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Theorem List for Metamath Proof Explorer - 14401-14500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempwsmnd 14401 The structure power of a monoid is a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theorempws0g 14402 Zero in a product of monoids. (Contributed by Mario Carneiro, 11-Jan-2015.)
s

Theoremimasmnd2 14403* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremimasmnd 14404* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremimasmndf1 14405 The image of a monoid under an injection is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
s

Theoremxpsmnd 14406 The binary product of monoids is a monoid. (Contributed by Mario Carneiro, 20-Aug-2015.)
s

10.1.2  Monoid homomorphisms and submonoids

Syntaxcmhm 14407 Hom-set generator class for monoids.
MndHom

Syntaxcsubmnd 14408 Class function taking a monoid to its lattice of submonoids.
SubMnd

Definitiondf-mhm 14409* 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 14410* 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 14411* Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmrcl1 14412 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmrcl2 14413 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmf 14414 A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhmpropd 14415* 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 14416 A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremmhm0 14417 A monoid homorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
MndHom

Theoremsubmrcl 14418 Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremissubm 14419* Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremissubm2 14420 Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
s        SubMnd

Theoremsubmss 14421 Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremsubmid 14422 Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.)
SubMnd

Theoremsubm0cl 14423 Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
SubMnd

Theoremsubmcl 14424 Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
SubMnd

Theoremsubmmnd 14425 Submonoids are themselves monoids under the given operation. (Contributed by Mario Carneiro, 7-Mar-2015.)
s        SubMnd

Theoremsubmbas 14426 The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015.)
s        SubMnd

Theoremsubm0 14427 Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
s               SubMnd

Theoremsubsubm 14428 A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.)
s        SubMnd SubMnd SubMnd

Theorem0mhm 14429 The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
MndHom

Theoremresmhm 14430 Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
s        MndHom SubMnd MndHom

Theoremresmhm2 14431 One direction of resmhm2b 14432. (Contributed by Mario Carneiro, 18-Jun-2015.)
s        MndHom SubMnd MndHom

Theoremresmhm2b 14432 Restriction of a the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
s        SubMnd MndHom MndHom

Theoremmhmco 14433 The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
MndHom MndHom MndHom

Theoremmhmima 14434 The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
MndHom SubMnd SubMnd

Theoremmhmeql 14435 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 14436 Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.)
SubMnd ACS

Theoremprdspjmhm 14437* 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 14438* 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 14439* Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.)
s                      MndHom

Theorempwsco1mhm 14440* 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 14441* 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 13399. If order is not significant, it is simpler to use families instead.

Theoremgsumvallem1 14442* 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 14443* 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 14444 A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theoremgsumvalx 14445* Expand out the substitutions in df-gsum 13399. (Contributed by Mario Carneiro, 18-Sep-2015.)
g

Theoremgsumval 14446* Expand out the substitutions in df-gsum 13399. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumpropd 14447 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 14392 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.)
g g

Theoremgsumress 14448* 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 14449 Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.)
SubMnd              s        g g

Theoremgsumval1 14450* Value of the group sum operation when every element being summed is an identity of . (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsum0 14451 Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumz 14452* Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumval2a 14453* Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumval2 14454 Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
g

Theoremgsumwsubmcl 14455 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 14456 A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.)
g

Theoremgsumwcl 14457 Closure of the composite of a word in a structure . (Contributed by Stefan O'Rear, 15-Aug-2015.)
Word g

Theoremgsumccat 14458 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 14459 Valuation of a pair in a monoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
g

Theoremgsumspl 14460 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 14461 Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.)
MndHom Word g g

Theoremgsumwspan 14462* 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 14463 Extend class definition with the free monoid construction.
freeMnd

Syntaxcvrmd 14464 Extend class notation with free monoid injection.
varFMnd

Definitiondf-frmd 14465 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 14466* 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 14467 Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd       Word        concat

Theoremfrmdbas 14468 The base set of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd              Word

Theoremfrmdelbas 14469 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 14470 The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd                     concat

Theoremfrmdadd 14471 Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd                     concat

Theoremvrmdfval 14472* The canonical injection from the generating set to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
varFMnd

Theoremvrmdval 14473 The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
varFMnd

Theoremvrmdf 14474 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 14475 A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
freeMnd

Theoremfrmd0 14476 The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd

Theoremfrmdsssubm 14477 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 14478 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 14479 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 14480* 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 14481* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.)
freeMnd              Word g                             varFMnd

Theoremfrmdup3 14482* 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 14483* Define class of all groups. A group is a monoid (df-mnd 14361) 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 13147) and an internal group operation (notated per df-plusg 13215). 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 14489), associativity (so for any a, b, c, see grpass 14490), 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 14492). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 15086). Subgroups can often be formed from groups, see df-subg 14612. An example of an (Abelian) group is the set of complex numbers over the group operation (addition), as proven in cnaddablx 15152; an Abelian group is a group as proven in ablgrp 15088. Other structures include groups, including unital rings (df-rng 15334) and fields (df-field 15509). (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)

Definitiondf-minusg 14484* Define inverse of group element. (Contributed by NM, 24-Aug-2011.)

Definitiondf-sbg 14485* Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)

Definitiondf-mulg 14486* Define the group multiple function, also known as group exponentiation when viewed multiplicatively. (Contributed by Mario Carneiro, 11-Dec-2014.)
.g

Theoremisgrp 14487* 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 14488 A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)

Theoremgrpcl 14489 Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.)

Theoremgrpass 14490 A group operation is associative. (Contributed by NM, 14-Aug-2011.)

Theoremgrpinvex 14491* Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremgrpideu 14492* 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 14493 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Theoremgrppropd 14494* 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 14495 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 14496 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 14497 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 15340, on the other hand, requires that we know in advance that is a ring.) (Contributed by NM, 11-Oct-2013.)

Theoremisgrpd2e 14498* Deduce a group from its properties. In this version of isgrpd2 14499, we don't assume there is an expression for the inverse of . (Contributed by NM, 10-Aug-2013.)

Theoremisgrpd2 14499* 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 2284, but we make an exception for theorems such as isgrpd2 14499, ismndd 14390, and islmodd 15627 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)

Theoremisgrpde 14500* Deduce a group from its properties. In this version of isgrpd 14501, we don't assume there is an expression for the inverse of . (Contributed by NM, 6-Jan-2015.)

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