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Theorem List for Metamath Proof Explorer - 16601-16700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmnd12g 16601 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 16602 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 16603 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 16604 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 16605 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 16606 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 16607 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 16608* 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 16609 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 16610 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 16611* 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 16612 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 16613* 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 16614  0g is unaffected by restriction. This is a bit more generic than submnd0 16615. (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 16615 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 16616 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 16617* 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 16618 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 16619 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 16620 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 16621 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 16622* 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 16623* 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 16624 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 16625 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 16626 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 16627 The (smallest) structure representing a trivial monoid consists of one element. (Contributed by AV, 28-Apr-2019.) Obsolete version of mnd1 16626 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 16628 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 16629 Hom-set generator class for monoids.
 class MndHom
 
Syntaxcsubmnd 16630 Class function taking a monoid to its lattice of submonoids.
 class SubMnd
 
Definitiondf-mhm 16631* 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 16632* 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 16633* 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 16634 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  ( F  e.  ( S MndHom  T )  ->  S  e.  Mnd )
 
Theoremmhmrcl2 16635 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  ( F  e.  ( S MndHom  T )  ->  T  e.  Mnd )
 
Theoremmhmf 16636 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 16637* 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 16638 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 16639 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 16640 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 16641 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 16642 Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  ( S  e.  (SubMnd `  M )  ->  M  e.  Mnd )
 
Theoremissubm 16643* 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 16644 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 16645* 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 16646 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 16647 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 16648 Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |- 
 .0.  =  ( 0g `  M )   =>    |-  ( S  e.  (SubMnd `  M )  ->  .0.  e.  S )
 
Theoremsubmcl 16649 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 16650 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 16651 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 16652 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 16653 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 16654 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 16655 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 16656 One direction of resmhm2b 16657. (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 16657 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 16658 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 16659 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 16660 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 16661 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 16662* (( 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 16663* 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 16664* 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 16665* 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 16666* 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 16667* 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 15390. If order is not significant, it is simpler to use families instead.

 
Theoremgsumvallem2 16668* 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 16669 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 16670* 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 16671 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 16672 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 16673 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 16674 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 ++ 
 X ) )  =  ( ( G  gsumg  W ) 
 .+  ( G  gsumg  X ) ) )
 
Theoremgsumws2 16675 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 16676 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 ++ 
 <" Z "> ) )  =  (
 ( G  gsumg 
 W )  .+  Z ) )
 
Theoremgsumspl 16677 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 16678 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 16679* 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 16680 Extend class definition with the free monoid construction.
 class freeMnd
 
Syntaxcvrmd 16681 Extend class notation with free monoid injection.
 class varFMnd
 
Definitiondf-frmd 16682 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 ) ,  ( ++  |`  (Word  i  X. Word  i
 ) ) >. } )
 
Definitiondf-vrmd 16683* 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 16684 Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   &    |-  ( I  e.  V  ->  B  = Word  I )   &    |-  .+  =  ( ++  |`  ( B  X.  B ) )   =>    |-  ( I  e.  V  ->  M  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. } )
 
Theoremfrmdbas 16685 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 16686 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 )
 
Theoremfrmdplusg 16687 The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   =>    |-  .+  =  ( ++  |`  ( B  X.  B ) )
 
Theoremfrmdadd 16688 Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   =>    |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( X ++  Y ) )
 
Theoremvrmdfval 16689* The canonical injection from the generating set  I to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  U  =  (varFMnd `  I )   =>    |-  ( I  e.  V  ->  U  =  ( j  e.  I  |->  <" j "> ) )
 
Theoremvrmdval 16690 The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  U  =  (varFMnd `  I )   =>    |-  ( ( I  e.  V  /\  A  e.  I )  ->  ( U `
  A )  = 
 <" A "> )
 
Theoremvrmdf 16691 The mapping from the index set to the generators is a function into the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  U  =  (varFMnd `  I )   =>    |-  ( I  e.  V  ->  U : I -->Word  I )
 
Theoremfrmdmnd 16692 A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  M  =  (freeMnd `  I
 )   =>    |-  ( I  e.  V  ->  M  e.  Mnd )
 
Theoremfrmd0 16693 The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   =>    |-  (/)  =  ( 0g `  M )
 
Theoremfrmdsssubm 16694 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.)
 |-  M  =  (freeMnd `  I
 )   =>    |-  ( ( I  e.  V  /\  J  C_  I )  -> Word  J  e.  (SubMnd `  M ) )
 
Theoremfrmdgsum 16695 Any word in a free monoid can be expressed as the sum of the singletons composing it. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   &    |-  U  =  (varFMnd `  I )   =>    |-  ( ( I  e.  V  /\  W  e. Word  I )  ->  ( M  gsumg  ( U  o.  W ) )  =  W )
 
Theoremfrmdss2 16696 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  J is Word  J". (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  M  =  (freeMnd `  I
 )   &    |-  U  =  (varFMnd `  I )   =>    |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M ) )  ->  ( ( U " J )  C_  A  <-> Word  J  C_  A )
 )
 
Theoremfrmdup1 16697* 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.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  G )   &    |-  E  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x ) ) )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  A : I --> B )   =>    |-  ( ph  ->  E  e.  ( M MndHom  G ) )
 
Theoremfrmdup2 16698* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  G )   &    |-  E  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x ) ) )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  A : I --> B )   &    |-  U  =  (varFMnd `  I )   &    |-  ( ph  ->  Y  e.  I )   =>    |-  ( ph  ->  ( E `  ( U `
  Y ) )  =  ( A `  Y ) )
 
Theoremfrmdup3lem 16699* Lemma for frmdup3 16700. (Contributed by Mario Carneiro, 18-Jul-2016.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  G )   &    |-  U  =  (varFMnd `  I )   =>    |-  ( ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I
 --> B )  /\  ( F  e.  ( M MndHom  G )  /\  ( F  o.  U )  =  A ) )  ->  F  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x ) ) ) )
 
Theoremfrmdup3 16700* Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 18-Jul-2016.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  G )   &    |-  U  =  (varFMnd `  I )   =>    |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B ) 
 ->  E! m  e.  ( M MndHom  G ) ( m  o.  U )  =  A )
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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