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Theorem List for Metamath Proof Explorer - 16101-16200   *Has distinct variable group(s)
TypeLabelDescription
Statement

10.6.4  Subspace sum; bases for a left module

Syntaxclbs 16101 Extend class notation with the set of bases for a vector space.
LBasis

Definitiondf-lbs 16102* Define the set of bases to a left module or left vector space. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis Scalar

Theoremislbs 16103* 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 16104 A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis

Theoremlbsel 16105 An element of a basis is a vector. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis

Theoremlbssp 16106 The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis

Theoremlbsind 16107 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 16108 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 16109 No proper subset of a basis spans the space. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis              Scalar

Theoremlsmcl 16110 The sum of two subspaces is a subspace. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)

Theoremlsmspsn 16111* Member of subspace sum of spans of singletons. (Contributed by NM, 8-Apr-2015.)
Scalar

Theoremlsmelval2 16112* 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 16113 Subspace sum in terms of span. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)

Theoremlsmsp2 16114 Subspace sum of spans of subsets is the span of their union. (spanuni 22999 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)

Theoremlsmssspx 16115 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 16116 The span of a pair of vectors equals the sum of the spans of their singletons. (Contributed by NM, 13-Jan-2015.)

Theoremlsppreli 16117 A vector expressed as a sum belongs to the span of its components. (Contributed by NM, 9-Apr-2015.)
Scalar

Theoremlsmelpr 16118 Two ways to say that a vector belongs to the span of a pair of vectors. (Contributed by NM, 14-Jan-2015.)

Theoremlsppr0 16119 The span of a vector paired with zero equals the span of the singleton of the vector. (Contributed by NM, 29-Aug-2014.)

Theoremlsppr 16120* Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.)
Scalar

Theoremlspprel 16121* Member of the span of a pair of vectors. (Contributed by NM, 10-Apr-2015.)
Scalar

Theoremlspprabs 16122 Absorption of vector sum into span of pair. (Contributed by NM, 27-Apr-2015.)

Theoremlspvadd 16123 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 16124 Triangle-type inequality for span of a singleton. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)

Theoremlspsntrim 16125 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 16126* 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 16127 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 16128 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 16129 Extend class notation with class of all left vector spaces.

Definitiondf-lvec 16130 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 16131 The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.)
Scalar

Theoremlvecdrng 16132 The set of scalars of a left vector space is a division ring. (Contributed by NM, 17-Apr-2014.)
Scalar

Theoremlveclmod 16133 A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)

Theoremlsslvec 16134 A vector subspace is a vector space. (Contributed by NM, 14-Mar-2015.)
s

Theoremlvecvs0or 16135 If a scalar product is zero, one of its factors must be zero. (hvmul0or 22480 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecvsn0 16136 A scalar product is nonzero iff both of its factors are nonzero. (Contributed by NM, 3-Jan-2015.)
Scalar

Theoremlssvs0or 16137 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 16138 Cancellation law for scalar multiplication. (hvmulcan 22527 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecvscan2 16139 Cancellation law for scalar multiplication. (hvmulcan2 22528 analog.) (Contributed by NM, 2-Jul-2014.)
Scalar

Theoremlvecinv 16140 Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.)
Scalar

Theoremlspsnvs 16141 A non-zero scalar product does not change the span of a singleton. (spansncol 23023 analog.) (Contributed by NM, 23-Apr-2014.)
Scalar

Theoremlspsneleq 16142 Membership relation that implies equality of spans. (spansneleq 23025 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspsncmp 16143 Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.)

Theoremlspsnne1 16144 Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.)

Theoremlspsnne2 16145 Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015.)

Theoremlspsnnecom 16146 Swap two vectors with different spans. (Contributed by NM, 20-May-2015.)

Theoremlspabs2 16147 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)

Theoremlspabs3 16148 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)

Theoremlspsneq 16149* Equal spans of singletons must have proportional vectors. See lspsnss2 16036 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.)
Scalar

Theoremlspsneu 16150* Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.)
Scalar

Theoremlspsnel4 16151 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 23028 analog.) (Contributed by NM, 4-Jul-2014.)

Theoremlspdisj 16152 The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.)

