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Theorem List for Intuitionistic Logic Explorer - 12801-12900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
7.1.3  Semigroups

A semigroup (Smgrp, see df-sgrp 12802) is a set together with an associative binary operation (see Wikipedia, Semigroup, 8-Jan-2020, https://en.wikipedia.org/wiki/Semigroup 12802). In other words, a semigroup is an associative magma. The notion of semigroup is a generalization of that of group where the existence of an identity or inverses is not required.

 
Syntaxcsgrp 12801 Extend class notation with class of all semigroups.
 class Smgrp
 
Definitiondf-sgrp 12802* A semigroup is a set equipped with an everywhere defined internal operation (so, a magma, see df-mgm 12769), whose operation is associative. Definition in section II.1 of [Bruck] p. 23, or of an "associative magma" in definition 5 of [BourbakiAlg1] p. 4 . (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
 |- Smgrp  =  { g  e. Mgm  |  [. ( Base `  g )  /  b ]. [. ( +g  `  g )  /  o ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( ( x o y ) o z )  =  ( x o ( y o z ) ) }
 
Theoremissgrp 12803* The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  ( M  e. Smgrp  <->  ( M  e. Mgm  /\ 
 A. x  e.  B  A. y  e.  B  A. z  e.  B  (
 ( x  .o.  y
 )  .o.  z )  =  ( x  .o.  (
 y  .o.  z )
 ) ) )
 
Theoremissgrpv 12804* The predicate "is a semigroup" for a structure which is a set. (Contributed by AV, 1-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  ( M  e.  V  ->  ( M  e. Smgrp  <->  A. x  e.  B  A. y  e.  B  ( ( x  .o.  y
 )  e.  B  /\  A. z  e.  B  ( ( x  .o.  y
 )  .o.  z )  =  ( x  .o.  (
 y  .o.  z )
 ) ) ) )
 
Theoremissgrpn0 12805* The predicate "is a semigroup" for a structure with a nonempty base set. (Contributed by AV, 1-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  ( A  e.  B  ->  ( M  e. Smgrp  <->  A. x  e.  B  A. y  e.  B  ( ( x  .o.  y
 )  e.  B  /\  A. z  e.  B  ( ( x  .o.  y
 )  .o.  z )  =  ( x  .o.  (
 y  .o.  z )
 ) ) ) )
 
Theoremisnsgrp 12806 A condition for a structure not to be a semigroup. (Contributed by AV, 30-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  (
 ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( ( X  .o.  Y )  .o. 
 Z )  =/=  ( X  .o.  ( Y  .o.  Z ) )  ->  M  e/ Smgrp ) )
 
Theoremsgrpmgm 12807 A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
 |-  ( M  e. Smgrp  ->  M  e. Mgm )
 
Theoremsgrpass 12808 A semigroup operation is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 30-Jan-2020.)
 |-  B  =  ( Base `  G )   &    |-  .o.  =  (
 +g  `  G )   =>    |-  (
 ( G  e. Smgrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .o.  Y )  .o. 
 Z )  =  ( X  .o.  ( Y  .o.  Z ) ) )
 
Theoremsgrp0 12809 Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.)
 |-  ( ( M  e.  V  /\  ( Base `  M )  =  (/) )  ->  M  e. Smgrp )
 
Theoremsgrp1 12810 The structure with one element and the only closed internal operation for a singleton is a semigroup. (Contributed by AV, 10-Feb-2020.)
 |-  M  =  { <. (
 Base `  ndx ) ,  { I } >. , 
 <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
 >. }   =>    |-  ( I  e.  V  ->  M  e. Smgrp )
 
7.1.4  Definition and basic properties of monoids

According to Wikipedia ("Monoid", https://en.wikipedia.org/wiki/Monoid, 6-Feb-2020,) "In abstract algebra [...] a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are semigroups with identity.". In the following, monoids are defined in the second way (as semigroups with identity), see df-mnd 12812, whereas many authors define magmas in the first way (as algebraic structure with a single associative binary operation and an identity element, i.e. without the need of a definition for/knowledge about semigroups), see ismnd 12814. See, for example, the definition in [Lang] p. 3: "A monoid is a set G, with a law of composition which is associative, and having a unit element".

