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Theorem List for Metamath Proof Explorer - 16201-16300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlspsncv0 16201* 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 16202 Lemma for lspprat 16208. 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 16198 (hence the name), which we use extensively below. In this lemma, we show that since , either or . (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem2 16203 Lemma for lspprat 16208. 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 16204 Lemma for lspprat 16208. In the first case of lsppratlem1 16202, since , also , and since and , we have as desired. (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem4 16205 Lemma for lspprat 16208. In the second case of lsppratlem1 16202, and implies and thus as well. (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem5 16206 Lemma for lspprat 16208. Combine the two cases and show a contradiction to under the assumptions on and . (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem6 16207 Lemma for lspprat 16208. Negating the assumption on , we arrive close to the desired conclusion. (Contributed by NM, 29-Aug-2014.)

Theoremlspprat 16208* 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 16209* 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 16210* 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 16211 Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 16209. (Contributed by David Moews, 1-May-2017.)
mrCls              mrInd       LBasis

Theoremlvecdim 16212 The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 16199 and lbsacsbs 16211 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 14592. (Contributed by David Moews, 1-May-2017.)
LBasis

Theoremlbsextlem1 16213* Lemma for lbsext 16218. 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 16214* Lemma for lbsext 16218. 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 16215* Lemma for lbsext 16218. A chain in has an upper bound in . (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis                                                               []

Theoremlbsextlem4 16216* Lemma for lbsext 16218. lbsextlem3 16215 satisfies the conditions for the application of Zorn's lemma zorn 8371 (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 16217* For any linearly independent subset of , there is a basis containing the vectors in . (Contributed by Mario Carneiro, 17-May-2015.)
LBasis

Theoremlbsext 16218* 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 16219 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 16220 Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis

Theoremlvecprop2d 16221* If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 16222 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
Scalar       Scalar

Theoremlvecpropd 16222* 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 16223 Extend class notation with the subring algebra generator.
subringAlg

Syntaxcrglmod 16224 Extend class notation with the left module induced by a ring over itself.
ringLMod

Syntaxclidl 16225 Ring left-ideal function.
LIdeal

Syntaxcrsp 16226 Ring span function.
RSpan

Definitiondf-sra 16227* 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 16228 Every ring can be viewed as a left module over itself. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod subringAlg

Definitiondf-lidl 16229 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 16230 Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
RSpan ringLMod

Theoremsraval 16231 Lemma for srabase 16233 through sravsca 16237. (Contributed by Mario Carneiro, 27-Nov-2014.)
subringAlg sSet Scalar s sSet

Theoremsralem 16232 Lemma for srabase 16233 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg               Slot

Theoremsrabase 16233 Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsraaddg 16234 Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsramulr 16235 Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsrasca 16236 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 16237 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 16238 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg               TopSet TopSet

Theoremsratopn 16239 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsrads 16240 Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsralmod 16241 The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
subringAlg        SubRing

Theoremsralmod0 16242 The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.)
subringAlg

Theoremissubgrpd2 16243* Prove a subgroup by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
s                                                         SubGrp

Theoremissubgrpd 16244* Prove a subgroup by closure. (Contributed by Stefan O'Rear, 7-Dec-2014.)
s

Theoremissubrngd2 16245* Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
s                                                                                     SubRing

Theoremrlmfn 16246 ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod

Theoremrlmval 16247 Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ringLMod subringAlg

Theoremlidlval 16248 Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
LIdeal ringLMod

Theoremrspval 16249 Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
RSpan ringLMod

Theoremrlmbas 16250 Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ringLMod

Theoremrlmplusg 16251 Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ringLMod

Theoremrlm0 16252 Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
ringLMod

Theoremrlmmulr 16253 Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
ringLMod

Theoremrlmsca 16254 Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ScalarringLMod

Theoremrlmsca2 16255 Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
ScalarringLMod

Theoremrlmvsca 16256 Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ringLMod

