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

Theoremmhm0 14701 A monoid homorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremsubmrcl 14702 Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremissubm 14703* Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremissubm2 14704 Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
s        SubMnd

Theoremsubmss 14705 Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremsubmid 14706 Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.)
SubMnd

Theoremsubm0cl 14707 Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremsubmcl 14708 Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
SubMnd

Theoremsubmmnd 14709 Submonoids are themselves monoids under the given operation. (Contributed by Mario Carneiro, 7-Mar-2015.)
s        SubMnd

Theoremsubmbas 14710 The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015.)
s        SubMnd

Theoremsubm0 14711 Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
s               SubMnd

Theoremsubsubm 14712 A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.)
s        SubMnd SubMnd SubMnd

Theorem0mhm 14713 The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
MndHom

Theoremresmhm 14714 Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
s        MndHom SubMnd MndHom

Theoremresmhm2 14715 One direction of resmhm2b 14716. (Contributed by Mario Carneiro, 18-Jun-2015.)
s        MndHom SubMnd MndHom

Theoremresmhm2b 14716 Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
s        SubMnd MndHom MndHom

Theoremmhmco 14717 The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
MndHom MndHom MndHom

Theoremmhmima 14718 The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
MndHom SubMnd SubMnd

Theoremmhmeql 14719 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 14720 Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.)
SubMnd ACS

Theoremprdspjmhm 14721* 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 14722* 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 14723* Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.)
s                      MndHom

Theorempwsco1mhm 14724* 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 14725* 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 13683. If order is not significant, it is simpler to use families instead.

Theoremgsumvallem1 14726* 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 14727* 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 14728 A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremgsumvalx 14729* Expand out the substitutions in df-gsum 13683. (Contributed by Mario Carneiro, 18-Sep-2015.)
g

Theoremgsumval 14730* Expand out the substitutions in df-gsum 13683. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumpropd 14731 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 14676 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.)
g g

Theoremgsumress 14732* 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 14733 Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.)
SubMnd              s        g g

Theoremgsumval1 14734* Value of the group sum operation when every element being summed is an identity of . (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsum0 14735 Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumz 14736* Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumval2a 14737* Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumval2 14738 Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumwsubmcl 14739 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 14740 A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.)
g

Theoremgsumwcl 14741 Closure of the composite of a word in a structure . (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word g

Theoremgsumccat 14742 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 14743 Valuation of a pair in a monoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
g

Theoremgsumspl 14744 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 14745 Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.)
MndHom Word g g

Theoremgsumwspan 14746* 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 14747 Extend class definition with the free monoid construction.
freeMnd

Syntaxcvrmd 14748 Extend class notation with free monoid injection.
varFMnd

Definitiondf-frmd 14749 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 14750* 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 14751 Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd       Word        concat

Theoremfrmdbas 14752 The base set of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd              Word

Theoremfrmdelbas 14753 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 14754 The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd                     concat

Theoremfrmdadd 14755 Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd                     concat

Theoremvrmdfval 14756* The canonical injection from the generating set to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
varFMnd

Theoremvrmdval 14757 The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
varFMnd

Theoremvrmdf 14758 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 14759 A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd

Theoremfrmd0 14760 The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd

Theoremfrmdsssubm 14761 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 14762 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 14763 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 14764* 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 14765* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd              Word g                             varFMnd

Theoremfrmdup3 14766* 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 14767* Define class of all groups. A group is a monoid (df-mnd 14645) 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 13429) and an internal group operation (notated per df-plusg 13497). 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 14773), associativity (so for any a, b, c, see grpass 14774), 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 14776). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 15370). Subgroups can often be formed from groups, see df-subg 14896. An example of an (Abelian) group is the set of complex numbers over the group operation (addition), as proven in cnaddablx 15436; an Abelian group is a group as proven in ablgrp 15372. Other structures include groups, including unital rings (df-rng 15618) and fields (df-field 15793). (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)

Definitiondf-minusg 14768* Define inverse of group element. (Contributed by NM, 24-Aug-2011.)

Definitiondf-sbg 14769* Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)

Definitiondf-mulg 14770* Define the group multiple function, also known as group exponentiation when viewed multiplicatively. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremisgrp 14771* 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 14772 A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremgrpcl 14773 Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.)

Theoremgrpass 14774 A group operation is associative. (Contributed by NM, 14-Aug-2011.)

Theoremgrpinvex 14775* Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrpideu 14776* 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 14777 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremgrppropd 14778* 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 14779 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 14780 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 14781 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 15624, on the other hand, requires that we know in advance that is a ring.) (Contributed by NM, 11-Oct-2013.)

Theoremisgrpd2e 14782* Deduce a group from its properties. In this version of isgrpd2 14783, we don't assume there is an expression for the inverse of . (Contributed by NM, 10-Aug-2013.)

Theoremisgrpd2 14783* 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 2404, but we make an exception for theorems such as isgrpd2 14783, ismndd 14674, and islmodd 15911 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)

Theoremisgrpde 14784* Deduce a group from its properties. In this version of isgrpd 14785, we don't assume there is an expression for the inverse of . (Contributed by NM, 6-Jan-2015.)

Theoremisgrpd 14785* Deduce a group from its properties. Unlike isgrpd2 14783, 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 14786* Properties that determine a group. (negative) is normally dependent on i.e. read it as . (Contributed by NM, 3-Sep-2011.)

Theoremisgrpix 14787* 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 14788 The identity element of a group belongs to the group. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremgrpbn0 14789 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Theoremgrplid 14790 The identity element of a group is a left identity. (Contributed by NM, 18-Aug-2011.)

Theoremgrprid 14791 The identity element of a group is a right identity. (Contributed by NM, 18-Aug-2011.)

Theoremgrpn0 14792 A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)

Theoremgrprcan 14793 Right cancellation law for groups. (Contributed by NM, 24-Aug-2011.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)

Theoremgrpinveu 14794* 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 14795 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 14796 Properties showing that an element is the identity element of a group. (Contributed by NM, 7-Aug-2013.)

Theoremgrpidd2 14797* Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 14785. (Contributed by Mario Carneiro, 14-Jun-2015.)

Theoremgrpinvfval 14798* The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)

Theoremgrpinvval 14799* The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)

Theoremgrpinvfn 14800 Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)

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