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Theorem List for Metamath Proof Explorer - 27301-27400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlnrfrlm 27301 Finite-dimensional free modules over a Noetherian ring are Noetherian. (Contributed by Stefan O'Rear, 3-Feb-2015.)
freeLMod        LNoeR LNoeM

Theoremlnrfg 27302 Finitely-generated modules over a Noetherian ring, being homomorphic images of free modules, are Noetherian. (Contributed by Stefan O'Rear, 7-Feb-2015.)
Scalar       LFinGen LNoeR LNoeM

Theoremlnrfgtr 27303 A submodule of a finitely generated module over a Noetherian ring is finitely generated. Often taken as the definition of Noetherian ring. (Contributed by Stefan O'Rear, 7-Feb-2015.)
Scalar              s        LFinGen LNoeR LFinGen

19.16.49  Hilbert's Basis Theorem

Syntaxcldgis 27304 The leading ideal sequence used in the Hilbert Basis Theorem.
ldgIdlSeq

Definitiondf-ldgis 27305* Define a function which carries polynomial ideals to the sequence of coefficient ideals of leading coefficients of degree- elements in the polynomial ideal. The proof that this map is strictly monotone is the core of the Hilbert Basis Theorem hbt 27313. (Contributed by Stefan O'Rear, 31-Mar-2015.)
ldgIdlSeq LIdealPoly1 deg1 coe1

Theoremhbtlem1 27306* Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)
Poly1       LIdeal       ldgIdlSeq       deg1        coe1

Theoremhbtlem2 27307 Leading coefficient ideals are ideals. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       LIdeal       ldgIdlSeq       LIdeal

Theoremhbtlem7 27308 Functionality of leading coefficient ideal sequence. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Poly1       LIdeal       ldgIdlSeq       LIdeal

Theoremhbtlem4 27309 The leading ideal function goes to increasing sequences. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       LIdeal       ldgIdlSeq

Theoremhbtlem3 27310 The leading ideal function is monotone. (Contributed by Stefan O'Rear, 31-Mar-2015.)
Poly1       LIdeal       ldgIdlSeq

Theoremhbtlem5 27311* The leading ideal function is strictly monotone. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       LIdeal       ldgIdlSeq

Theoremhbtlem6 27312* There is a finite set of polynomials matching any single stage of the image. (Contributed by Stefan O'Rear, 1-Apr-2015.)
Poly1       LIdeal       ldgIdlSeq       RSpan       LNoeR

Theoremhbt 27313 The Hilbert Basis Theorem - the ring of univariate polynomials over a Noetherian ring is a Noetherian ring. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Poly1       LNoeR LNoeR

19.16.50  Additional material on polynomials [DEPRECATED]

Syntaxcmnc 27314 Extend class notation with the class of monic polynomials.

Syntaxcplylt 27315 Extend class notatin with the class of limited-degree polynomials.
Poly<

Definitiondf-mnc 27316* Define the class of monic polynomials. (Contributed by Stefan O'Rear, 5-Dec-2014.)
Poly coeffdeg

Definitiondf-plylt 27317* Define the class of limited-degree polynomials. (Contributed by Stefan O'Rear, 8-Dec-2014.)
Poly< Poly deg

Theoremdgrsub2 27318 Subtracting two polynomials with the same degree and top coefficient gives a polynomial of strictly lower degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
deg       Poly Poly deg coeff coeff deg

Theoremdgrnznn 27319 A nonzero polynomial with a root has positive degree. TODO: use in aaliou2 20259. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Poly deg

Theoremelmnc 27320 Property of a monic polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
Poly coeffdeg

Theoremmncply 27321 A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
Poly

Theoremmnccoe 27322 A monic polynomial has leading coefficient 1. (Contributed by Stefan O'Rear, 5-Dec-2014.)
coeffdeg

Theoremmncn0 27323 A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.)

