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Theorem List for Metamath Proof Explorer - 14301-14400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
9.2.6  Posets and lattices as relations
 
Syntaxcps 14301 Extend class notation with the class of all posets.
 class  PosetRel
 
Syntaxctsr 14302 Extend class notation with the class of all totally ordered sets.
 class  TosetRel
 
Syntaxcspw 14303 Extend class notation with the supremum of an ordered set.
 class  sup w
 
Syntaxcinf 14304 Extend class notation with the infimum of an ordered set.
 class  inf w
 
Syntaxcla 14305 Extend class notation with the class of all lattices.
 class  LatRel
 
Definitiondf-ps 14306 Define the class of all posets (partially ordered sets) with weak ordering (e.g. "less than or equal to" instead of "less than"). A poset is a relation which is transitive, reflexive, and antisymmetric. (Contributed by NM, 11-May-2008.)
 |-  PosetRel 
 =  { r  |  ( Rel  r  /\  ( r  o.  r
 )  C_  r  /\  ( r  i^i  `' r
 )  =  (  _I  |`  U. U. r ) ) }
 
Definitiondf-tsr 14307 Define the class of all totally ordered sets. (Contributed by FL, 1-Nov-2009.)
 |-  TosetRel 
 =  { r  e.  PosetRel 
 |  ( dom  r  X.  dom  r )  C_  ( r  u.  `' r
 ) }
 
Definitiondf-spw 14308* Define suprema under weak orderings. Unlike df-sup 7194 for strong orderings,  sup w is evaluates to a member of the field of  R iff the supremum exists. Read 
R  sup w  A as the  R-supremum of set  A. (Contributed by NM, 13-May-2008.)
 |- 
 sup w  =  (
 r  e.  PosetRel ,  x  e.  _V  |->  ( iota_ y  e. 
 U. U. r ( A. z  e.  x  z
 r y  /\  A. z  e.  U. U. r
 ( A. w  e.  x  w r z  ->  y r z ) ) ) )
 
Definitiondf-nfw 14309* Define the class of all infima of a weak ordering relation. (Contributed by FL, 6-Sep-2009.)
 |- 
 inf w  =  (
 r  e.  _V ,  x  e.  _V  |->  ( `' r  sup w  x ) )
 
Definitiondf-lar 14310* Define the class of all lattices, which are posets in which every two elements have upper and lower bounds. (Contributed by NM, 12-Jun-2008.)
 |-  LatRel  =  { r  e.  PosetRel 
 |  A. x  e.  dom  r A. y  e.  dom  r ( ( r 
 sup w  { x ,  y } )  e. 
 dom  r  /\  (
 r  inf w  { x ,  y } )  e. 
 dom  r ) }
 
Theoremisps 14311 The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation. (Contributed by NM, 11-May-2008.)
 |-  ( R  e.  A  ->  ( R  e.  PosetRel  <->  ( Rel  R  /\  ( R  o.  R )  C_  R  /\  ( R  i^i  `' R )  =  (  _I  |`  U. U. R ) ) ) )
 
Theorempsrel 14312 A poset is a relation. (Contributed by NM, 12-May-2008.)
 |-  ( A  e.  PosetRel  ->  Rel 
 A )
 
Theorempsref2 14313 A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)
 |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |` 
 U. U. R ) )
 
Theorempstr2 14314 A poset is transitive. (Contributed by FL, 3-Aug-2009.)
 |-  ( R  e.  PosetRel  ->  ( R  o.  R ) 
 C_  R )
 
Theorempslem 14315 Lemma for psref 14317 and others. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( R  e.  PosetRel  ->  ( ( ( A R B  /\  B R C )  ->  A R C )  /\  ( A  e.  U.
 U. R  ->  A R A )  /\  (
 ( A R B  /\  B R A ) 
 ->  A  =  B ) ) )
 
Theorempsdmrn 14316 The domain and range of a poset equal its field. (Contributed by NM, 13-May-2008.)
 |-  ( R  e.  PosetRel  ->  ( dom  R  =  U. U. R  /\  ran  R  =  U. U. R ) )
 
Theorempsref 14317 A poset is reflexive. (Contributed by NM, 13-May-2008.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  PosetRel  /\  A  e.  X )  ->  A R A )
 
Theorempsrn 14318 The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.)
 |-  X  =  dom  R   =>    |-  ( R  e.  PosetRel  ->  X  =  ran  R )
 
Theorempsasym 14319 A poset is antisymmetric. (Contributed by NM, 12-May-2008.)
 |-  ( ( R  e.  PosetRel  /\  A R B  /\  B R A )  ->  A  =  B )
 
Theorempstr 14320 A poset is transitive. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  ( ( R  e.  PosetRel  /\  A R B  /\  B R C )  ->  A R C )
 
