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Theorem List for Metamath Proof Explorer - 14401-14500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfn0g 14401 The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |- 
 0g  Fn  _V
 
Theorem0g0 14402 The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)
 |-  (/)  =  ( 0g `  (/) )
 
Theoremismgmid 14403* The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  E. e  e.  B  A. x  e.  B  ( ( e 
 .+  x )  =  x  /\  ( x 
 .+  e )  =  x ) )   =>    |-  ( ph  ->  ( ( U  e.  B  /\  A. x  e.  B  ( ( U  .+  x )  =  x  /\  ( x  .+  U )  =  x )
 ) 
 <->  .0.  =  U ) )
 
Theoremmgmidcl 14404* The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  E. e  e.  B  A. x  e.  B  ( ( e 
 .+  x )  =  x  /\  ( x 
 .+  e )  =  x ) )   =>    |-  ( ph  ->  .0. 
 e.  B )
 
Theoremmgmlrid 14405* The identity element of a magma, if it exists, is a left and right identity. (Contributed by Mario Carneiro, 27-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  E. e  e.  B  A. x  e.  B  ( ( e 
 .+  x )  =  x  /\  ( x 
 .+  e )  =  x ) )   =>    |-  ( ( ph  /\  X  e.  B ) 
 ->  ( (  .0.  .+  X )  =  X  /\  ( X  .+  .0.  )  =  X )
 )
 
Theoremismgmid2 14406* Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  U  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( U  .+  x )  =  x )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( x  .+  U )  =  x )   =>    |-  ( ph  ->  U  =  .0.  )
 
Theoremmndidcl 14407 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 )
 
Theoremmndlrid 14408 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 14409 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 14410 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 )
 
Theoremgrpidd 14411* Deduce the identity element of a monoid from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  .0.  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  (  .0.  .+  x )  =  x )   &    |-  (
 ( ph  /\  x  e.  B )  ->  ( x  .+  .0.  )  =  x )   =>    |-  ( ph  ->  .0.  =  ( 0g `  G ) )
 
Theoremismndd 14412* 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 )
 
Theoremmndfo 14413 The addition operation of a monoid is an onto function (assuming it is a function). (Contributed by Mario Carneiro, 11-Oct-2013.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  .+  Fn  ( B  X.  B ) )  ->  .+  : ( B  X.  B )
 -onto-> B )
 
Theoremmndpropd 14414* If two structures have the same group components (properties), one is a monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( 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 ) )
 
Theoremgrpidpropd 14415* If two structures have the same group components (properties), they have the same identity element. (Contributed by Mario Carneiro, 27-Nov-2014.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   =>    |-  ( ph  ->  ( 0g `  K )  =  ( 0g `  L ) )
 
Theoremmndprop 14416 If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)
 |-  ( Base `  K )  =  ( Base `  L )   &    |-  ( +g  `  K )  =  ( +g  `  L )   =>    |-  ( K  e.  Mnd  <->  L  e.  Mnd )
 
Theoremissubmnd 14417* 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 ) )
 
Theoremsubmnd0 14418 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 14419 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 14420* 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 14421 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 14422 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 14423 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 14424 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 14425* 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 14426* 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 14427 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 14428 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 )
 
10.1.2  Monoid homomorphisms and submonoids
 
Syntaxcmhm 14429 Hom-set generator class for monoids.
 class MndHom
 
Syntaxcsubmnd 14430 Class function taking a monoid to its lattice of submonoids.
 class SubMnd
 
Definitiondf-mhm 14431* 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 14432* 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 14433* 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 14434 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  ( F  e.  ( S MndHom  T )  ->  S  e.  Mnd )
 
Theoremmhmrcl2 14435 Reverse closure of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  ( F  e.  ( S MndHom  T )  ->  T  e.  Mnd )
 
Theoremmhmf 14436 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 14437* 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 14438 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 14439 A monoid homorphism preserves zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |- 
 .0.  =  ( 0g `  S )   &    |-  Y  =  ( 0g `  T )   =>    |-  ( F  e.  ( S MndHom  T )  ->  ( F `  .0.  )  =  Y )
 
Theoremsubmrcl 14440 Reverse closure for submonoids. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  ( S  e.  (SubMnd `  M )  ->  M  e.  Mnd )
 
Theoremissubm 14441* 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 14442 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 )
 ) )
 
Theoremsubmss 14443 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 14444 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 14445 Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |- 
 .0.  =  ( 0g `  M )   =>    |-  ( S  e.  (SubMnd `  M )  ->  .0.  e.  S )
 
Theoremsubmcl 14446 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 14447 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 14448 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 14449 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 14450 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 14451 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 14452 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 14453 One direction of resmhm2b 14454. (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 14454 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 14455 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 14456 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 14457 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 14458 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 )
 )
 
Theoremprdspjmhm 14459* 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 14460* 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 14461* 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 14462* 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 14463* 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 ) )
 
10.1.3  Ordered group sum operation

One important use of words is as formal composites in cases where order is significant, using the general sum operator df-gsum 13421. If order is not significant, it is simpler to use families instead.