 
Syntaxcmnd 12811 Extend class notation with class of all monoids.
 class  Mnd
 
Definitiondf-mnd 12812* A monoid is a semigroup, which has a two-sided neutral element. Definition 2 in [BourbakiAlg1] p. 12. In other words (according to the definition in [Lang] p. 3), a monoid is a set equipped with an everywhere defined internal operation (see mndcl 12818), whose operation is associative (see mndass 12819) and has a two-sided neutral element (see mndid 12820), see also ismnd 12814. (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
 |- 
 Mnd  =  { g  e. Smgrp  |  [. ( Base `  g )  /  b ]. [. ( +g  `  g
 )  /  p ]. E. e  e.  b  A. x  e.  b  ( ( e p x )  =  x  /\  ( x p e )  =  x ) }
 
Theoremismnddef 12813* The predicate "is a monoid", corresponding 1-to-1 to the definition. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 1-Feb-2020.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e.  Mnd  <->  ( G  e. Smgrp  /\  E. e  e.  B  A. a  e.  B  ( ( e 
 .+  a )  =  a  /\  ( a 
 .+  e )  =  a ) ) )
 
Theoremismnd 12814* The predicate "is a monoid". This is the defining theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl 12818), whose operation is associative (so, a semigroup, see also mndass 12819) and has a two-sided neutral element (see mndid 12820). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e.  Mnd  <->  (
 A. a  e.  B  A. b  e.  B  ( ( a  .+  b
 )  e.  B  /\  A. c  e.  B  ( ( a  .+  b
 )  .+  c )  =  ( a  .+  (
 b  .+  c )
 ) )  /\  E. e  e.  B  A. a  e.  B  ( ( e 
 .+  a )  =  a  /\  ( a 
 .+  e )  =  a ) ) )
 
Theoremsgrpidmndm 12815* A semigroup with an identity element which is inhabited is a monoid. Of course there could be monoids with the empty set as identity element, but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e. Smgrp  /\ 
 E. e  e.  B  ( E. w  w  e.  e  /\  e  =  .0.  ) )  ->  G  e.  Mnd )
 
Theoremmndsgrp 12816 A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.)
 |-  ( G  e.  Mnd  ->  G  e. Smgrp )
 
Theoremmndmgm 12817 A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.)
 |-  ( M  e.  Mnd  ->  M  e. Mgm )
 
Theoremmndcl 12818 Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Proof shortened by AV, 8-Feb-2020.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
 
Theoremmndass 12819 A monoid operation is associative. (Contributed by NM, 14-Aug-2011.) (Proof shortened by AV, 8-Feb-2020.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .+  Y )  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
 
Theoremmndid 12820* A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e.  Mnd 
 ->  E. u  e.  B  A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x ) )
 
Theoremmndideu 12821* 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.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e.  Mnd 
 ->  E! u  e.  B  A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x ) )
 
Theoremmnd32g 12822 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  ( Y  .+  Z )  =  ( Z  .+  Y ) )   =>    |-  ( ph  ->  (
 ( X  .+  Y )  .+  Z )  =  ( ( X  .+  Z )  .+  Y ) )
 
Theoremmnd12g 12823 Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  ( X  .+  Y )  =  ( Y  .+  X ) )   =>    |-  ( ph  ->  ( X  .+  ( Y  .+  Z ) )  =  ( Y  .+  ( X  .+  Z ) ) )
 
Theoremmnd4g 12824 Commutative/associative law for commutative monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  W  e.  B )   &    |-  ( ph  ->  ( Y  .+  Z )  =  ( Z  .+  Y ) )   =>    |-  ( ph  ->  ( ( X  .+  Y )  .+  ( Z  .+  W ) )  =  ( ( X  .+  Z )  .+  ( Y 
 .+  W ) ) )
 
Theoremmndidcl 12825 The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Mnd  ->  .0.  e.  B )
 
Theoremmndbn0 12826 The base set of a monoid is not empty. (It is also inhabited, as seen at mndidcl 12825). Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Mnd  ->  B  =/=  (/) )
 
