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

Theoremlsmelval2 15801* 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 15802 Subspace sum in terms of span. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)

Theoremlsmsp2 15803 Subspace sum of spans of subsets is the span of their union. (spanuni 22084 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)

Theoremlsmssspx 15804 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 15805 The span of a pair of vectors equals the sum of the spans of their singletons. (Contributed by NM, 13-Jan-2015.)

Theoremlsppreli 15806 A vector expressed as a sum belongs to the span of its components. (Contributed by NM, 9-Apr-2015.)
Scalar

Theoremlsmelpr 15807 Two ways to say that a vector belongs to the span of a pair of vectors. (Contributed by NM, 14-Jan-2015.)

Theoremlsppr0 15808 The span of a vector paired with zero equals the span of the singleton of the vector. (Contributed by NM, 29-Aug-2014.)

Theoremlsppr 15809* Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.)
Scalar

Theoremlspprel 15810* Member of the span of a pair of vectors. (Contributed by NM, 10-Apr-2015.)
Scalar

Theoremlspprabs 15811 Absorption of vector sum into span of pair. (Contributed by NM, 27-Apr-2015.)

Theoremlspvadd 15812 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 15813 Triangle-type inequality for span of a singleton. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)

Theoremlspsntrim 15814 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 15815* 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 15816 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 15817 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 15818 Extend class notation with class of all left vector spaces.

Definitiondf-lvec 15819 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 15820 The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.)
Scalar

Theoremlvecdrng 15821 The set of scalars of a left vector space is a division ring. (Contributed by NM, 17-Apr-2014.)
Scalar

Theoremlveclmod 15822 A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)

Theoremlsslvec 15823 A vector subspace is a vector space. (Contributed by NM, 14-Mar-2015.)
s

Theoremlvecvs0or 15824 If a scalar product is zero, one of its factors must be zero. (hvmul0or 21565 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecvsn0 15825 A scalar product is nonzero iff both of its factors are nonzero. (Contributed by NM, 3-Jan-2015.)
Scalar

Theoremlssvs0or 15826 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 15827 Cancellation law for scalar multiplication. (hvmulcan 21612 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecvscan2 15828 Cancellation law for scalar multiplication. (hvmulcan2 21613 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecinv 15829 Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.)
Scalar

Theoremlspsnvs 15830 A non-zero scalar product does not change the span of a singleton. (spansncol 22108 analog.) (Contributed by NM, 23-Apr-2014.)
Scalar

Theoremlspsneleq 15831 Membership relation that implies equality of spans. (spansneleq 22110 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspsncmp 15832 Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.)

Theoremlspsnne1 15833 Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.)

Theoremlspsnne2 15834 Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015.)

Theoremlspsnnecom 15835 Swap two vectors with different spans. (Contributed by NM, 20-May-2015.)

Theoremlspabs2 15836 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)

Theoremlspabs3 15837 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)

Theoremlspsneq 15838* Equal spans of singletons must have proportional vectors. See lspsnss2 15725 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.)
Scalar

Theoremlspsneu 15839* Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.)
Scalar

Theoremlspsnel4 15840 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 22113 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspdisj 15841 The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.)

Theoremlspdisjb 15842 The a nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.)

Theoremlspdisj2 15843 Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.)

Theoremlspfixed 15844* 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 15845 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 15846 vs. lspexchn2 15847); look for lspexch 15845 and prcom 3679 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 15846 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 15845 to see if this will shorten proofs. (Contributed by NM, 20-May-2015.)

Theoremlspexchn2 15847 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 15845 to see if this will shorten proofs. (Contributed by NM, 24-May-2015.)

Theoremlspindpi 15848 Partial independence property. (Contributed by NM, 23-Apr-2015.)

Theoremlspindp1 15849 Alternate way to say 3 vectors are mutually independent (swap 1st and 2nd). (Contributed by NM, 11-Apr-2015.)

Theoremlspindp2l 15850 Alternate way to say 3 vectors are mutually independent (rotate left). (Contributed by NM, 10-May-2015.)

Theoremlspindp2 15851 Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.)

Theoremlspindp3 15852 Independence of 2 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)

Theoremlspindp4 15853 (Partial) independence of 3 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)

Theoremlvecindp 15854 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 15855 Sums of independent vectors must have equal coefficients. (Contributed by NM, 22-Mar-2015.)
Scalar

Theoremlspsnsubn0 15856 Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.)

Theoremlsmcv 15857 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 22192 analog.) TODO: ugly proof; can it be shortened? (Contributed by NM, 2-Oct-2014.)

Theoremlspsolvlem 15858* Lemma for lspsolv 15859. (Contributed by Mario Carneiro, 25-Jun-2014.)
Scalar

Theoremlspsolv 15859 If is in the span of but not , then is in the span of . (Contributed by Mario Carneiro, 25-Jun-2014.)