 
Theoremgsumvallem1 14464* Lemma for properties of the set of identities of  G. Either  G has no identities, and  O  =  (/), or it has one and this identity is unique and identified by the 
0g function. (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.  V  ->  O  C_  {  .0.  } )
 
Theoremgsumvallem2 14465* 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.  } )
 
Theoremfisuppfi 14466 A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  ( `' F " C )  e. 
 Fin )
 
Theoremgsumvalx 14467* Expand out the substitutions in df-gsum 13421. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  O  =  { s  e.  B  |  A. t  e.  B  ( ( s 
 .+  t )  =  t  /\  ( t 
 .+  s )  =  t ) }   &    |-  ( ph  ->  W  =  ( `' F " ( _V  \  O ) ) )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  F  e.  X )   &    |-  ( ph  ->  dom  F  =  A )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ... ,  ( iota x E. m E. n  e.  ( ZZ>=
 `  m ) ( A  =  ( m
 ... n )  /\  x  =  (  seq  m (  .+  ,  F ) `  n ) ) ) ,  ( iota
 x E. f ( f : ( 1
 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq  1 ( 
 .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) ) )
 
Theoremgsumval 14468* Expand out the substitutions in df-gsum 13421. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  O  =  { s  e.  B  |  A. t  e.  B  ( ( s 
 .+  t )  =  t  /\  ( t 
 .+  s )  =  t ) }   &    |-  ( ph  ->  W  =  ( `' F " ( _V  \  O ) ) )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ... ,  ( iota x E. m E. n  e.  ( ZZ>=
 `  m ) ( A  =  ( m
 ... n )  /\  x  =  (  seq  m (  .+  ,  F ) `  n ) ) ) ,  ( iota
 x E. f ( f : ( 1
 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq  1 ( 
 .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) ) )
 
Theoremgsumpropd 14469 The group sum depends only on the base set and additive operation. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 14414 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  ( Base `  G )  =  ( Base `  H )
 )   &    |-  ( ph  ->  ( +g  `  G )  =  ( +g  `  H ) )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsumress 14470* The group sum in a substructure is the same as the group sum in the original structure. The only requirement on the substructure is that it contain the identity element; neither  G nor 
H need be groups. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  H  =  ( Gs  S )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  S  C_  B )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  .0.  e.  S )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x )
 )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsumsubm 14471 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 ) )
 
Theoremgsumval1 14472* Value of the group sum operation when every element being summed is an identity of  G. (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 ) }   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  A  e.  W )   &    |-  ( ph  ->  F : A --> O )   =>    |-  ( ph  ->  ( G  gsumg  F )  =  .0.  )
 
Theoremgsum0 14473 Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( G  gsumg  (/) )  =  .0.
 
Theoremgsumz 14474* 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.  )
 
Theoremgsumval2a 14475* Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  F : ( M ... N ) --> B )   &    |-  O  =  { x  e.  B  |  A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y ) }   &    |-  ( ph  ->  -.  ran  F 
 C_  O )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( 
 seq  M (  .+  ,  F ) `  N ) )
 
Theoremgsumval2 14476 Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  F : ( M ... N ) --> B )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( 
 seq  M (  .+  ,  F ) `  N ) )
 
Theoremgsumwsubmcl 14477 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 14478 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 14479 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 14480 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 14481 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 ) )
 
Theoremgsumspl 14482 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 14483 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 14484* 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 ) ) )
 
10.1.4  Free monoids
 
Syntaxcfrmd 14485 Extend class definition with the free monoid construction.
 class freeMnd
 
Syntaxcvrmd 14486 Extend class notation with free monoid injection.
 class varFMnd
 
Definitiondf-frmd 14487 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 14488* 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 14489 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 14490 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 14491 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 14492 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 )   =>    |-  .+  =  ( concat  |`  ( B  X.  B ) )
 
Theoremfrmdadd 14493 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 concat  Y )
 )
 
Theoremvrmdfval 14494* 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 14495 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 14496 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 14497 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 14498 The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   =>    |-  (/)  =  ( 0g `  M )
 
Theoremfrmdsssubm 14499 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 14500 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 )
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