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Theorem List for Metamath Proof Explorer - 15901-16000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremismnddef 15901* 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. SGrp  /\  E. e  e.  B  A. a  e.  B  ( ( e 
 .+  a )  =  a  /\  ( a 
 .+  e )  =  a ) ) )
 
Theoremismnd 15902* The predicate "is a monoid". This is the definig 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 15908), whose operation is associative (so, a semigroup, see also mndass 15909) and has a two-sided neutral element (see mndid 15912). (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 ) ) )
 
Theoremisnmnd 15903* A condition for a structure not to be a monoid: every element of the base set is not a left identity for at least one element of the base set. (Contributed by AV, 4-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  ( A. z  e.  B  E. x  e.  B  ( z  .o.  x )  =/=  x  ->  M  e/  Mnd )
 
Definitiondf-mndOLD 15904* Obsolete version of df-mnd 15900 as of 6-Feb-2020. Definition of a monoid. A monoid is a set equipped with an everywhere defined internal operation (so, a magma, see mndcl 15908), whose operation is associative (so, a semigroup, see mndass 15909) and has a two-sided neutral element (see mndid 15912). (Contributed by Mario Carneiro, 6-Jan-2015.) (New usage is discouraged.)
 |- MndOLD  =  { g  |  [. ( Base `  g )  /  b ]. [. ( +g  `  g )  /  p ]. ( A. x  e.  b  A. y  e.  b  A. z  e.  b  ( ( x p y )  e.  b  /\  ( ( x p y ) p z )  =  ( x p ( y p z ) ) )  /\  E. e  e.  b  A. x  e.  b  (
 ( e p x )  =  x  /\  ( x p e )  =  x ) ) }
 
TheoremismndOLD 15905* Obsolete version of ismnd 15902 as of 6-Feb-2020. The predicate "is a monoid." (Contributed by Mario Carneiro, 6-Jan-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e. MndOLD  <->  ( A. a  e.  B  A. b  e.  B  A. c  e.  B  (
 ( a  .+  b
 )  e.  B  /\  ( ( a  .+  b )  .+  c )  =  ( a  .+  ( b  .+  c ) ) )  /\  E. e  e.  B  A. a  e.  B  ( ( e 
 .+  a )  =  a  /\  ( a 
 .+  e )  =  a ) ) )
 
Theoremmndsgrp 15906 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. SGrp )
 
Theoremmndmgm 15907 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 15908 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 15909 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 ) ) )
 
TheoremmndclOLD 15910 Obsolete version of mndcl 15908 as of 8-Feb-2020. Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 6-Feb-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
 
TheoremmndassOLD 15911 Obsolete version of mndass 15909 as of 8-Feb-2020. A monoid operation is associative. (Contributed by NM, 14-Aug-2011.) (Revised by AV, 6-Feb-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .+  Y )  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
 
Theoremmndid 15912* 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 15913* 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 15914 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 15915 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 15916 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 15917 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 )
 
Theoremmndplusf 15918 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 15919 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 15920 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 15921 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 15922* 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 15923 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 15924 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 15925* 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 15926 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 15927* 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 15928  0g is unaffected by restriction. This is a bit more generic than submnd0 15929 (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 15929 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 ) )
 
Theoremprdsplusgcl 15930 Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  .+  =  ( +g  `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R : I --> Mnd )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( F  .+  G )  e.  B )
 
Theoremprdsidlem 15931* Characterization of identity in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  .+  =  ( +g  `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R : I --> Mnd )   &    |-  .0.  =  ( 0g  o.  R )   =>    |-  ( ph  ->  (  .0.  e.  B  /\  A. x  e.  B  (
 (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x ) ) )
 
Theoremprdsmndd 15932 The product of a family of monoids is a monoid. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Mnd )   =>    |-  ( ph  ->  Y  e.  Mnd )
 
Theoremprds0g 15933 Zero in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Mnd )   =>    |-  ( ph  ->  ( 0g  o.  R )  =  ( 0g `  Y ) )
 
Theorempwsmnd 15934 The structure power of a monoid is a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  Mnd  /\  I  e.  V )  ->  Y  e.  Mnd )
 
Theorempws0g 15935 Zero in a product of monoids. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Mnd  /\  I  e.  V )  ->  ( I  X.  {  .0.  } )  =  ( 0g `  Y ) )
 
