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Theorem List for Metamath Proof Explorer - 14201-14300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcmg 14201 Extend class notation with a function mapping a group operation to the power operation for the group.
.g

Definitiondf-mnd 14202* Definition of a monoid. A monoid is a set equipped with an everywhere defined internal operation (so, a magma, see mndcl 14207), whose operation is associative (so, a semigroup, see mndass 14208) and has a two-sided neutral element (see mndid 14209). (Contributed by Mario Carneiro, 6-Jan-2015.)

Definitiondf-plusf 14203* Define group addition function. Usually we will use directly instead of , and they have the same behavior in most cases. The main advantage of is that it is a guaranteed function (mndplusf 14218), while only has closure (mndcl 14207). (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremismnd 14204* The predicate "is a monoid." (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremmgmidmo 14205* A two-sided identity element is unique (if it exists) in any magma. (Contributed by Mario Carneiro, 7-Dec-2014.)

Theoremmndlem1 14206 Lemma for monoid properties. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremmndcl 14207 Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremmndass 14208 A monoid operation is associative. (Contributed by NM, 14-Aug-2011.)

Theoremmndid 14209* A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.)

Theoremmndideu 14210* 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 14211 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)

Theoremmnd12g 14212 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)

Theoremmnd4g 14213 Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)

Theoremplusffval 14214* The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremplusfval 14215 The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremplusfeq 14216 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 14217 The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)

Theoremmndplusf 14218 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremgrpidval 14219* The value of the identity element of a group. (Contributed by NM, 20-Aug-2011.) (Revised by Mario Carneiro, 2-Oct-2015.)

Theoremfn0g 14220 The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)

Theorem0g0 14221 The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)

Theoremismgmid 14222* The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)

Theoremmgmidcl 14223* The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)

Theoremmgmlrid 14224* The identity element of a magma, if it exists, is a left and right identity. (Contributed by Mario Carneiro, 27-Dec-2014.)

Theoremismgmid2 14225* Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.)

Theoremmndidcl 14226 The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Theoremmndlrid 14227 A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011.)

Theoremmndlid 14228 The identity element of a monoid is a left identity. (Contributed by NM, 18-Aug-2011.)

Theoremmndrid 14229 The identity element of a monoid is a right identity. (Contributed by NM, 18-Aug-2011.)

Theoremgrpidd 14230* Deduce the identity element of a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremismndd 14231* Deduce a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremmndfo 14232 The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.)

Theoremmndpropd 14233* 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 14234* If two structures have the same group components (properties), they have the same identity element. (Contributed by Mario Carneiro, 27-Nov-2014.)

Theoremmndprop 14235 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 14236* Characterize a submonoid by closure properties. (Contributed by Mario Carneiro, 10-Jan-2015.)
s

Theoremsubmnd0 14237 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 14238 Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theoremprdsidlem 14239* Characterization of identity in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
s

Theoremprdsmndd 14240 The product of a family of monoids is a monoid. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theoremprds0g 14241 Zero in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
s

Theorempwsmnd 14242 The structure power of a monoid is a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theorempws0g 14243 Zero in a product of monoids. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theoremimasmnd2 14244* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremimasmnd 14245* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremimasmndf1 14246 The image of a monoid under an injection is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremxpsmnd 14247 The binary product of monoids is a monoid. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

10.1.2  Monoid homomorphisms and submonoids

Syntaxcmhm 14248 Hom-set generator class for monoids.
MndHom

Syntaxcsubmnd 14249 Class function taking a monoid to its lattice of submonoids.
SubMnd

Definitiondf-mhm 14250* 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 14251* 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 14252* Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmrcl1 14253 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmrcl2 14254 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmf 14255 A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmpropd 14256* 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 14257 A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhm0 14258 A monoid homorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremsubmrcl 14259 Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremissubm 14260* Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremissubm2 14261 Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
s        SubMnd

Theoremsubmss 14262 Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremsubmid 14263 Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.)
SubMnd

Theoremsubm0cl 14264 Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremsubmcl 14265 Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
SubMnd

Theoremsubmmnd 14266 Submonoids are themselves monoids under the given operation. (Contributed by Mario Carneiro, 7-Mar-2015.)
s        SubMnd

Theoremsubmbas 14267 The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015.)
s        SubMnd

Theoremsubm0 14268 Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
s               SubMnd

Theoremsubsubm 14269 A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.)
s        SubMnd SubMnd SubMnd

Theorem0mhm 14270 The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
MndHom

Theoremresmhm 14271 Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
s        MndHom SubMnd MndHom

Theoremresmhm2 14272 One direction of resmhm2b 14273. (Contributed by Mario Carneiro, 18-Jun-2015.)
s        MndHom SubMnd MndHom

Theoremresmhm2b 14273 Restriction of a the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
s        SubMnd MndHom MndHom

Theoremmhmco 14274 The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
MndHom MndHom MndHom

Theoremmhmima 14275 The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
MndHom SubMnd SubMnd

Theoremmhmeql 14276 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 14277 Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.)
SubMnd ACS

Theoremprdspjmhm 14278* 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 14279* 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 14280* Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.)
s                      MndHom

Theorempwsco1mhm 14281* 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 14282* 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 13279. If order is not significant, it is simpler to use families instead.

Theoremgsumvallem1 14283* 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 14284* 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 14285 A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremgsumvalx 14286* Expand out the substitutions in df-gsum 13279. (Contributed by Mario Carneiro, 18-Sep-2015.)
g

Theoremgsumval 14287* Expand out the substitutions in df-gsum 13279. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumpropd 14288 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 14233 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.)
g g

Theoremgsumress 14289* 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 14290 Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.)
SubMnd              s        g g

Theoremgsumval1 14291* Value of the group sum operation when every element being summed is an identity of . (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsum0 14292 Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumz 14293* Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumval2a 14294* Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumval2 14295 Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumwsubmcl 14296 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 14297 A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.)
g

Theoremgsumwcl 14298 Closure of the composite of a word in a structure . (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word g

Theoremgsumccat 14299 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 14300 Valuation of a pair in a monoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
g

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