Theoremcnvps 14321 The converse of a poset is a poset. In the general case  ( `' R  e.  PosetRel  ->  R  e.  PosetRel ) is not true. See cnvpsb 14322 for a special case where the property holds. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
 
Theoremcnvpsb 14322 The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)
 |-  ( Rel  R  ->  ( R  e.  PosetRel  <->  `' R  e.  PosetRel ) )
 
Theorempsss 14323 Any subset of a partially ordered set is partially ordered. (Contributed by FL, 24-Jan-2010.)
 |-  ( R  e.  PosetRel  ->  ( R  i^i  ( A  X.  A ) )  e.  PosetRel )
 
Theorempsssdm2 14324 Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   =>    |-  ( R  e.  PosetRel  ->  dom  ( R  i^i  ( A  X.  A ) )  =  ( X  i^i  A ) )
 
Theorempsssdm 14325 Field of a subposet. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  PosetRel  /\  A  C_  X )  ->  dom  ( R  i^i  ( A  X.  A ) )  =  A )
 
Theoremistsr 14326 The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  X  =  dom  R   =>    |-  ( R  e.  TosetRel  <->  ( R  e.  PosetRel  /\  ( X  X.  X )  C_  ( R  u.  `' R ) ) )
 
Theoremistsr2 14327* The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  X  =  dom  R   =>    |-  ( R  e.  TosetRel  <->  ( R  e.  PosetRel  /\ 
 A. x  e.  X  A. y  e.  X  ( x R y  \/  y R x ) ) )
 
Theoremtsrlin 14328 A toset is a linear order. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  TosetRel  /\  A  e.  X  /\  B  e.  X )  ->  ( A R B  \/  B R A ) )
 
Theoremtsrlemax 14329 Two ways of saying a number is less than or equal to the maximum of two others. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  TosetRel  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A R if ( B R C ,  C ,  B )  <->  ( A R B  \/  A R C ) ) )
 
Theoremtsrps 14330 A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
 
Theoremcnvtsr 14331 The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  TosetRel  ->  `' R  e.  TosetRel  )
 
Theoremtsrss 14332 Any subset of a totally ordered set is totally ordered. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Nov-2013.)
 |-  ( R  e.  TosetRel  ->  ( R  i^i  ( A  X.  A ) )  e.  TosetRel  )
 
Theoremspwval2 14333* Value of supremum under a weak ordering. Read  R  sup w  A as "the  R-supremum of  A."  U. U. R is the field of a relation  R by relfld 5198. Unlike df-sup 7194 for strong orderings, the supremum exists iff  R  sup w  A belongs to the field. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)
 |-  X  =  U. U. R   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  V ) 
 ->  ( R  sup w  A )  =  ( iota_ x  e.  X ( A. y  e.  A  y R x  /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) ) )
 
Theoremspwval 14334* Value of supremum under a weak ordering. Read  R  sup w  A as "the  R-supremum of  A."  U. U. R is the field of a relation  R by relfld 5198. Unlike df-sup 7194 for strong orderings, the supremum exists iff  R  sup w  A belongs to the field. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  PosetRel  /\  A  e.  V )  ->  ( R  sup w  A )  =  ( iota_ x  e.  X ( A. y  e.  A  y R x  /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) ) )
 
Theoremspwmo 14335* A poset has at most one supremum. (Contributed by NM, 13-May-2008.) (Revised by NM, 16-Jun-2017.)
 |-  ( ph  <->  ( A. y  e.  A  y R x 
 /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )   =>    |-  ( R  e.  PosetRel  ->  E* x  e.  X ph )
 
Theoremspweu 14336* A supremum is unique. (Contributed by NM, 15-May-2008.)
 |-  ( ph  <->  ( A. y  e.  A  y R x 
 /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )   =>    |-  ( ( R  e.  PosetRel  /\ 
 E. x  e.  X  ph )  ->  E! x  e.  X  ph )
 
Theoremspwpr2 14337* Property of supremum defining condition for an unordered pair. (Contributed by NM, 24-Jun-2008.)
 |-  ( ph  <->  ( A. y  e.  A  y R x 
 /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )   =>    |-  ( ( ( R  e.  T  /\  A  =  { B ,  C } )  /\  ( B  e.  U  /\  C  e.  W ) )  ->  ( ph  <->  ( ( B R x  /\  C R x )  /\  A. y  e.  X  (
 ( B R y 
 /\  C R y )  ->  x R y ) ) ) )
 
Theoremspwex 14338* A supremum exists iff  R  sup w  A belongs to the domain of  R. (Contributed by NM, 15-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)
 |-  X  =  dom  R   &    |-  ( ph 
 <->  ( A. y  e.  A  y R x 
 /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  V ) 
 ->  ( E. x  e.  X  ph  <->  ( R  sup w  A )  e.  X ) )
 