Theoremimasmnd2 15936* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |- 
 .+  =  ( +g  `  R )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  (
 a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `  a
 )  =  ( F `
  p )  /\  ( F `  b )  =  ( F `  q ) )  ->  ( F `  ( a 
 .+  b ) )  =  ( F `  ( p  .+  q ) ) ) )   &    |-  ( ph  ->  R  e.  W )   &    |-  ( ( ph  /\  x  e.  V  /\  y  e.  V )  ->  ( x  .+  y )  e.  V )   &    |-  ( ( ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  ( F `
  ( ( x 
 .+  y )  .+  z ) )  =  ( F `  ( x  .+  ( y  .+  z ) ) ) )   &    |-  ( ph  ->  .0. 
 e.  V )   &    |-  (
 ( ph  /\  x  e.  V )  ->  ( F `  (  .0.  .+  x ) )  =  ( F `  x ) )   &    |-  ( ( ph  /\  x  e.  V ) 
 ->  ( F `  ( x  .+  .0.  ) )  =  ( F `  x ) )   =>    |-  ( ph  ->  ( U  e.  Mnd  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
 
Theoremimasmnd 15937* The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |- 
 .+  =  ( +g  `  R )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  (
 a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `  a
 )  =  ( F `
  p )  /\  ( F `  b )  =  ( F `  q ) )  ->  ( F `  ( a 
 .+  b ) )  =  ( F `  ( p  .+  q ) ) ) )   &    |-  ( ph  ->  R  e.  Mnd )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ph  ->  ( U  e.  Mnd  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
 
Theoremimasmndf1 15938 The image of a monoid under an injection is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  U  =  ( F 
 "s 
 R )   &    |-  V  =  (
 Base `  R )   =>    |-  ( ( F : V -1-1-> B  /\  R  e.  Mnd )  ->  U  e.  Mnd )
 
Theoremxpsmnd 15939 The binary product of monoids is a monoid. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  T  =  ( R  X.s  S )   =>    |-  ( ( R  e.  Mnd  /\  S  e.  Mnd )  ->  T  e.  Mnd )
 
Theoremmnd1 15940 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 )
 
Theoremmnd1OLD 15941 The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) Obsolete version of mnd1 15940 as of 11-Feb-2020. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  M  =  { <. (
 Base `  ndx ) ,  { I } >. , 
 <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
 >. }   =>    |-  ( I  e.  V  ->  M  e.  Mnd )
 
Theoremmnd1id 15942 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 )
 
10.1.6  Monoid homomorphisms and submonoids
 
Syntaxcmhm 15943 Hom-set generator class for monoids.
 class MndHom
 
Syntaxcsubmnd 15944 Class function taking a monoid to its lattice of submonoids.
 class SubMnd
 
Definitiondf-mhm 15945* 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 15946* 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 15947* 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 15948 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  ( F  e.  ( S MndHom  T )  ->  S  e.  Mnd )
 
Theoremmhmrcl2 15949 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  ( F  e.  ( S MndHom  T )  ->  T  e.  Mnd )
 
Theoremmhmf 15950 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 15951* 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 15952 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 15953 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 15954 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 15955 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 15956 Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  ( S  e.  (SubMnd `  M )  ->  M  e.  Mnd )
 
Theoremissubm 15957* 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 15958 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 15959* 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 ) )
 
Theoremsubmss 15960 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 15961 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 15962 Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |- 
 .0.  =  ( 0g `  M )   =>    |-  ( S  e.  (SubMnd `  M )  ->  .0.  e.  S )
 
Theoremsubmcl 15963 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 )
 
Theoremsubmmnd 15964 Submonoids are themselves monoids under the given operation. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  H  =  ( Ms  S )   =>    |-  ( S  e.  (SubMnd `  M )  ->  H  e.  Mnd )
 
Theoremsubmbas 15965 The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015.)
 |-  H  =  ( Ms  S )   =>    |-  ( S  e.  (SubMnd `  M )  ->  S  =  ( Base `  H )
 )
 
Theoremsubm0 15966 Submonoids have the same identity. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  H  =  ( Ms  S )   &    |-  .0.  =  ( 0g `  M )   =>    |-  ( S  e.  (SubMnd `  M )  ->  .0.  =  ( 0g `  H ) )
 
