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Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlsppratlem2 15901 Lemma for lspprat 15906. 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 15902 Lemma for lspprat 15906. In the first case of lsppratlem1 15900, since , also , and since and , we have as desired. (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem4 15903 Lemma for lspprat 15906. In the second case of lsppratlem1 15900, and implies and thus as well. (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem5 15904 Lemma for lspprat 15906. Combine the two cases and show a contradiction to under the assumptions on and . (Contributed by NM, 29-Aug-2014.)

Theoremlsppratlem6 15905 Lemma for lspprat 15906. Negating the assumption on , we arrive close to the desired conclusion. (Contributed by NM, 29-Aug-2014.)

Theoremlspprat 15906* 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 15907* 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 15908* 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 15909 Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 15907. (Contributed by David Moews, 1-May-2017.)
mrCls              mrInd       LBasis

Theoremlvecdim 15910 The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 15897 and lbsacsbs 15909 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 14286. (Contributed by David Moews, 1-May-2017.)
LBasis

Theoremlbsextlem1 15911* Lemma for lbsext 15916. 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 15912* Lemma for lbsext 15916. 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 15913* Lemma for lbsext 15916. A chain in has an upper bound in . (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis                                                               []

Theoremlbsextlem4 15914* Lemma for lbsext 15916. lbsextlem3 15913 satisfies the conditions for the application of Zorn's lemma zorn 8134 (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 15915* For any linearly independent subset of , there is a basis containing the vectors in . (Contributed by Mario Carneiro, 17-May-2015.)
LBasis

Theoremlbsext 15916* 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 15917 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 15918 Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014.)
LBasis

Theoremlvecprop2d 15919* If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 15920 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
Scalar       Scalar

Theoremlvecpropd 15920* 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 15921 Extend class notation with the subring algebra generator.
subringAlg

Syntaxcrglmod 15922 Extend class notation with the left module induced by a ring over itself.
ringLMod

Syntaxclidl 15923 Ring left-ideal function.
LIdeal

Syntaxcrsp 15924 Ring span function.
RSpan

Definitiondf-sra 15925* 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 15926 Every ring can be viewed as a left module over itself. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod subringAlg

Definitiondf-lidl 15927 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 15928 Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
RSpan ringLMod

Theoremsraval 15929 Lemma for srabase 15931 through sravsca 15935. (Contributed by Mario Carneiro, 27-Nov-2014.)
subringAlg sSet Scalar s sSet

Theoremsralem 15930 Lemma for srabase 15931 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg               Slot

Theoremsrabase 15931 Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsraaddg 15932 Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsramulr 15933 Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsrasca 15934 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 15935 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 15936 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg               TopSet TopSet

Theoremsratopn 15937 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsrads 15938 Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.)
subringAlg

Theoremsralmod 15939 The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
subringAlg        SubRing

Theoremsralmod0 15940 The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.)
subringAlg

Theoremissubgrpd2 15941* Prove a subgroup by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
s                                                         SubGrp

Theoremissubgrpd 15942* Prove a subgroup by closure. (Contributed by Stefan O'Rear, 7-Dec-2014.)
s

Theoremissubrngd2 15943* Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
s                                                                                     SubRing

Theoremrlmfn 15944 ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod

Theoremrlmval 15945 Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ringLMod subringAlg

Theoremlidlval 15946 Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
LIdeal ringLMod

Theoremrspval 15947 Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
RSpan ringLMod

Theoremrlmbas 15948 Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ringLMod

Theoremrlmplusg 15949 Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ringLMod

Theoremrlm0 15950 Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
ringLMod

Theoremrlmmulr 15951 Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
ringLMod

Theoremrlmsca 15952 Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ScalarringLMod

Theoremrlmsca2 15953 Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
ScalarringLMod

Theoremrlmvsca 15954 Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ringLMod

Theoremrlmtopn 15955 Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
ringLMod

Theoremrlmds 15956 Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
ringLMod

Theoremrlmlmod 15957 The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod

Theoremrlmlvec 15958 The ring module over a division ring is a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
ringLMod

Theoremrlmvneg 15959 Vector negation in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 5-Jun-2015.)
ringLMod

Theoremrlmscaf 15960 Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
mulGrp ringLMod

Theoremlidlss 15961 An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.)
LIdeal

TheoremlidlssOLD 15962 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 15963* Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.)
LIdeal

Theoremlidl0cl 15964 An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlacl 15965 An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlnegcl 15966 An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlsubg 15967 An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       SubGrp

Theoremlidlsubcl 15968 An ideal is closed under subtraction. (Contributed by Stefan O'Rear, 28-Mar-2015.)
LIdeal

Theoremlidlmcl 15969 An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidl1el 15970 An ideal contains 1 iff it is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidl0 15971 Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidl1 15972 Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlacs 15973 The ideal system is an algebraic closure system on the base set. (Contributed by Stefan O'Rear, 4-Apr-2015.)
LIdeal       ACS

Theoremrspcl 15974 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 15975 The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
RSpan

Theoremrsp1 15976 The span of the identity element is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
RSpan

Theoremrsp0 15977 The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
RSpan

Theoremrspssp 15978 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 15979 Moore closure generalizes ideal span. (Contributed by Stefan O'Rear, 4-Apr-2015.)
LIdeal       RSpan       mrCls

Theoremlidlnz 15980* A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremdrngnidl 15981 A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LIdeal

Theoremlidlrsppropd 15982* 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 15983 Ring two-sided ideal function.
2Ideal

Definitiondf-2idl 15984 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 15985 Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       oppr       LIdeal       2Ideal

Theorem2idlcpbl 15986 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
~QG        2Ideal

Theoremdivs1 15987 The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
s ~QG        2Ideal              ~QG

Theoremdivsrng 15988 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 15989* 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 15990 In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
LIdeal       oppr       LIdeal

Theoremcrng2idl 15991 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 15992 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 15993 Ring left-principal-ideal function.
LPIdeal

Syntaxclpir 15994 Class of left principal ideal rings.
LPIR

Definitiondf-lpidl 15995* 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 15996 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 15997* Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       RSpan

Theoremislpidl 15998* Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal       RSpan

Theoremlpi0 15999 The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal

Theoremlpi1 16000 The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal

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