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Theorem List for Metamath Proof Explorer - 14701-14800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremresmhm2 14701 One direction of resmhm2b 14702. (Contributed by Mario Carneiro, 18-Jun-2015.)
s        MndHom SubMnd MndHom

Theoremresmhm2b 14702 Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
s        SubMnd MndHom MndHom

Theoremmhmco 14703 The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
MndHom MndHom MndHom

Theoremmhmima 14704 The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
MndHom SubMnd SubMnd

Theoremmhmeql 14705 The equalizer of two monoid homomorphisms is a submonoid. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
MndHom MndHom SubMnd

Theoremsubmacs 14706 Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.)
SubMnd ACS

Theoremprdspjmhm 14707* A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
s                                          MndHom

Theorempwspjmhm 14708* A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 15-Jun-2015.)
s               MndHom

Theorempwsdiagmhm 14709* Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.)
s                      MndHom

Theorempwsco1mhm 14710* Right composition with a function on the index sets yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
s        s                                           MndHom

Theorempwsco2mhm 14711* Left composition with a monoid homomorphism yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
s        s                      MndHom        MndHom

10.1.3  Ordered group sum operation

One important use of words is as formal composites in cases where order is significant, using the general sum operator df-gsum 13669. If order is not significant, it is simpler to use families instead.

Theoremgsumvallem1 14712* Lemma for properties of the set of identities of . Either has no identities, and , or it has one and this identity is unique and identified by the function. (Contributed by Mario Carneiro, 7-Dec-2014.)

Theoremgsumvallem2 14713* Lemma for properties of the set of identities of . The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014.)

Theoremfisuppfi 14714 A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremgsumvalx 14715* Expand out the substitutions in df-gsum 13669. (Contributed by Mario Carneiro, 18-Sep-2015.)
g

Theoremgsumval 14716* Expand out the substitutions in df-gsum 13669. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumpropd 14717 The group sum depends only on the base set and additive operation. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 14662 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.)
g g

Theoremgsumress 14718* The group sum in a substructure is the same as the group sum in the original structure. The only requirement on the substructure is that it contain the identity element; neither nor need be groups. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
s                                                  g g

Theoremgsumsubm 14719 Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.)
SubMnd              s        g g

Theoremgsumval1 14720* Value of the group sum operation when every element being summed is an identity of . (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsum0 14721 Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumz 14722* Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumval2a 14723* Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumval2 14724 Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumwsubmcl 14725 Closure of the composite in any submonoid. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
SubMnd Word g

Theoremgsumws1 14726 A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.)
g

Theoremgsumwcl 14727 Closure of the composite of a word in a structure . (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word g

Theoremgsumccat 14728 Homomorphic property of composites. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
Word Word g concat g g

Theoremgsumws2 14729 Valuation of a pair in a monoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
g

Theoremgsumspl 14730 The primary purpose of the splice construction is to enable local rewrites. Thus, in any monoidal valuation, if a splice does not cause a local change it does not cause a global change. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Word                      Word        Word        g g        g splice g splice

Theoremgsumwmhm 14731 Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.)
MndHom Word g g

Theoremgsumwspan 14732* The submonoid generated by a set of elements is precisely the set of elements which can be expressed as finite products of the generator. (Contributed by Stefan O'Rear, 22-Aug-2015.)
mrClsSubMnd       Word g

10.1.4  Free monoids

Syntaxcfrmd 14733 Extend class definition with the free monoid construction.
freeMnd

Syntaxcvrmd 14734 Extend class notation with free monoid injection.
varFMnd

Definitiondf-frmd 14735 Define a free monoid over a set of generators, defined as the set of finite strings on with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd Word concat Word Word

Definitiondf-vrmd 14736* Define a free monoid over a set of generators, defined as the set of finite strings on with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.)
varFMnd

Theoremfrmdval 14737 Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd       Word        concat

Theoremfrmdbas 14738 The base set of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd              Word

Theoremfrmdelbas 14739 An element of the base set of a free monoid is a string on the generators. (Contributed by Mario Carneiro, 27-Feb-2016.)
freeMnd              Word

Theoremfrmdplusg 14740 The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd                     concat

Theoremfrmdadd 14741 Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd                     concat

Theoremvrmdfval 14742* The canonical injection from the generating set to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
varFMnd

Theoremvrmdval 14743 The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
varFMnd

Theoremvrmdf 14744 The mapping from the index set to the generators is a function into the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
varFMnd       Word

Theoremfrmdmnd 14745 A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd

Theoremfrmd0 14746 The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd

Theoremfrmdsssubm 14747 The set of words taking values in a subset is a (free) submonoid of the free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd       Word SubMnd

Theoremfrmdgsum 14748 Any word in a free monoid can be expressed as the sum of the singletons composing it. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd       varFMnd       Word g

Theoremfrmdss2 14749 A subset of generators is contained in a submonoid iff the set of words on the generators is in the submonoid. This can be viewed as an elementary way of saying "the monoidal closure of is Word ". (Contributed by Mario Carneiro, 2-Oct-2015.)
freeMnd       varFMnd       SubMnd Word

Theoremfrmdup1 14750* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd              Word g                             MndHom

Theoremfrmdup2 14751* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd              Word g                             varFMnd

Theoremfrmdup3 14752* Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd              varFMnd       MndHom