19.16.51  Degree and minimal polynomial of algebraic numbers

Syntaxcdgraa 27324 Extend class notation to include the degree function for algebraic numbers.
degAA

Syntaxcmpaa 27325 Extend class notation to include the minimal polynomial for an algebraic number.
minPolyAA

Definitiondf-dgraa 27326* Define the degree of an algebraic number as the smallest degree of any nonzero polynomial which has said number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.)
degAA Poly deg

Definitiondf-mpaa 27327* Define the minimal polynomial of an algebraic number as the unique monic polynomial which achieves the minimum of degAA. (Contributed by Stefan O'Rear, 25-Nov-2014.)
minPolyAA Polydeg degAA coeffdegAA

Theoremdgraaval 27328* Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
degAA Poly deg

Theoremdgraalem 27329* Properties of the degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
degAA Poly deg degAA

Theoremdgraacl 27330 Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
degAA

Theoremdgraaf 27331 Degree function on algebraic numbers is a function. (Contributed by Stefan O'Rear, 25-Nov-2014.)
degAA

Theoremdgraaub 27332 Upper bound on degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Poly degAA deg

Theoremdgraa0p 27333 A rational polynomial of degree less than an algebraic number cannot be zero at that number unless it is the zero polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Poly deg degAA

Theoremmpaaeu 27334* An algebraic number has exactly one monic polynomial of the least degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Polydeg degAA coeffdegAA

Theoremmpaaval 27335* Value of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
minPolyAA Polydeg degAA coeffdegAA

Theoremmpaalem 27336 Properties of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
minPolyAA Poly degminPolyAA degAA minPolyAA coeffminPolyAAdegAA

Theoremmpaacl 27337 Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
minPolyAA Poly

Theoremmpaadgr 27338 Minimal polynomial has degree the degree of the number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
degminPolyAA degAA

Theoremmpaaroot 27339 The minimal polynomial of an algebraic number has the number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.)
minPolyAA

Theoremmpaamn 27340 Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.)

19.16.52  Algebraic integers I

Syntaxcitgo 27341 Extend class notation with the integral-over predicate.
IntgOver

Syntaxcza 27342 Extend class notation with the class of algebraic integers.

Definitiondf-itgo 27343* A complex number is said to be integral over a subset if it is the root of a monic polynomial with coefficients from the subset. This definition is typically not used for fields but it works there, see aaitgo 27346. This definition could work for subsets of an arbitrary ring with a more general definition of polynomials. TODO: use (Contributed by Stefan O'Rear, 27-Nov-2014.)
IntgOver Poly coeffdeg

Definitiondf-za 27344 Define an algebraic integer as a complex number which is the root of a monic integer polynomial. (Contributed by Stefan O'Rear, 30-Nov-2014.)
IntgOver

Theoremitgoval 27345* Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014.)
IntgOver Poly coeffdeg

Theoremaaitgo 27346 The standard algebraic numbers are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
IntgOver

Theoremitgoss 27347 An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
IntgOver IntgOver

Theoremitgocn 27348 All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
IntgOver

Theoremcnsrexpcl 27349 Exponentiation is closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRingfld

Theoremfsumcnsrcl 27350* Finite sums are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRingfld

Theoremcnsrplycl 27351 Polynomials are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRingfld       Poly

Theoremrgspnval 27352* Value of the ring-span of a set of elements in a ring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
RingSpan              SubRing

Theoremrgspncl 27353 The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
RingSpan              SubRing

Theoremrgspnssid 27354 The ring-span of a set contains the set. (Contributed by Stefan O'Rear, 30-Nov-2014.)
RingSpan

Theoremrgspnmin 27355 The ring-span is contained in all subspaces which contain all the generators. (Contributed by Stefan O'Rear, 30-Nov-2014.)
RingSpan              SubRing

Theoremrgspnid 27356 The span of a subring is itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRing       RingSpan

Theoremrngunsnply 27357* Adjoining one element to a ring results in a set of polynomial evaluations. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRingfld              RingSpanfld        Poly

Theoremflcidc 27358* Finite linear combinations with an indicator function. (Contributed by Stefan O'Rear, 5-Dec-2014.)

19.16.53  Finite cardinality [SO]

Theoremen1uniel 27359 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)

Theoremen2eleq 27360 Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremen2other2 27361 Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)

19.16.54  Words in monoids and ordered group sum

One important use of words is as formal composites in cases where order is significant, using the general sum operator df-gsum 13730. If order is not significant, it is simpler to use families instead.