Theoremhashfinmndnn 12827 A finite monoid has positive integer size. (Contributed by Rohan Ridenour, 3-Aug-2023.)
 |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  B  e.  Fin )   =>    |-  ( ph  ->  ( `  B )  e.  NN )
 
Theoremmndplusf 12828 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 3-Feb-2020.)
 |-  B  =  ( Base `  G )   &    |-  .+^  =  ( +f `  G )   =>    |-  ( G  e.  Mnd  ->  .+^ 
 : ( B  X.  B ) --> B )
 
Theoremmndlrid 12829 A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Mnd  /\  X  e.  B ) 
 ->  ( (  .0.  .+  X )  =  X  /\  ( X  .+  .0.  )  =  X )
 )
 
Theoremmndlid 12830 The identity element of a monoid is a left identity. (Contributed by NM, 18-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Mnd  /\  X  e.  B ) 
 ->  (  .0.  .+  X )  =  X )
 
Theoremmndrid 12831 The identity element of a monoid is a right identity. (Contributed by NM, 18-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Mnd  /\  X  e.  B ) 
 ->  ( X  .+  .0.  )  =  X )
 
Theoremismndd 12832* Deduce a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ph  ->  .0. 
 e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( x  .+  .0.  )  =  x )   =>    |-  ( ph  ->  G  e.  Mnd )
 
Theoremmndpfo 12833 The addition operation of a monoid as a function is an onto function. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 11-Oct-2013.) (Revised by AV, 23-Jan-2020.)
 |-  B  =  ( Base `  G )   &    |-  .+^  =  ( +f `  G )   =>    |-  ( G  e.  Mnd  ->  .+^ 
 : ( B  X.  B ) -onto-> B )
 
Theoremmndfo 12834 The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.) (Proof shortened by AV, 23-Jan-2020.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  .+  : ( B  X.  B )
 -onto-> B )
 
Theoremmndpropd 12835* If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  Mnd  <->  L  e.  Mnd ) )
 
Theoremmndprop 12836 If two structures have the same group components (properties), one is a monoid iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)
 |-  ( Base `  K )  =  ( Base `  L )   &    |-  ( +g  `  K )  =  ( +g  `  L )   =>    |-  ( K  e.  Mnd  <->  L  e.  Mnd )
 
Theoremissubmnd 12837* Characterize a submonoid by closure properties. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  H  =  ( Gs  S )   =>    |-  ( ( G  e.  Mnd  /\  S  C_  B  /\  .0.  e.  S )  ->  ( H  e.  Mnd  <->  A. x  e.  S  A. y  e.  S  ( x  .+  y )  e.  S ) )
 
Theoremress0g 12838  0g is unaffected by restriction. This is a bit more generic than submnd0 12839. (Contributed by Thierry Arnoux, 23-Oct-2017.)
 |-  S  =  ( Rs  A )   &    |-  B  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Mnd  /\  .0.  e.  A  /\  A  C_  B )  ->  .0.  =  ( 0g `  S ) )
 
Theoremsubmnd0 12839 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.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  H  =  ( Gs  S )   =>    |-  ( ( ( G  e.  Mnd  /\  H  e.  Mnd )  /\  ( S 
 C_  B  /\  .0.  e.  S ) )  ->  .0.  =  ( 0g `  H ) )
 
Theoremmndinvmod 12840* Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  E* w  e.  B  ( ( w  .+  A )  =  .0.  /\  ( A  .+  w )  =  .0.  ) )
 
Theoremmnd1 12841 The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) (Proof shortened by AV, 11-Feb-2020.)
 |-  M  =  { <. (
 Base `  ndx ) ,  { I } >. , 
 <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
 >. }   =>    |-  ( I  e.  V  ->  M  e.  Mnd )
 
Theoremmnd1id 12842 The singleton element of a trivial monoid is its identity element. (Contributed by AV, 23-Jan-2020.)
 |-  M  =  { <. (
 Base `  ndx ) ,  { I } >. , 
 <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
 >. }   =>    |-  ( I  e.  V  ->  ( 0g `  M )  =  I )
 
