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Theorem List for Metamath Proof Explorer - 26501-26600   *Has distinct variable group(s)
TypeLabelDescription
Statement

16.15.50  Additional material on polynomials [DEPRECATED]

Syntaxcmnc 26501 Extend class notation with the class of monic polynomials.

Syntaxcplylt 26502 Extend class notatin with the class of limited-degree polynomials.
Poly<

Definitiondf-mnc 26503* Define the class of monic polynomials. (Contributed by Stefan O'Rear, 5-Dec-2014.)
Poly coeffdeg

Definitiondf-plylt 26504* Define the class of limited-degree polynomials. (Contributed by Stefan O'Rear, 8-Dec-2014.)
Poly< Poly deg

Theoremdgrsub2 26505 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 26506 A nonzero polynomial with a root has positive degree. TODO: use in aaliou2 19552. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Poly deg

Theoremelmnc 26507 Property of a monic polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
Poly coeffdeg

Theoremmncply 26508 A monic polynomial is a polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
Poly

Theoremmnccoe 26509 A monic polynomial has leading coefficient 1. (Contributed by Stefan O'Rear, 5-Dec-2014.)
coeffdeg

Theoremmncn0 26510 A monic polynomial is not zero. (Contributed by Stefan O'Rear, 5-Dec-2014.)

16.15.51  Degree and minimal polynomial of algebraic numbers

Syntaxcdgraa 26511 Extend class notation to include the degree function for algebraic numbers.
degAA

Syntaxcmpaa 26512 Extend class notation to include the minimal polynomial for an algebraic number.
minPolyAA

Definitiondf-dgraa 26513* 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 26514* 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 26515* Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
degAA Poly deg

Theoremdgraalem 26516* Properties of the degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
degAA Poly deg degAA

Theoremdgraacl 26517 Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014.)
degAA

Theoremdgraaf 26518 Degree function on algebraic numbers is a function. (Contributed by Stefan O'Rear, 25-Nov-2014.)
degAA

Theoremdgraaub 26519 Upper bound on degree of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Poly degAA deg

Theoremdgraa0p 26520 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 26521* An algebraic number has exactly one monic polynomial of least degree. (Contributed by Stefan O'Rear, 25-Nov-2014.)
Polydeg degAA coeffdegAA

Theoremmpaaval 26522* Value of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
minPolyAA Polydeg degAA coeffdegAA

Theoremmpaalem 26523 Properties of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
minPolyAA Poly degminPolyAA degAA minPolyAA coeffminPolyAAdegAA

Theoremmpaacl 26524 Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014.)
minPolyAA Poly

Theoremmpaadgr 26525 Minimal polynomial has degree the degree of the number. (Contributed by Stefan O'Rear, 25-Nov-2014.)
degminPolyAA degAA

Theoremmpaaroot 26526 The minimal polynomial of an algebraic number has the number as a root. (Contributed by Stefan O'Rear, 25-Nov-2014.)
minPolyAA

Theoremmpaamn 26527 Minimal polynomial is monic. (Contributed by Stefan O'Rear, 25-Nov-2014.)

16.15.52  Algebraic integers I

Syntaxcitgo 26528 Extend class notation with the integral-over predicate.
IntgOver

Syntaxcza 26529 Extend class notation with the class of algebraic integers.

Definitiondf-itgo 26530* 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 26533. 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 26531 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 26532* Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014.)
IntgOver Poly coeffdeg

Theoremaaitgo 26533 The standard algebraic numbers are generated by IntgOver. (Contributed by Stefan O'Rear, 27-Nov-2014.)
IntgOver

Theoremitgoss 26534 An integral element is integral over a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
IntgOver IntgOver

Theoremitgocn 26535 All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
IntgOver

Theoremcnsrexpcl 26536 Exponentiation is closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRingfld

Theoremfsumcnsrcl 26537* Finite sums are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRingfld

Theoremcnsrplycl 26538 Polynomials are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRingfld       Poly

Theoremrgspnval 26539* Value of the ring-span of a set of elements in a ring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
RingSpan              SubRing

Theoremrgspncl 26540 The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
RingSpan              SubRing

Theoremrgspnssid 26541 The ring-span of a set contains the set. (Contributed by Stefan O'Rear, 30-Nov-2014.)
RingSpan

Theoremrgspnmin 26542 The ring-span is contained in all subspaces which contain all the generators. (Contributed by Stefan O'Rear, 30-Nov-2014.)
RingSpan              SubRing

Theoremrgspnid 26543 The span of a subring is itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRing       RingSpan

Theoremrngunsnply 26544* 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 26545* Finite linear combinations with an indicator function. (Contributed by Stefan O'Rear, 5-Dec-2014.)

