoremlsmsp 15839 Subspace sum in terms of span. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)

Theoremlsmsp2 15840 Subspace sum of spans of subsets is the span of their union. (spanuni 22123 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)

Theoremlsmssspx 15841 Subspace sum (in its extended domain) is a subset of the span of the union of its arguments. (Contributed by NM, 6-Aug-2014.)

Theoremlsmpr 15842 The span of a pair of vectors equals the sum of the spans of their singletons. (Contributed by NM, 13-Jan-2015.)

Theoremlsppreli 15843 A vector expressed as a sum belongs to the span of its components. (Contributed by NM, 9-Apr-2015.)
Scalar

Theoremlsmelpr 15844 Two ways to say that a vector belongs to the span of a pair of vectors. (Contributed by NM, 14-Jan-2015.)

Theoremlsppr0 15845 The span of a vector paired with zero equals the span of the singleton of the vector. (Contributed by NM, 29-Aug-2014.)

Theoremlsppr 15846* Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.)
Scalar

Theoremlspprel 15847* Member of the span of a pair of vectors. (Contributed by NM, 10-Apr-2015.)
Scalar

Theoremlspprabs 15848 Absorption of vector sum into span of pair. (Contributed by NM, 27-Apr-2015.)

Theoremlspvadd 15849 The span of a vector sum is included in the span of its arguments. (Contributed by NM, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)

Theoremlspsntri 15850 Triangle-type inequality for span of a singleton. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)

Theoremlspsntrim 15851 Triangle-type inequality for span of a singleton of vector difference. (Contributed by NM, 25-Apr-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)

Theoremlbspropd 15852* If two structures have the same components (properties), they have the same set of bases. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
Scalar       Scalar                                          LBasis LBasis

Theorempj1lmhm 15853 The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
s LMHom

Theorempj1lmhm2 15854 The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
s LMHom s

10.7  Vector spaces

10.7.1  Definition and basic properties

Syntaxclvec 15855 Extend class notation with class of all left vector spaces.

Definitiondf-lvec 15856 Define the class of all left vector spaces. A left vector space over a division ring is an Abelian group (vectors) together with a division ring (scalars) and a left scalar product connecting them. Some authors call this a "left module over a division ring", reserving "vector space" for those where the division ring multiplication is commutative i.e. a field. (Contributed by NM, 11-Nov-2013.)
Scalar

Theoremislvec 15857 The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.)
Scalar

Theoremlvecdrng 15858 The set of scalars of a left vector space is a division ring. (Contributed by NM, 17-Apr-2014.)
Scalar

Theoremlveclmod 15859 A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)

Theoremlsslvec 15860 A vector subspace is a vector space. (Contributed by NM, 14-Mar-2015.)
s

Theoremlvecvs0or 15861 If a scalar product is zero, one of its factors must be zero. (hvmul0or 21604 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecvsn0 15862 A scalar product is nonzero iff both of its factors are nonzero. (Contributed by NM, 3-Jan-2015.)
Scalar

Theoremlssvs0or 15863 If a scalar product belongs to a subspace, either the scalar component is zero or the vector component also belongs. (Contributed by NM, 5-Apr-2015.)
Scalar

Theoremlvecvscan 15864 Cancellation law for scalar multiplication. (hvmulcan 21651 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecvscan2 15865 Cancellation law for scalar multiplication. (hvmulcan2 21652 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecinv 15866 Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.)
Scalar

Theoremlspsnvs 15867 A non-zero scalar product does not change the span of a singleton. (spansncol 22147 analog.) (Contributed by NM, 23-Apr-2014.)
Scalar

Theoremlspsneleq 15868 Membership relation that implies equality of spans. (spansneleq 22149 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspsncmp 15869 Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.)

Theoremlspsnne1 15870 Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.)

Theoremlspsnne2 15871 Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015.)

Theoremlspsnnecom 15872 Swap two vectors with different spans. (Contributed by NM, 20-May-2015.)