7.1.5  Monoid homomorphisms and submonoids
 
Syntaxcmhm 12843 Hom-set generator class for monoids.
 class MndHom
 
Syntaxcsubmnd 12844 Class function taking a monoid to its lattice of submonoids.
 class SubMnd
 
Definitiondf-mhm 12845* 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  =  ( s  e.  Mnd ,  t  e.  Mnd  |->  { f  e.  ( ( Base `  t
 )  ^m  ( Base `  s ) )  |  ( A. x  e.  ( Base `  s ) A. y  e.  ( Base `  s ) ( f `  ( x ( +g  `  s
 ) y ) )  =  ( ( f `
  x ) (
 +g  `  t )
 ( f `  y
 ) )  /\  (
 f `  ( 0g `  s ) )  =  ( 0g `  t
 ) ) } )
 
Definitiondf-submnd 12846* 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  =  ( s  e.  Mnd  |->  { t  e.  ~P ( Base `  s )  |  ( ( 0g `  s )  e.  t  /\  A. x  e.  t  A. y  e.  t  ( x ( +g  `  s
 ) y )  e.  t ) } )
 
Theoremismhm 12847* Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  (
 Base `  T )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  Y  =  ( 0g
 `  T )   =>    |-  ( F  e.  ( S MndHom  T )  <->  ( ( S  e.  Mnd  /\  T  e.  Mnd )  /\  ( F : B --> C  /\  A. x  e.  B  A. y  e.  B  ( F `  ( x  .+  y ) )  =  ( ( F `  x )  .+^  ( F `
  y ) ) 
 /\  ( F `  .0.  )  =  Y ) ) )
 
Theoremmhmrcl1 12848 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  ( F  e.  ( S MndHom  T )  ->  S  e.  Mnd )
 
Theoremmhmrcl2 12849 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  ( F  e.  ( S MndHom  T )  ->  T  e.  Mnd )
 
Theoremmhmf 12850 A monoid homomorphism is a function. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  (
 Base `  T )   =>    |-  ( F  e.  ( S MndHom  T )  ->  F : B --> C )
 
Theoremmhmpropd 12851* Monoid homomorphism depends only on the monoidal attributes of structures. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 7-Nov-2015.)
 |-  ( ph  ->  B  =  ( Base `  J )
 )   &    |-  ( ph  ->  C  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  C  =  ( Base `  M )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  J )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  C )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  M ) y ) )   =>    |-  ( ph  ->  ( J MndHom  K )  =  ( L MndHom  M ) )
 
Theoremmhmlin 12852 A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  B  =  ( Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   =>    |-  (
 ( F  e.  ( S MndHom  T )  /\  X  e.  B  /\  Y  e.  B )  ->  ( F `
  ( X  .+  Y ) )  =  ( ( F `  X )  .+^  ( F `
  Y ) ) )
 
Theoremmhm0 12853 A monoid homomorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |- 
 .0.  =  ( 0g `  S )   &    |-  Y  =  ( 0g `  T )   =>    |-  ( F  e.  ( S MndHom  T )  ->  ( F `  .0.  )  =  Y )
 
Theoremidmhm 12854 The identity homomorphism on a monoid. (Contributed by AV, 14-Feb-2020.)
 |-  B  =  ( Base `  M )   =>    |-  ( M  e.  Mnd  ->  (  _I  |`  B )  e.  ( M MndHom  M )
 )
 
Theoremmhmf1o 12855 A monoid homomorphism is bijective iff its converse is also a monoid homomorphism. (Contributed by AV, 22-Oct-2019.)
 |-  B  =  ( Base `  R )   &    |-  C  =  (
 Base `  S )   =>    |-  ( F  e.  ( R MndHom  S )  ->  ( F : B -1-1-onto-> C  <->  `' F  e.  ( S MndHom  R ) ) )
 
Theoremsubmrcl 12856 Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  ( S  e.  (SubMnd `  M )  ->  M  e.  Mnd )
 