Theoremsubsubm 15967 A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015.)
 |-  H  =  ( Gs  S )   =>    |-  ( S  e.  (SubMnd `  G )  ->  ( A  e.  (SubMnd `  H ) 
 <->  ( A  e.  (SubMnd `  G )  /\  A  C_  S ) ) )
 
Theorem0mhm 15968 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 )
 )
 
Theoremresmhm 15969 Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  U  =  ( Ss  X )   =>    |-  ( ( F  e.  ( S MndHom  T )  /\  X  e.  (SubMnd `  S ) )  ->  ( F  |`  X )  e.  ( U MndHom  T ) )
 
Theoremresmhm2 15970 One direction of resmhm2b 15971. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  U  =  ( Ts  X )   =>    |-  ( ( F  e.  ( S MndHom  U )  /\  X  e.  (SubMnd `  T ) )  ->  F  e.  ( S MndHom  T ) )
 
Theoremresmhm2b 15971 Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  U  =  ( Ts  X )   =>    |-  ( ( X  e.  (SubMnd `  T )  /\  ran 
 F  C_  X )  ->  ( F  e.  ( S MndHom  T )  <->  F  e.  ( S MndHom  U ) ) )
 
Theoremmhmco 15972 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 15973 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 15974 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 )
 )
 
Theoremsubmacs 15975 Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS `  B )
 )
 
Theoremmrcmndind 15976* (( From SO's determinants branch )). TODO: Appropriate description to be added! (Contributed by SO, 14-Jul-2018.)
 |-  ( x  =  y 
 ->  ( ps  <->  ch ) )   &    |-  ( x  =  ( y  .+  z )  ->  ( ps 
 <-> 
 th ) )   &    |-  ( x  =  .0.  ->  ( ps  <->  ta ) )   &    |-  ( x  =  A  ->  ( ps  <->  et ) )   &    |-  .0.  =  ( 0g `  M )   &    |- 
 .+  =  ( +g  `  M )   &    |-  B  =  (
 Base `  M )   &    |-  ( ph  ->  M  e.  Mnd )   &    |-  ( ph  ->  G  C_  B )   &    |-  ( ph  ->  B  =  ( (mrCls `  (SubMnd `  M ) ) `
  G ) )   &    |-  ( ph  ->  ta )   &    |-  (
 ( ( ph  /\  y  e.  B  /\  z  e.  G )  /\  ch )  ->  th )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  et )
 
Theoremprdspjmhm 15977* A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  S  e.  X )   &    |-  ( ph  ->  R : I --> Mnd )   &    |-  ( ph  ->  A  e.  I
 )   =>    |-  ( ph  ->  ( x  e.  B  |->  ( x `
  A ) )  e.  ( Y MndHom  ( R `  A ) ) )
 
Theorempwspjmhm 15978* A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   =>    |-  ( ( R  e.  Mnd  /\  I  e.  V  /\  A  e.  I )  ->  ( x  e.  B  |->  ( x `
  A ) )  e.  ( Y MndHom  R ) )
 
Theorempwsdiagmhm 15979* Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  R )   &    |-  F  =  ( x  e.  B  |->  ( I  X.  { x } ) )   =>    |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  F  e.  ( R MndHom  Y )
 )
 
Theorempwsco1mhm 15980* Right composition with a function on the index sets yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Y  =  ( R 
 ^s 
 A )   &    |-  Z  =  ( R  ^s  B )   &    |-  C  =  (
 Base `  Z )   &    |-  ( ph  ->  R  e.  Mnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  (
 g  e.  C  |->  ( g  o.  F ) )  e.  ( Z MndHom  Y ) )
 
Theorempwsco2mhm 15981* Left composition with a monoid homomorphism yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Y  =  ( R 
 ^s 
 A )   &    |-  Z  =  ( S  ^s  A )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F  e.  ( R MndHom  S )
 )   =>    |-  ( ph  ->  (
 g  e.  B  |->  ( F  o.  g ) )  e.  ( Y MndHom  Z ) )
 
10.1.7  Ordered sums in a monoid

One important use of words is as formal composites in cases where order is significant, using the general sum operator df-gsum 14822. If order is not significant, it is simpler to use families instead.