Theoremlssacsex 15860* In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 15687 by lspsolv 15859. (Contributed by David Moews, 1-May-2017.)
mrCls              ACS

Theoremlspsnat 15861 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 22121 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)

Theoremlspsncv0 15862* 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 15863 Lemma for lspprat 15869. 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 15859 (hence the name), which we use extensively below. In this lemma, we show that since , either or . (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem2 15864 Lemma for lspprat 15869. Show that if and are both in (which will be our goal for each of the two cases above), then , contradicting the hypothesis for . (Contributed by NM, 29-Aug-2014.) (Revised by Mario Carneiro, 5-Sep-2014.)

Theoremlsppratlem3 15865 Lemma for lspprat 15869. In the first case of lsppratlem1 15863, since , also , and since and , we have as desired. (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem4 15866 Lemma for lspprat 15869. In the second case of lsppratlem1 15863, and implies and thus as well. (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem5 15867 Lemma for lspprat 15869. Combine the two cases and show a contradiction to under the assumptions on and . (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem6 15868 Lemma for lspprat 15869. Negating the assumption on , we arrive close to the desired conclusion. (Contributed by NM, 29-Aug-2014.)

Theoremlspprat 15869* A proper subspace of the span of a pair of vectors is the span of a singleton (an atom) or the zero subspace (if is zero). Proof suggested by Mario Carneiro, 28-Aug-2014. (Contributed by NM, 29-Aug-2014.)

Theoremislbs2 15870* An equivalent formulation of the basis predicate in a vector space: a subset is a basis iff no element is in the span of the rest of the set. (Contributed by Mario Carneiro, 14-Jan-2015.)
LBasis

Theoremislbs3 15871* An equivalent formulation of the basis predicate: a subset is a basis iff it is a minimal spanning set. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis

Theoremlbsacsbs 15872 Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 15870. (Contributed by David Moews, 1-May-2017.)
mrCls              mrInd       LBasis

Theoremlvecdim 15873 The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 15860 and lbsacsbs 15872 to show that being a basis for a vector space is equivalent to being a basis for the associated algebraic closure system, and then using acsexdimd 14249. (Contributed by David Moews, 1-May-2017.)
LBasis

Theoremlbsextlem1 15874* Lemma for lbsext 15879. The set is the set of all linearly independent sets containing ; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis

Theoremlbsextlem2 15875* Lemma for lbsext 15879. Since is a chain (actually, we only need it to be closed under binary union), the union of the spans of each individual element of is a subspace, and it contains all of (except for our target vector - we are trying to make a linear combination of all the other vectors in some set from ). (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis                                                               []

Theoremlbsextlem3 15876* Lemma for lbsext 15879. A chain in has an upper bound in . (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis                                                               []

Theoremlbsextlem4 15877* Lemma for lbsext 15879. lbsextlem3 15876 satisfies the conditions for the application of Zorn's lemma zorn 8102 (thus invoking AC), and so there is a maximal linearly independent set extending . Here we prove that such a set is a basis. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis

Theoremlbsextg 15878* For any linearly independent subset of , there is a basis containing the vectors in . (Contributed by Mario Carneiro, 17-May-2015.)
LBasis

Theoremlbsext 15879* For any linearly independent subset of , there is a basis containing the vectors in . (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 17-May-2015.)
LBasis

Theoremlbsexg 15880 Every vector space has a basis. This theorem is an AC equivalent; this is the forward implication. (Contributed by Mario Carneiro, 17-May-2015.)
LBasis       CHOICE

Theoremlbsex 15881 Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis

Theoremlvecprop2d 15882* If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 15883 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
Scalar       Scalar

Theoremlvecpropd 15883* If two structures have the same components (properties), one is a left vector space iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.)
Scalar       Scalar

10.8  Ideals

10.8.1  The subring algebra; ideals

Syntaxcsra 15884 Extend class notation with the subring algebra generator.
subringAlg

Syntaxcrglmod 15885 Extend class notation with the left module induced by a ring over itself.
ringLMod

Syntaxclidl 15886 Ring left-ideal function.
LIdeal

Syntaxcrsp 15887 Ring span function.
RSpan

Definitiondf-sra 15888* Given any subring of a ring, we can construct a left-algebra by regarding the elements of the subring as scalars and the ring itself as a set of vectors. (Contributed by Mario Carneiro, 27-Nov-2014.)
subringAlg sSet Scalar s sSet

Definitiondf-rgmod 15889 Every ring can be viewed as a left module over itself. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod subringAlg

Definitiondf-lidl 15890 Define the class of left ideals of a given ring. An ideal is a submodule of the ring viewed as a module over itself. (Contributed by Stefan O'Rear, 31-Mar-2015.)
LIdeal ringLMod

Definitiondf-rsp 15891 Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
RSpan ringLMod

Theoremsraval 15892 Lemma for srabase 15894 through sravsca 15898. (Contributed by Mario Carneiro, 27-Nov-2014.)
subringAlg sSet Scalar s sSet

Theoremsralem 15893 Lemma for srabase 15894 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg               Slot

Theoremsrabase 15894 Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsraaddg 15895 Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsramulr 15896 Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsrasca 15897 The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg               s Scalar

Theoremsravsca 15898 The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsratset 15899 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg               TopSet TopSet

Theoremsratopn 15900 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg

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