Theoremissubm 12857* Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  B  =  ( Base `  M )   &    |-  .0.  =  ( 0g `  M )   &    |-  .+  =  ( +g  `  M )   =>    |-  ( M  e.  Mnd  ->  ( S  e.  (SubMnd `  M )  <->  ( S  C_  B  /\  .0.  e.  S  /\  A. x  e.  S  A. y  e.  S  ( x  .+  y )  e.  S ) ) )
 
Theoremissubm2 12858 Submonoids are subsets that are also monoids with the same zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  B  =  ( Base `  M )   &    |-  .0.  =  ( 0g `  M )   &    |-  H  =  ( Ms  S )   =>    |-  ( M  e.  Mnd  ->  ( S  e.  (SubMnd `  M )  <->  ( S  C_  B  /\  .0.  e.  S  /\  H  e.  Mnd )
 ) )
 
Theoremissubmd 12859* Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  .0.  =  ( 0g `  M )   &    |-  ( ph  ->  M  e.  Mnd )   &    |-  ( ph  ->  ch )   &    |-  ( ( ph  /\  ( ( x  e.  B  /\  y  e.  B )  /\  ( th  /\  ta ) ) )  ->  et )   &    |-  (
 z  =  .0.  ->  ( ps  <->  ch ) )   &    |-  (
 z  =  x  ->  ( ps  <->  th ) )   &    |-  (
 z  =  y  ->  ( ps  <->  ta ) )   &    |-  (
 z  =  ( x 
 .+  y )  ->  ( ps  <->  et ) )   =>    |-  ( ph  ->  { z  e.  B  |  ps }  e.  (SubMnd `  M ) )
 
Theoremmndissubm 12860 If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. (Contributed by AV, 17-Feb-2024.)
 |-  B  =  ( Base `  G )   &    |-  S  =  (
 Base `  H )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Mnd  /\  H  e.  Mnd )  ->  ( ( S  C_  B  /\  .0.  e.  S  /\  ( +g  `  H )  =  ( ( +g  `  G )  |`  ( S  X.  S ) ) )  ->  S  e.  (SubMnd `  G )
 ) )
 
Theoremsubmss 12861 Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  B  =  ( Base `  M )   =>    |-  ( S  e.  (SubMnd `  M )  ->  S  C_  B )
 
Theoremsubmid 12862 Every monoid is trivially a submonoid of itself. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  B  =  ( Base `  M )   =>    |-  ( M  e.  Mnd  ->  B  e.  (SubMnd `  M ) )
 
Theoremsubm0cl 12863 Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |- 
 .0.  =  ( 0g `  M )   =>    |-  ( S  e.  (SubMnd `  M )  ->  .0.  e.  S )
 
Theoremsubmcl 12864 Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |- 
 .+  =  ( +g  `  M )   =>    |-  ( ( S  e.  (SubMnd `  M )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .+  Y )  e.  S )
 
Theorem0subm 12865 The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Mnd  ->  {  .0.  }  e.  (SubMnd `  G ) )
 
Theoreminsubm 12866 The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.)
 |-  ( ( A  e.  (SubMnd `  M )  /\  B  e.  (SubMnd `  M ) )  ->  ( A  i^i  B )  e.  (SubMnd `  M )
 )
 
Theorem0mhm 12867 The constant zero linear function between two monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- 
 .0.  =  ( 0g `  N )   &    |-  B  =  (
 Base `  M )   =>    |-  ( ( M  e.  Mnd  /\  N  e.  Mnd )  ->  ( B  X.  {  .0.  } )  e.  ( M MndHom  N )
 )
 
Theoremmhmco 12868 The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  ( ( F  e.  ( T MndHom  U )  /\  G  e.  ( S MndHom  T ) )  ->  ( F  o.  G )  e.  ( S MndHom  U )
 )
 
Theoremmhmima 12869 The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M ) )  ->  ( F
 " X )  e.  (SubMnd `  N )
 )
 
Theoremmhmeql 12870 The equalizer of two monoid homomorphisms is a submonoid. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( ( F  e.  ( S MndHom  T )  /\  G  e.  ( S MndHom  T ) )  ->  dom  ( F  i^i  G )  e.  (SubMnd `  S )
 )
 
