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Theorem List for Metamath Proof Explorer - 26701-26800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremitgocn 26701 All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
IntgOver

Theoremcnsrexpcl 26702 Exponentiation is closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRingfld

Theoremfsumcnsrcl 26703* Finite sums are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRingfld

Theoremcnsrplycl 26704 Polynomials are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRingfld       Poly

Theoremrgspnval 26705* Value of the ring-span of a set of elements in a ring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
RingSpan              SubRing

Theoremrgspncl 26706 The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
RingSpan              SubRing

Theoremrgspnssid 26707 The ring-span of a set contains the set. (Contributed by Stefan O'Rear, 30-Nov-2014.)
RingSpan

Theoremrgspnmin 26708 The ring-span is contained in all subspaces which contain all the generators. (Contributed by Stefan O'Rear, 30-Nov-2014.)
RingSpan              SubRing

Theoremrgspnid 26709 The span of a subring is itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
SubRing       RingSpan

Theoremrngunsnply 26710* 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 26711* Finite linear combinations with an indicator function. (Contributed by Stefan O'Rear, 5-Dec-2014.)

16.16.53  Finite cardinality [SO]

Theoremen1uniel 26712 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)

Theoremen2eleq 26713 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 26714 Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)

16.16.54  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 13332. If order is not significant, it is simpler to use families instead.

Theoremissubmd 26715* Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
SubMnd

16.16.55  Transpositions in the symmetric group

Syntaxcpmtr 26716 Syntax for the transposition generator function.
pmTrsp

Definitiondf-pmtr 26717* Define a function that generates the transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theoremf1omvdmvd 26718 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 26719 A permutation and its inverse move the same points. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremmvdco 26720 Composing two permutations moves at most the union of the points. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremf1omvdconj 26721 Conjugation of a permutation takes the image of the moved subclass. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremf1otrspeq 26722 A transposition is characterized by the points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Theoremf1omvdco2 26723 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 26724 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 26725* The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrval 26726* A generated transposition, expressed in a symmetric form. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrfv 26727 General value of mapping a point under a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrprfv 26728 In a transposition of two given points, each maps to the other. (Contributed by Stefan O'Rear, 25-Aug-2015.)
pmTrsp

Theorempmtrf 26729 Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrmvd 26730 A transposition moves precisely the transposed points. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp

Theorempmtrrn 26731 Transposing two points gives a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfrn 26732 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 26733 Mapping of a point under a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfinv 26734 A transposition function is an involution. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfmvdn0 26735 A transpositon moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.)
pmTrsp

Theorempmtrff1o 26736 A transposition function is a permutation. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfcnv 26737 A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfb 26738 An intrinsic characterization of the transposition permutations. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theorempmtrfconj 26739 Any conjugate of a transposition is a transposition. (Contributed by Stefan O'Rear, 22-Aug-2015.)
pmTrsp

Theoremsymgsssg 26740* The symmetric group has subgroups restricting the set of non-fixed points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
SubGrp

Theoremsymgfisg 26741* The symmetric group has a subgroup of permutations that move finitely many points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
SubGrp

Theoremsymgtrf 26742 Transpositions are elements of the symmetric group. (Contributed by Stefan O'Rear, 23-Aug-2015.)
pmTrsp

Theoremsymggen 26743* The span of the transpositions is the subgroup that moves finitely many points. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp                     mrClsSubMnd

Theoremsymggen2 26744 A finite permutation group is generated by the transpositions. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp                     mrClsSubMnd

Theoremsymgtrinv 26745 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

16.16.56  The sign of a permutation

Syntaxcpsgn 26746 Syntax for the sign of a permutation.
pmSgn

Definitiondf-psgn 26747* 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 26748* Lemma for psgnuni 26754. 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 26749* Lemma for psgnuni 26754. 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 26750* Lemma for psgnuni 26754. 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 26751* Lemma for psgnuni 26754. 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 26752 Lemma for psgnuni 26754. 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 26753 Addition and subtraction of parities are the same. (Contributed by Stefan O'Rear, 27-Aug-2015.)