10.2  Groups

10.2.1  Definition and basic properties

Definitiondf-grp 14753* Define class of all groups. A group is a monoid (df-mnd 14631) whose internal operation is such that every element admits a left inverse (which can be proven to be a two-sided inverse). Thus, a group is an algebraic structure formed from a base set of elements (notated per df-base 13415) and an internal group operation (notated per df-plusg 13483). The operation combines any two elements of the group base set and must satisfy the 4 group axioms: closure (the result of the group operation must always be a member of the base set, see grpcl 14759), associativity (so for any a, b, c, see grpass 14760), identity (there must be an element such that for any a), and inverse (for each element a in the base set, there must be an element in the base set such that ). It can be proven that the identity element is unique (grpideu 14762). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 15356). Subgroups can often be formed from groups, see df-subg 14882. An example of an (Abelian) group is the set of complex numbers over the group operation (addition), as proven in cnaddablx 15422; an Abelian group is a group as proven in ablgrp 15358. Other structures include groups, including unital rings (df-rng 15604) and fields (df-field 15779). (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)

Definitiondf-minusg 14754* Define inverse of group element. (Contributed by NM, 24-Aug-2011.)

Definitiondf-sbg 14755* Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)

Definitiondf-mulg 14756* Define the group multiple function, also known as group exponentiation when viewed multiplicatively. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremisgrp 14757* The predicate "is a group." (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrpmnd 14758 A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremgrpcl 14759 Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.)

Theoremgrpass 14760 A group operation is associative. (Contributed by NM, 14-Aug-2011.)

Theoremgrpinvex 14761* Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrpideu 14762* The two-sided identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 8-Dec-2014.)

Theoremgrpplusf 14763 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremgrppropd 14764* If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Theoremgrpprop 14765 If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)

Theoremgrppropstr 14766 Generalize a specific 2-element group to show that any set with the same (relevant) properties is also a group. (Contributed by NM, 28-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrpss 14767 Show that a structure extending a constructed group (e.g. a ring) is also a group. This allows us to prove that a constructed potential ring is a group before we know that it is also a ring. (Theorem rnggrp 15610, on the other hand, requires that we know in advance that is a ring.) (Contributed by NM, 11-Oct-2013.)

Theoremisgrpd2e 14768* Deduce a group from its properties. In this version of isgrpd2 14769, we don't assume there is an expression for the inverse of . (Contributed by NM, 10-Aug-2013.)

Theoremisgrpd2 14769* Deduce a group from its properties. (negative) is normally dependent on i.e. read it as . Note: normally we don't use a antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2401, but we make an exception for theorems such as isgrpd2 14769, ismndd 14660, and islmodd 15897 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)

Theoremisgrpde 14770* Deduce a group from its properties. In this version of isgrpd 14771, we don't assume there is an expression for the inverse of . (Contributed by NM, 6-Jan-2015.)

Theoremisgrpd 14771* Deduce a group from its properties. Unlike isgrpd2 14769, this one goes straight from the base properties rather than going through . (negative) is normally dependent on i.e. read it as . (Contributed by NM, 6-Jun-2013.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremisgrpi 14772* Properties that determine a group. (negative) is normally dependent on i.e. read it as . (Contributed by NM, 3-Sep-2011.)

Theoremisgrpix 14773* Properties that determine a group. Read as . Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.)

Theoremgrpidcl 14774 The identity element of a group belongs to the group. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremgrpbn0 14775 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Theoremgrplid 14776 The identity element of a group is a left identity. (Contributed by NM, 18-Aug-2011.)

Theoremgrprid 14777 The identity element of a group is a right identity. (Contributed by NM, 18-Aug-2011.)

Theoremgrpn0 14778 A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)

Theoremgrprcan 14779 Right cancellation law for groups. (Contributed by NM, 24-Aug-2011.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)

Theoremgrpinveu 14780* The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 24-Aug-2011.)

Theoremgrpid 14781 Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.)

Theoremisgrpid2 14782 Properties showing that an element is the identity element of a group. (Contributed by NM, 7-Aug-2013.)

Theoremgrpidd2 14783* Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 14771. (Contributed by Mario Carneiro, 14-Jun-2015.)

Theoremgrpinvfval 14784* The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)

Theoremgrpinvval 14785* The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)

Theoremgrpinvfn 14786 Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Theoremgrpinvfvi 14787 The group inverse function is compatible with identity-function protection. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Theoremgrpsubfval 14788* Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.)

Theoremgrpsubval 14789 Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 13-Dec-2014.)

Theoremgrpinvf 14790 The group inversion operation is a function on the base set. (Contributed by Mario Carneiro, 4-May-2015.)

Theoremgrpinvcl 14791 A group element's inverse is a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 4-May-2015.)

Theoremgrplinv 14792 The left inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrprinv 14793 The right inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrpinvid1 14794 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)

Theoremgrpinvid2 14795 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)

Theoremisgrpinv 14796* Properties showing that a function is the inverse function of a group. (Contributed by NM, 7-Aug-2013.) (Revised by Mario Carneiro, 2-Oct-2015.)

Theoremgrpinvid 14797 The inverse of the identity element of a group. (Contributed by NM, 24-Aug-2011.)

Theoremgrplcan 14798 Left cancellation law for groups. (Contributed by NM, 25-Aug-2011.)

Theoremgrpinvinv 14799 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.)

Theoremgrpinvcnv 14800 The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.)

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