Theoremlspdisjb 16153 A nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.)

Theoremlspdisj2 16154 Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.)

Theoremlspfixed 16155* 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 16156 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 16157 vs. lspexchn2 16158); look for lspexch 16156 and prcom 3842 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 16157 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 16156 to see if this will shorten proofs. (Contributed by NM, 20-May-2015.)

Theoremlspexchn2 16158 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 16156 to see if this will shorten proofs. (Contributed by NM, 24-May-2015.)

Theoremlspindpi 16159 Partial independence property. (Contributed by NM, 23-Apr-2015.)

Theoremlspindp1 16160 Alternate way to say 3 vectors are mutually independent (swap 1st and 2nd). (Contributed by NM, 11-Apr-2015.)

Theoremlspindp2l 16161 Alternate way to say 3 vectors are mutually independent (rotate left). (Contributed by NM, 10-May-2015.)

Theoremlspindp2 16162 Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.)

Theoremlspindp3 16163 Independence of 2 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)

Theoremlspindp4 16164 (Partial) independence of 3 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)

Theoremlvecindp 16165 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 16166 Sums of independent vectors must have equal coefficients. (Contributed by NM, 22-Mar-2015.)
Scalar

Theoremlspsnsubn0 16167 Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.)

Theoremlsmcv 16168 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 23107 analog.) TODO: ugly proof; can it be shortened? (Contributed by NM, 2-Oct-2014.)

Theoremlspsolvlem 16169* Lemma for lspsolv 16170. (Contributed by Mario Carneiro, 25-Jun-2014.)
Scalar

Theoremlspsolv 16170 If is in the span of but not , then is in the span of . (Contributed by Mario Carneiro, 25-Jun-2014.)

Theoremlssacsex 16171* In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 15998 by lspsolv 16170. (Contributed by David Moews, 1-May-2017.)
mrCls              ACS

Theoremlspsnat 16172 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 23036 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)

Theoremlspsncv0 16173* 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 16174 Lemma for lspprat 16180. 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 16170 (hence the name), which we use extensively below. In this lemma, we show that since , either or . (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem2 16175 Lemma for lspprat 16180. 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 16176 Lemma for lspprat 16180. In the first case of lsppratlem1 16174, since , also , and since and , we have as desired. (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem4 16177 Lemma for lspprat 16180. In the second case of lsppratlem1 16174, and implies and thus as well. (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem5 16178 Lemma for lspprat 16180. Combine the two cases and show a contradiction to under the assumptions on and . (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem6 16179 Lemma for lspprat 16180. Negating the assumption on , we arrive close to the desired conclusion. (Contributed by NM, 29-Aug-2014.)

Theoremlspprat 16180* 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 16181* 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 16182* 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 16183 Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 16181. (Contributed by David Moews, 1-May-2017.)
mrCls              mrInd       LBasis

Theoremlvecdim 16184 The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 16171 and lbsacsbs 16183 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 14564. (Contributed by David Moews, 1-May-2017.)
LBasis

Theoremlbsextlem1 16185* Lemma for lbsext 16190. 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 16186* Lemma for lbsext 16190. 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 16187* Lemma for lbsext 16190. A chain in has an upper bound in . (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis                                                               []

Theoremlbsextlem4 16188* Lemma for lbsext 16190. lbsextlem3 16187 satisfies the conditions for the application of Zorn's lemma zorn 8343 (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 16189* For any linearly independent subset of , there is a basis containing the vectors in . (Contributed by Mario Carneiro, 17-May-2015.)
LBasis

Theoremlbsext 16190* 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 16191 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 16192 Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis

Theoremlvecprop2d 16193* If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 16194 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
Scalar       Scalar

Theoremlvecpropd 16194* 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 16195 Extend class notation with the subring algebra generator.
subringAlg

Syntaxcrglmod 16196 Extend class notation with the left module induced by a ring over itself.
ringLMod

Syntaxclidl 16197 Ring left-ideal function.
LIdeal

Syntaxcrsp 16198 Ring span function.
RSpan

Definitiondf-sra 16199* 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 16200 Every ring can be viewed as a left module over itself. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod subringAlg

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