Theoremlspabs2 15873 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)

Theoremlspabs3 15874 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)

Theoremlspsneq 15875* Equal spans of singletons must have proportional vectors. See lspsnss2 15762 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.)
Scalar

Theoremlspsneu 15876* Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.)
Scalar

Theoremlspsnel4 15877 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 22152 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspdisj 15878 The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.)

Theoremlspdisjb 15879 The a nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.)

Theoremlspdisj2 15880 Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.)

Theoremlspfixed 15881* Show membership in the span of the sum of two vectors, one of which () is fixed in advance. (Contributed by NM, 27-May-2015.)

Theoremlspexch 15882 Exchange property for span of a pair. TODO: see if a version with Y,Z and X,Z reversed will shorten proofs (analogous to lspexchn1 15883 vs. lspexchn2 15884); look for lspexch 15882 and prcom 3705 in same proof. TODO: would a hypothesis of instead of { Z } ) ` be better overall? This would be shorter and also satisfy the condition. Here and also lspindp* and all proofs affected by them (all in NM's mathbox); there are 58 hypotheses with the pattern as of 24-May-2015. (Contributed by NM, 11-Apr-2015.)

Theoremlspexchn1 15883 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 15882 to see if this will shorten proofs. (Contributed by NM, 20-May-2015.)

Theoremlspexchn2 15884 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 15882 to see if this will shorten proofs. (Contributed by NM, 24-May-2015.)

Theoremlspindpi 15885 Partial independence property. (Contributed by NM, 23-Apr-2015.)

Theoremlspindp1 15886 Alternate way to say 3 vectors are mutually independent (swap 1st and 2nd). (Contributed by NM, 11-Apr-2015.)

Theoremlspindp2l 15887 Alternate way to say 3 vectors are mutually independent (rotate left). (Contributed by NM, 10-May-2015.)

Theoremlspindp2 15888 Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.)

Theoremlspindp3 15889 Independence of 2 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)

Theoremlspindp4 15890 (Partial) independence of 3 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)

Theoremlvecindp 15891 Compute the coefficient in a sum with an independent vector (first conjunct), which can then be removed to continue with the remaining vectors summed in expressions and (second conjunct). Typically is the span of the remaining vectors. (Contributed by NM, 5-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Scalar

Theoremlvecindp2 15892 Sums of independent vectors must have equal coefficients. (Contributed by NM, 22-Mar-2015.)
Scalar

Theoremlspsnsubn0 15893 Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.)

Theoremlsmcv 15894 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 22231 analog.) TODO: ugly proof; can it be shortened? (Contributed by NM, 2-Oct-2014.)

Theoremlspsolvlem 15895* Lemma for lspsolv 15896. (Contributed by Mario Carneiro, 25-Jun-2014.)
Scalar

Theoremlspsolv 15896 If is in the span of but not , then is in the span of . (Contributed by Mario Carneiro, 25-Jun-2014.)

Theoremlssacsex 15897* In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 15724 by lspsolv 15896. (Contributed by David Moews, 1-May-2017.)
mrCls              ACS

Theoremlspsnat 15898 There is no subspace strictly between the zero subspace and the span of a vector (i.e. a 1-dimensional subspace is an atom). (h1datomi 22160 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)

Theoremlspsncv0 15899* The span of a singleton covers the zero subspace, using Definition 3.2.18 of [PtakPulmannova] p. 68 for "covers".) (Contributed by NM, 12-Aug-2014.)

Theoremlsppratlem1 15900 Lemma for lspprat 15906. Let (if there is no such then is the zero subspace), and let (assuming the conclusion is false). The goal is to write , in terms of , , which would normally be done by solving the system of linear equations. The span equivalent of this process is lspsolv 15896 (hence the name), which we use extensively below. In this lemma, we show that since , either or . (Contributed by NM, 29-Aug-2014.)

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