7.2  Groups
 
7.2.1  Definition and basic properties
 
Syntaxcgrp 12871 Extend class notation with class of all groups.
 class  Grp
 
Syntaxcminusg 12872 Extend class notation with inverse of group element.
 class  invg
 
Syntaxcsg 12873 Extend class notation with group subtraction (or division) operation.
 class  -g
 
Definitiondf-grp 12874* Define class of all groups. A group is a monoid (df-mnd 12812) 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  G is an algebraic structure formed from a base set of elements (notated  ( Base `  G
) per df-base 12462) and an internal group operation (notated  ( +g  `  G
) per df-plusg 12543). 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 12879), associativity (so  ( (
a +g  b ) +g  c )  =  ( a +g  ( b +g  c ) ) for any a, b, c, see grpass 12880), identity (there must be an element  e  =  ( 0g `  G
) such that  e +g  a  =  a +g  e  =  a for any a), and inverse (for each element a in the base set, there must be an element  b  =  invg a in the base set such that  a +g  b  =  b +g  a  =  e). It can be proven that the identity element is unique (grpideu 12882). Groups need not be commutative; a commutative group is an Abelian group. Subgroups can often be formed from groups. An example of an (Abelian) group is the set of complex numbers  CC over the group operation  + (addition). Other structures include groups, including unital rings and fields. (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |- 
 Grp  =  { g  e.  Mnd  |  A. a  e.  ( Base `  g ) E. m  e.  ( Base `  g ) ( m ( +g  `  g
 ) a )  =  ( 0g `  g
 ) }
 
Definitiondf-minusg 12875* Define inverse of group element. (Contributed by NM, 24-Aug-2011.)
 |- 
 invg  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g )  |->  ( iota_ w  e.  ( Base `  g
 ) ( w (
 +g  `  g ) x )  =  ( 0g `  g ) ) ) )
 
Definitiondf-sbg 12876* Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)
 |-  -g  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( +g  `  g
 ) ( ( invg `  g ) `
  y ) ) ) )
 
Theoremisgrp 12877* 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.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  <->  ( G  e.  Mnd  /\  A. a  e.  B  E. m  e.  B  ( m  .+  a )  =  .0.  ) )
 
Theoremgrpmnd 12878 A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( G  e.  Grp  ->  G  e.  Mnd )
 
Theoremgrpcl 12879 Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
 
Theoremgrpass 12880 A group operation is associative. (Contributed by NM, 14-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .+  Y )  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
 
Theoremgrpinvex 12881* Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  E. y  e.  B  ( y  .+  X )  =  .0.  )
 
Theoremgrpideu 12882* 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.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  ->  E! u  e.  B  A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x ) )
 
Theoremgrpmndd 12883 A group is a monoid. (Contributed by SN, 1-Jun-2024.)
 |-  ( ph  ->  G  e.  Grp )   =>    |-  ( ph  ->  G  e.  Mnd )
 
Theoremgrpcld 12884 Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  B )
 
Theoremgrpplusf 12885 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  F  =  ( +f `  G )   =>    |-  ( G  e.  Grp  ->  F : ( B  X.  B ) --> B )
 
Theoremgrpplusfo 12886 The group addition operation is a function onto the base set/set of group elements. (Contributed by NM, 30-Oct-2006.) (Revised by AV, 30-Aug-2021.)
 |-  B  =  ( Base `  G )   &    |-  F  =  ( +f `  G )   =>    |-  ( G  e.  Grp  ->  F : ( B  X.  B ) -onto-> B )
 
Theoremgrppropd 12887* 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.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
 
Theoremgrpprop 12888 If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)
 |-  ( Base `  K )  =  ( Base `  L )   &    |-  ( +g  `  K )  =  ( +g  `  L )   =>    |-  ( K  e.  Grp  <->  L  e.  Grp )
 
Theoremgrppropstrg 12889 Generalize a specific 2-element group  L to show that any set  K with the same (relevant) properties is also a group. (Contributed by NM, 28-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  ( Base `  K )  =  B   &    |-  ( +g  `  K )  =  .+   &    |-  L  =  { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. }   =>    |-  ( K  e.  V  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
 