 
Theoremgsumvallem2 15982* Lemma for properties of the set of identities of  G. The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  O  =  { x  e.  B  |  A. y  e.  B  ( ( x 
 .+  y )  =  y  /\  ( y 
 .+  x )  =  y ) }   =>    |-  ( G  e.  Mnd 
 ->  O  =  {  .0.  } )
 
Theoremgsumsubm 15983 Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  S  e.  (SubMnd `  G ) )   &    |-  ( ph  ->  F : A --> S )   &    |-  H  =  ( Gs  S )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsumz 15984* Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Mnd  /\  A  e.  V ) 
 ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
 
Theoremgsumwsubmcl 15985 Closure of the composite in any submonoid. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
 |-  ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  ->  ( G  gsumg 
 W )  e.  S )
 
Theoremgsumws1 15986 A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  B  =  ( Base `  G )   =>    |-  ( S  e.  B  ->  ( G  gsumg 
 <" S "> )  =  S )
 
Theoremgsumwcl 15987 Closure of the composite of a word in a structure  G. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  B  =  ( Base `  G )   =>    |-  ( ( G  e.  Mnd  /\  W  e. Word  B )  ->  ( G  gsumg 
 W )  e.  B )
 
Theoremgsumccat 15988 Homomorphic property of composites. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  W  e. Word  B 
 /\  X  e. Word  B )  ->  ( G  gsumg  ( W concat  X ) )  =  ( ( G  gsumg  W ) 
 .+  ( G  gsumg  X ) ) )
 
Theoremgsumws2 15989 Valuation of a pair in a monoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  S  e.  B  /\  T  e.  B )  ->  ( G  gsumg  <" S T "> )  =  ( S  .+  T ) )
 
Theoremgsumccatsn 15990 Homomorphic property of composites with a singleton. (Contributed by AV, 20-Jan-2019.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  W  e. Word  B 
 /\  Z  e.  B )  ->  ( G  gsumg  ( W concat  <" Z "> ) )  =  (
 ( G  gsumg 
 W )  .+  Z ) )
 
Theoremgsumspl 15991 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.)
 |-  B  =  ( Base `  M )   &    |-  ( ph  ->  M  e.  Mnd )   &    |-  ( ph  ->  S  e. Word  B )   &    |-  ( ph  ->  F  e.  ( 0 ... T ) )   &    |-  ( ph  ->  T  e.  ( 0 ... ( # `  S ) ) )   &    |-  ( ph  ->  X  e. Word  B )   &    |-  ( ph  ->  Y  e. Word  B )   &    |-  ( ph  ->  ( M  gsumg 
 X )  =  ( M  gsumg 
 Y ) )   =>    |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  X >. ) )  =  ( M 
 gsumg  ( S splice  <. F ,  T ,  Y >. ) ) )
 
Theoremgsumwmhm 15992 Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  B  =  ( Base `  M )   =>    |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  ( H `  ( M 
 gsumg  W ) )  =  ( N  gsumg  ( H  o.  W ) ) )
 
Theoremgsumwspan 15993* 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.)
 |-  B  =  ( Base `  M )   &    |-  K  =  (mrCls `  (SubMnd `  M )
 )   =>    |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  ( K `  G )  =  ran  ( w  e. Word  G  |->  ( M 
 gsumg  w ) ) )
 
10.1.8  Free monoids
 
Syntaxcfrmd 15994 Extend class definition with the free monoid construction.
 class freeMnd
 
Syntaxcvrmd 15995 Extend class notation with free monoid injection.
 class varFMnd
 
Definitiondf-frmd 15996 Define a free monoid over a set  i of generators, defined as the set of finite strings on  I with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |- freeMnd  =  ( i  e.  _V  |->  {
 <. ( Base `  ndx ) , Word 
 i >. ,  <. ( +g  ` 
 ndx ) ,  ( concat  |`  (Word  i  X. Word  i )
 ) >. } )
 
Definitiondf-vrmd 15997* Define a free monoid over a set  i of generators, defined as the set of finite strings on  I with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |- varFMnd  =  ( i  e.  _V  |->  ( j  e.  i  |-> 
 <" j "> ) )
 
Theoremfrmdval 15998 Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   &    |-  ( I  e.  V  ->  B  = Word  I )   &    |-  .+  =  ( concat  |`  ( B  X.  B ) )   =>    |-  ( I  e.  V  ->  M  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. } )
 
Theoremfrmdbas 15999 The base set of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  M )   =>    |-  ( I  e.  V  ->  B  = Word  I )
 
Theoremfrmdelbas 16000 An element of the base set of a free monoid is a string on the generators. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  M )   =>    |-  ( X  e.  B  ->  X  e. Word  I )
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