Theorempsgnuni 26754 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 26755* Function definition of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgnfn 26756* Functionality and domain of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmSgn

Theorempsgndmsubg 26757 The finitary permutations are a subgroup. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmSgn       SubGrp

Theorempsgneldm 26758 Property of being a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmSgn

Theorempsgneldm2 26759* The finitary permutations are the span of the transpositons. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgneldm2i 26760 A sequence of transpositions describes a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgneu 26761* A finitary permutation has exactly one parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgnval 26762* Value of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgnvali 26763* A finitary permutation has at least one representation for its parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgnvalii 26764 Any representation of a permutation is length matching the permutation sign. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmTrsp       pmSgn       Word g

Theorempsgnpmtr 26765 All transpositions are odd. (Contributed by Stefan O'Rear, 29-Aug-2015.)
pmTrsp       pmSgn

Theoremcnmsgnsubg 26766 The signs form a multiplicative subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.)
mulGrpflds        SubGrp

Theoremcnmsgnbas 26767 The base set of the sign subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.)
mulGrpflds

Theoremcnmsgngrp 26768 The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.)
mulGrpflds

Theorempsgnghm 26769 The sign is a homomorphism from the finitary permutation group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmSgn       s        mulGrpflds

Theorempsgnghm2 26770 The sign is a homomorphism from the finite symmetric group to the numeric signs. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmSgn       mulGrpflds

16.16.57  The matrix algebra

Syntaxcmmul 26771 Syntax for the matrix multiplication operator.
maMul

Syntaxcmat 26772 Syntax for the square matrix algebra.
Mat

Definitiondf-mamu 26773* The operator which multiplies an MxN matrix with an NxP matrix. Note that it is not generally possible to recover the dimensions from the matrix, since all Nx0 and all 0xN matrices are represented by the empty set. (Contributed by Stefan O'Rear, 4-Sep-2015.)
maMul g

Definitiondf-mat 26774* The algebra of NxN matrices over a ring... (Contributed by Stefan O'Rear, 31-Aug-2015.)
Mat freeLMod sSet maMul

Theoremmamufval 26775* Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.)
maMul                                                  g

Theoremmamuval 26776* Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
maMul                                                                g

Theoremmamufv 26777* A cell in the multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
maMul                                                                              g

Theoremmndvcl 26778 Tuple-wise additive closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremmndvass 26779 Tuple-wise associativity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremmndvlid 26780 Tuple-wise left identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremmndvrid 26781 Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremgrpvlinv 26782 Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremgrpvrinv 26783 Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.)

Theoremmhmvlin 26784 Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015.)
MndHom

Theoremrngvcl 26785 Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Theoremgsumcom3 26786* A commutative law for finitely supported iterated sums. (Contributed by Stefan O'Rear, 2-Nov-2015.)
CMnd                                          g g g g

Theoremgsumcom3fi 26787* A commutative law for finite iterated sums. (Contributed by Stefan O'Rear, 5-Sep-2015.)
CMnd                            g g g g

Theoremmamucl 26788 Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.)
maMul

Theoremmamudiagcl 26789* Diagonal matrices are matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)

Theoremmamulid 26790* Diagonal matrices are left identities. (Contributed by Stefan O'Rear, 3-Sep-2015.)
maMul

Theoremmamurid 26791* Diagonal matrices are right identities. (Contributed by Stefan O'Rear, 3-Sep-2015.)
maMul

Theoremmamuass 26792 Matrix multiplication is associative. (Contributed by Stefan O'Rear, 5-Sep-2015.)
maMul        maMul        maMul        maMul

Theoremmamudi 26793 Matrix multiplication distributes over addition on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
maMul

Theoremmamudir 26794 Matrix multiplication distributes over addition on the right. (Contributed by Stefan O'Rear, 5-Sep-2015.)
maMul

Theoremmamuvs1 26795 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
maMul

Theoremmamuvs2 26796 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
maMul

Theoremmatval 26797 Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Mat        freeLMod        maMul        sSet

Theoremmatrcl 26798 Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Mat

Theoremmatmulr 26799 Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Mat        maMul

Theoremmatbas 26800 The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
Mat        freeLMod

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