Theoremissubmd 27362* Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
SubMnd

19.16.55  Transpositions in the symmetric group

Syntaxcpmtr 27363 Syntax for the transposition generator function.
pmTrsp

Definitiondf-pmtr 27364* Define a function that generates the transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theoremf1omvdmvd 27365 A permutation of any class moves a point which is moved to a different point which is moved. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremf1omvdcnv 27366 A permutation and its inverse move the same points. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremmvdco 27367 Composing two permutations moves at most the union of the points. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremf1omvdconj 27368 Conjugation of a permutation takes the image of the moved subclass. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremf1otrspeq 27369 A transposition is characterized by the points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremf1omvdco2 27370 If exactly one of two permutations is limited to a set of points, then the composition will not be. (Contributed by Stefan O'Rear, 23-Aug-2015.)

Theoremf1omvdco3 27371 If a point is moved by exactly one of two permutations, then it will be moved by their composite. (Contributed by Stefan O'Rear, 23-Aug-2015.)

Theorempmtrfval 27372* The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrval 27373* A generated transposition, expressed in a symmetric form. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrfv 27374 General value of mapping a point under a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrprfv 27375 In a transposition of two given points, each maps to the other. (Contributed by Stefan O'Rear, 25-Aug-2015.)
pmTrsp

Theorempmtrf 27376 Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrmvd 27377 A transposition moves precisely the transposed points. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrrn 27378 Transposing two points gives a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfrn 27379 A transposition (as a kind of function) is the function transposing the two points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrffv 27380 Mapping of a point under a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfinv 27381 A transposition function is an involution. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfmvdn0 27382 A transpositon moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.)
pmTrsp

Theorempmtrff1o 27383 A transposition function is a permutation. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfcnv 27384 A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfb 27385 An intrinsic characterization of the transposition permutations. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfconj 27386 Any conjugate of a transposition is a transposition. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theoremsymgsssg 27387* The symmetric group has subgroups restricting the set of non-fixed points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
SubGrp

Theoremsymgfisg 27388* The symmetric group has a subgroup of permutations that move finitely many points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
SubGrp

Theoremsymgtrf 27389 Transpositions are elements of the symmetric group. (Contributed by Stefan O'Rear, 23-Aug-2015.)
pmTrsp

Theoremsymggen 27390* The span of the transpositions is the subgroup that moves finitely many points. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp                     mrClsSubMnd

Theoremsymggen2 27391 A finite permutation group is generated by the transpositions. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp                     mrClsSubMnd

Theoremsymgtrinv 27392 To invert a permutation represented as a sequence of transpositions, reverse the sequence. (Contributed by Stefan O'Rear, 27-Aug-2015.)
pmTrsp                     Word g g reverse

19.16.56  The sign of a permutation

Syntaxcpsgn 27393 Syntax for the sign of a permutation.
pmSgn

Definitiondf-psgn 27394* Define a function which takes the value for even permutations and for odd. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmSgn Word pmTrsp g

Theorempsgnunilem1 27395* Lemma for psgnuni 27401. Given two consequtive transpositions in a representation of a permutation, either they are equal and therefore equivalent to the identity, or they are not and it is possible to commute them such that a chosen point in the left transposition is preserved in the right. By repeating this process, a point can be removed from a representation of the identity. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempsgnunilem5 27396* Lemma for psgnuni 27401. It is impossible to shift a transposition off the end because if the active transposition is at the right end, it is the only transposition moving in contradiction to this being a representation of the identity. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
pmTrsp              Word        g               ..^              ..^        ..^

Theorempsgnunilem2 27397* Lemma for psgnuni 27401. Induction step for moving a transposition as far to the right as possible. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
pmTrsp              Word        g               ..^              ..^        Word g        Word g ..^ ..^

Theorempsgnunilem3 27398* Lemma for psgnuni 27401. Any nonempty representation of the identity can be incrementally transformed into a representation two shorter. (Contributed by Stefan O'Rear, 25-Aug-2015.)
pmTrsp              Word                      g        Word g

Theorempsgnunilem4 27399 Lemma for psgnuni 27401. An odd-length representation of the identity is impossible, as it could be repeatedly shortened to a length of 1, but a length 1 permutation must be a transposition. (Contributed by Stefan O'Rear, 25-Aug-2015.)
pmTrsp              Word        g

Theoremm1expaddsub 27400 Addition and subtraction of parities are the same. (Contributed by Stefan O'Rear, 27-Aug-2015.)

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