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

Theoremmndass 14701 A monoid operation is associative. (Contributed by NM, 14-Aug-2011.)

Theoremmndid 14702* A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.)

Theoremmndideu 14703* The two-sided identity element of a monoid is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by Mario Carneiro, 8-Dec-2014.)

Theoremmnd32g 14704 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)

Theoremmnd12g 14705 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)

Theoremmnd4g 14706 Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)

Theoremplusffval 14707* The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremplusfval 14708 The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremplusfeq 14709 If the addition operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremplusffn 14710 The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)

Theoremmndplusf 14711 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremgrpidval 14712* The value of the identity element of a group. (Contributed by NM, 20-Aug-2011.) (Revised by Mario Carneiro, 2-Oct-2015.)

Theoremfn0g 14713 The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)

Theorem0g0 14714 The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)

Theoremismgmid 14715* The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)

Theoremmgmidcl 14716* The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)

Theoremmgmlrid 14717* The identity element of a magma, if it exists, is a left and right identity. (Contributed by Mario Carneiro, 27-Dec-2014.)

Theoremismgmid2 14718* Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.)

Theoremmndidcl 14719 The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremmndlrid 14720 A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011.)

Theoremmndlid 14721 The identity element of a monoid is a left identity. (Contributed by NM, 18-Aug-2011.)

Theoremmndrid 14722 The identity element of a monoid is a right identity. (Contributed by NM, 18-Aug-2011.)

Theoremgrpidd 14723* Deduce the identity element of a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremismndd 14724* Deduce a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremmndfo 14725 The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.)

Theoremmndpropd 14726* If two structures have the same group components (properties), one is a monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremgrpidpropd 14727* If two structures have the same group components (properties), they have the same identity element. (Contributed by Mario Carneiro, 27-Nov-2014.)

Theoremmndprop 14728 If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)

Theoremissubmnd 14729* Characterize a submonoid by closure properties. (Contributed by Mario Carneiro, 10-Jan-2015.)
s

Theoremsubmnd0 14730 The zero of a submonoid is the same as the zero in the parent monoid. (Note that we must add the condition that the zero of the parent monoid is actually contained in the submonoid, because it is possible to have "subsets that are monoids" which are not submonoids because they have a different identity element.) (Contributed by Mario Carneiro, 10-Jan-2015.)
s

Theoremprdsplusgcl 14731 Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theoremprdsidlem 14732* Characterization of identity in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
s

Theoremprdsmndd 14733 The product of a family of monoids is a monoid. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theoremprds0g 14734 Zero in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theorempwsmnd 14735 The structure power of a monoid is a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theorempws0g 14736 Zero in a product of monoids. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theoremimasmnd2 14737* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremimasmnd 14738* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremimasmndf1 14739 The image of a monoid under an injection is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremxpsmnd 14740 The binary product of monoids is a monoid. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

10.1.2  Monoid homomorphisms and submonoids

Syntaxcmhm 14741 Hom-set generator class for monoids.
MndHom

Syntaxcsubmnd 14742 Class function taking a monoid to its lattice of submonoids.
SubMnd

Definitiondf-mhm 14743* A monoid homomorphism is a function on the base sets which preserves the binary operation and the identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Definitiondf-submnd 14744* A submonoid is a subset of a monoid which contains the identity and is closed under the operation. Such subsets are themselves monoids with the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremismhm 14745* Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmrcl1 14746 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmrcl2 14747 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmf 14748 A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmpropd 14749* Monoid homomorphism depends only on the monoidal attributes of structures. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 7-Nov-2015.)
MndHom MndHom

Theoremmhmlin 14750 A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhm0 14751 A monoid homorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremsubmrcl 14752 Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremissubm 14753* Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremissubm2 14754 Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
s        SubMnd

Theoremsubmss 14755 Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremsubmid 14756 Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.)
SubMnd

Theoremsubm0cl 14757 Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremsubmcl 14758 Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
SubMnd

Theoremsubmmnd 14759 Submonoids are themselves monoids under the given operation. (Contributed by Mario Carneiro, 7-Mar-2015.)
s        SubMnd

Theoremsubmbas 14760 The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015.)
s        SubMnd

Theoremsubm0 14761 Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
s               SubMnd

Theoremsubsubm 14762 A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.)
s        SubMnd SubMnd SubMnd

Theorem0mhm 14763 The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
MndHom

Theoremresmhm 14764 Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
s        MndHom SubMnd MndHom

Theoremresmhm2 14765 One direction of resmhm2b 14766. (Contributed by Mario Carneiro, 18-Jun-2015.)
s        MndHom SubMnd MndHom

Theoremresmhm2b 14766 Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
s        SubMnd MndHom MndHom

Theoremmhmco 14767 The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
MndHom MndHom MndHom

Theoremmhmima 14768 The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
MndHom SubMnd SubMnd

Theoremmhmeql 14769 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 14770 Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.)
SubMnd ACS

Theoremprdspjmhm 14771* 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 14772* 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 14773* Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.)
s                      MndHom

Theorempwsco1mhm 14774* 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 14775* 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 13733. If order is not significant, it is simpler to use families instead.

Theoremgsumvallem1 14776* 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 14777* 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 14778 A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremgsumvalx 14779* Expand out the substitutions in df-gsum 13733. (Contributed by Mario Carneiro, 18-Sep-2015.)
g

Theoremgsumval 14780* Expand out the substitutions in df-gsum 13733. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumpropd 14781 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 14726 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.)
g g

Theoremgsumress 14782* 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 14783 Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.)
SubMnd              s        g g

Theoremgsumval1 14784* Value of the group sum operation when every element being summed is an identity of . (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsum0 14785 Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumz 14786* Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumval2a 14787* Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumval2 14788 Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumwsubmcl 14789 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 14790 A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.)
g

Theoremgsumwcl 14791 Closure of the composite of a word in a structure . (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word g

Theoremgsumccat 14792 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 14793 Valuation of a pair in a monoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
g

Theoremgsumspl 14794 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 14795 Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.)
MndHom Word g g

Theoremgsumwspan 14796* 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 14797 Extend class definition with the free monoid construction.
freeMnd

Syntaxcvrmd 14798 Extend class notation with free monoid injection.
varFMnd

Definitiondf-frmd 14799 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 14800* 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

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