16.15.53  Finite cardinality [SO]

Theoremen1uniel 26546 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)

Theoremen2eleq 26547 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 26548 Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)

16.15.54  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 13279. If order is not significant, it is simpler to use families instead.

Theoremissubmd 26549* Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
SubMnd

16.15.55  Transpositions in the symmetric group

Syntaxcpmtr 26550 Syntax for the transposition generator function.
pmTrsp

Definitiondf-pmtr 26551* Define a function that generates the transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theoremf1omvdmvd 26552 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 26553 A permutation and its inverse move the same points. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremmvdco 26554 Composing two permutations moves at most the union of the points. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremf1omvdconj 26555 Conjugation of a permutation takes the image of the moved subclass. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremf1otrspeq 26556 A transposition is characterized by the points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremf1omvdco2 26557 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 26558 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 26559* The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrval 26560* A generated transposition, expressed in a symmetric form. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrfv 26561 General value of mapping a point under a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrprfv 26562 In a transposition of two given points, each maps to the other. (Contributed by Stefan O'Rear, 25-Aug-2015.)
pmTrsp

Theorempmtrf 26563 Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrmvd 26564 A transposition moves precisely the transposed points. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrrn 26565 Transposing two points gives a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfrn 26566 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 26567 Mapping of a point under a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfinv 26568 A transposition function is an involution. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfmvdn0 26569 A transpositon moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.)
pmTrsp

Theorempmtrff1o 26570 A transposition function is a permutation. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfcnv 26571 A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfb 26572 An intrinsic characterization of the transposition permutations. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfconj 26573 Any conjugate of a transposition is a transposition. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theoremsymgsssg 26574* The symmetric group has subgroups restricting the set of non-fixed points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
SubGrp

Theoremsymgfisg 26575* The symmetric group has a subgroup of permutations that move finitely many points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
SubGrp

Theoremsymgtrf 26576 Transpositions are elements of the symmetric group. (Contributed by Stefan O'Rear, 23-Aug-2015.)
pmTrsp

Theoremsymggen 26577* The span of the transpositions is the subgroup that moves finitely many points. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp                     mrClsSubMnd

Theoremsymggen2 26578 A finite permutation group is generated by the transpositions. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp                     mrClsSubMnd

Theoremsymgtrinv 26579 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

16.15.56  The sign of a permutation

Syntaxcpsgn 26580 Syntax for the sign of a permutation.
pmSgn

Definitiondf-psgn 26581* 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 26582* Lemma for psgnuni 26588. 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 26583* Lemma for psgnuni 26588. 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 26584* Lemma for psgnuni 26588. 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 26585* Lemma for psgnuni 26588. 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 26586 Lemma for psgnuni 26588. 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 26587 Addition and subtraction of parities are the same. (Contributed by Stefan O'Rear, 27-Aug-2015.)

Theorempsgnuni 26588 If the same permutation can be written in more than one way as a product of transpositions, the parity of those products must agree; otherwise the product of one with the inverse of the other would be an odd representation of the identity. (Contributed by Stefan O'Rear, 27-Aug-2015.)
pmTrsp              Word        Word        g g

Theorempsgnfval 26589* Function definition of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgnfn 26590* Functionality and domain of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmSgn

Theorempsgndmsubg 26591 The finitary permutations are a subgroup. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmSgn       SubGrp

Theorempsgneldm 26592 Property of being a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmSgn

Theorempsgneldm2 26593* The finitary permutations are the span of the transpositons. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgneldm2i 26594 A sequence of transpositions describes a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgneu 26595* A finitary permutation has exactly one parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgnval 26596* Value of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgnvali 26597* A finitary permutation has at least one representation for its parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgnvalii 26598 Any representation of a permutation is length matching the permutation sign. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgnpmtr 26599 All transpositions are odd. (Contributed by Stefan O'Rear, 29-Aug-2015.)
pmTrsp       pmSgn

Theoremcnmsgnsubg 26600 The signs form a multiplicative subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.)
mulGrpflds        SubGrp

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