Theoremrlmtopn 16257 Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
ringLMod

Theoremrlmds 16258 Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
ringLMod

Theoremrlmlmod 16259 The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod

Theoremrlmlvec 16260 The ring module over a division ring is a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
ringLMod

Theoremrlmvneg 16261 Vector negation in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 5-Jun-2015.)
ringLMod

Theoremrlmscaf 16262 Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
mulGrp ringLMod

Theoremlidlss 16263 An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.)
LIdeal

TheoremlidlssOLD 16264 An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
LIdeal

Theoremislidl 16265* Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.)
LIdeal

Theoremlidl0cl 16266 An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlacl 16267 An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlnegcl 16268 An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlsubg 16269 An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       SubGrp

Theoremlidlsubcl 16270 An ideal is closed under subtraction. (Contributed by Stefan O'Rear, 28-Mar-2015.)
LIdeal

Theoremlidlmcl 16271 An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidl1el 16272 An ideal contains 1 iff it is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidl0 16273 Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidl1 16274 Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlacs 16275 The ideal system is an algebraic closure system on the base set. (Contributed by Stefan O'Rear, 4-Apr-2015.)
LIdeal       ACS

Theoremrspcl 16276 The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
RSpan              LIdeal

Theoremrspssid 16277 The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
RSpan

Theoremrsp1 16278 The span of the identity element is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
RSpan

Theoremrsp0 16279 The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
RSpan

Theoremrspssp 16280 The ideal span of a set of elements in a ring is contained in any subring which contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
RSpan       LIdeal

Theoremmrcrsp 16281 Moore closure generalizes ideal span. (Contributed by Stefan O'Rear, 4-Apr-2015.)
LIdeal       RSpan       mrCls

Theoremlidlnz 16282* A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremdrngnidl 16283 A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlrsppropd 16284* The left ideals and ring span of a ring depend only on the ring components. Here is expected to be either (when closure is available) or (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal LIdeal RSpan RSpan

10.8.2  Two-sided ideals and quotient rings

Syntaxc2idl 16285 Ring two-sided ideal function.
2Ideal

Definitiondf-2idl 16286 Define the class of two-sided ideals of a ring. A two-sided ideal is a left ideal which is also a right ideal (or a left ideal over the opposite ring). (Contributed by Mario Carneiro, 14-Jun-2015.)
2Ideal LIdeal LIdealoppr

Theorem2idlval 16287 Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       oppr       LIdeal       2Ideal

Theorem2idlcpbl 16288 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
~QG        2Ideal

Theoremdivs1 16289 The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
s ~QG        2Ideal              ~QG

Theoremdivsrng 16290 If is a two-sided ideal in , then is a ring, called the quotient ring of by . (Contributed by Mario Carneiro, 14-Jun-2015.)
s ~QG        2Ideal

Theoremdivsrhm 16291* If is a two-sided ideal in , then the "natural map" from elements to their cosets is a ring homomorphism from to . (Contributed by Mario Carneiro, 15-Jun-2015.)
s ~QG        2Ideal              ~QG        RingHom

Theoremcrngridl 16292 In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       oppr       LIdeal

Theoremcrng2idl 16293 In a commutative ring, a two-sided ideal is the same as a left ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       2Ideal

Theoremdivscrng 16294 The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
s ~QG        LIdeal

10.8.3  Principal ideal rings. Divisibility in the integers

Syntaxclpidl 16295 Ring left-principal-ideal function.
LPIdeal

Syntaxclpir 16296 Class of left principal ideal rings.
LPIR

Definitiondf-lpidl 16297* Define the class of left principal ideals of a ring, which are ideals with a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal RSpan

Definitiondf-lpir 16298 Define the class of left principal ideal rings, rings where every left ideal has a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIR LIdeal LPIdeal

Theoremlpival 16299* Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       RSpan

Theoremislpidl 16300* Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       RSpan

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