Theoremisgrpd2e 12890* Deduce a group from its properties. In this version of isgrpd2 12891, we don't assume there is an expression for the inverse of  x. (Contributed by NM, 10-Aug-2013.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  .0.  =  ( 0g `  G ) )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  (
 ( ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )   =>    |-  ( ph  ->  G  e.  Grp )
 
Theoremisgrpd2 12891* Deduce a group from its properties. 
N (negative) is normally dependent on  x i.e. read it as  N ( x ). Note: normally we don't use a  ph antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2177, but we make an exception for theorems such as isgrpd2 12891 and ismndd 12832 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  .0.  =  ( 0g `  G ) )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  (
 ( ph  /\  x  e.  B )  ->  N  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( N  .+  x )  =  .0.  )   =>    |-  ( ph  ->  G  e.  Grp )
 
Theoremisgrpde 12892* Deduce a group from its properties. In this version of isgrpd 12893, we don't assume there is an expression for the inverse of  x. (Contributed by NM, 6-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ph  ->  .0. 
 e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  B  ( y  .+  x )  =  .0.  )   =>    |-  ( ph  ->  G  e.  Grp )
 
Theoremisgrpd 12893* Deduce a group from its properties. Unlike isgrpd2 12891, this one goes straight from the base properties rather than going through  Mnd.  N (negative) is normally dependent on  x i.e. read it as  N ( x ). (Contributed by NM, 6-Jun-2013.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ph  ->  .0. 
 e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  N  e.  B )   &    |-  ( ( ph  /\  x  e.  B )  ->  ( N  .+  x )  =  .0.  )   =>    |-  ( ph  ->  G  e.  Grp )
 
Theoremisgrpi 12894* Properties that determine a group. 
N (negative) is normally dependent on  x i.e. read it as  N ( x ). (Contributed by NM, 3-Sep-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  (
 ( x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  (
 ( x  .+  y
 )  .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  .0.  e.  B   &    |-  ( x  e.  B  ->  (  .0.  .+  x )  =  x )   &    |-  ( x  e.  B  ->  N  e.  B )   &    |-  ( x  e.  B  ->  ( N  .+  x )  =  .0.  )   =>    |-  G  e.  Grp
 
Theoremgrpsgrp 12895 A group is a semigroup. (Contributed by AV, 28-Aug-2021.)
 |-  ( G  e.  Grp  ->  G  e. Smgrp )
 
Theoremdfgrp2 12896* Alternate definition of a group as semigroup with a left identity and a left inverse for each element. This "definition" is weaker than df-grp 12874, based on the definition of a monoid which provides a left and a right identity. (Contributed by AV, 28-Aug-2021.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e.  Grp  <->  ( G  e. Smgrp  /\  E. n  e.  B  A. x  e.  B  ( ( n 
 .+  x )  =  x  /\  E. i  e.  B  ( i  .+  x )  =  n ) ) )
 
Theoremdfgrp2e 12897* Alternate definition of a group as a set with a closed, associative operation, a left identity and a left inverse for each element. Alternate definition in [Lang] p. 7. (Contributed by NM, 10-Oct-2006.) (Revised by AV, 28-Aug-2021.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e.  Grp  <->  (
 A. x  e.  B  A. y  e.  B  ( ( x  .+  y
 )  e.  B  /\  A. z  e.  B  ( ( x  .+  y
 )  .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )  /\  E. n  e.  B  A. x  e.  B  ( ( n 
 .+  x )  =  x  /\  E. i  e.  B  ( i  .+  x )  =  n ) ) )
 
Theoremgrpidcl 12898 The identity element of a group belongs to the group. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  ->  .0.  e.  B )
 
Theoremgrpbn0 12899 The base set of a group is not empty. It is also inhabited (see grpidcl 12898). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Grp  ->  B  =/=  (/) )
 
Theoremgrplid 12900 The identity element of a group is a left identity. (Contributed by NM, 18-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  (  .0.  .+  X )  =  X )
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