Theoremspwcl 14339* Closure of a supremum. (Contributed by NM, 15-May-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)
 |-  X  =  dom  R   &    |-  ( ph 
 <->  ( A. y  e.  A  y R x 
 /\  A. y  e.  X  ( A. z  e.  A  z R y  ->  x R y ) ) )   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  V  /\  E. x  e.  X  ph )  ->  ( R  sup w  A )  e.  X )
 
Theoremspwpr4 14340* Supremum of an unordered pair. (Contributed by NM, 7-Jul-2008.) (Revised by Mario Carneiro, 20-Nov-2013.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  PosetRel  /\  ( A R C  /\  B R C )  /\  A. x  e.  X  ( ( A R x 
 /\  B R x )  ->  C R x ) )  ->  ( R  sup w  { A ,  B }
 )  =  C )
 
Theoremspwpr4c 14341 Supremum of an unordered pair of comparable elements. (Contributed by NM, 7-Jul-2008.)
 |-  ( ( R  e.  PosetRel  /\  A R B ) 
 ->  ( R  sup w  { A ,  B }
 )  =  B )
 
Theoremisla 14342* The predicate "is a lattice" i.e. a poset in which any two elements have upper and lower bounds. (Contributed by NM, 12-Jun-2008.)
 |-  X  =  dom  R   =>    |-  ( R  e.  LatRel  <->  ( R  e.  PosetRel  /\ 
 A. x  e.  X  A. y  e.  X  ( ( R  sup w  { x ,  y }
 )  e.  X  /\  ( R  inf w  { x ,  y }
 )  e.  X ) ) )
 
Theoremlaspwcl 14343 Closure of the supremum (join) of two lattice elements. (Contributed by NM, 12-Jun-2008.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  LatRel  /\  A  e.  X  /\  B  e.  X )  ->  ( R  sup w  { A ,  B }
 )  e.  X )
 
Theoremlanfwcl 14344 Closure of the infimum (meet) of two lattice elements. (Contributed by NM, 20-Jun-2008.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  LatRel  /\  A  e.  X  /\  B  e.  X )  ->  ( R  inf w  { A ,  B }
 )  e.  X )
 
Theoremlaps 14345 A lattice is a poset. (Contributed by NM, 12-Jun-2008.)
 |-  ( R  e.  LatRel  ->  R  e.  PosetRel )
 
Theoremledm 14346 domain of  <_ is  RR*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  RR*  =  dom  <_
 
Theoremlern 14347 The range of  <_ is  RR*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  RR*  =  ran  <_
 
Theoremlefld 14348 The field of the 'less or equal to' relationship on the extended real. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  RR*  =  U. U.  <_
 
Theoremletsr 14349 The "less than or equal to" relationship on the extended reals is a toset. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |- 
 <_  e.  TosetRel
 
9.2.7  Directed sets, nets
 
Syntaxcdir 14350 Extend class notation with the class of all directed sets.
 class  DirRel
 
Syntaxctail 14351 Extend class notation with the tail function.
 class  tail
 
Definitiondf-dir 14352 Define the class of all directed sets/directions. (Contributed by Jeff Hankins, 25-Nov-2009.)
 |-  DirRel  =  { r  |  ( ( Rel  r  /\  (  _I  |`  U. U. r )  C_  r ) 
 /\  ( ( r  o.  r )  C_  r  /\  ( U. U. r  X.  U. U. r
 )  C_  ( `' r  o.  r ) ) ) }
 
Definitiondf-tail 14353* Define the tail function for directed sets. (Contributed by Jeff Hankins, 25-Nov-2009.)
 |- 
 tail  =  ( r  e.  DirRel  |->  ( x  e. 
 U. U. r  |->  ( r
 " { x }
 ) ) )
 
Theoremisdir 14354 A condition for a relation to be a direction. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  A  =  U. U. R   =>    |-  ( R  e.  V  ->  ( R  e.  DirRel  <->  (
 ( Rel  R  /\  (  _I  |`  A )  C_  R )  /\  (
 ( R  o.  R )  C_  R  /\  ( A  X.  A )  C_  ( `' R  o.  R ) ) ) ) )
 
Theoremreldir 14355 A direction is a relation. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  ( R  e.  DirRel  ->  Rel  R )
 
Theoremdirdm 14356 A direction's domain is equal to its field. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  ( R  e.  DirRel  ->  dom  R  =  U. U. R )
 
Theoremdirref 14357 A direction is reflexive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  DirRel  /\  A  e.  X ) 
 ->  A R A )
 
Theoremdirtr 14358 A direction is transitive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  ( ( ( R  e.  DirRel  /\  C  e.  V )  /\  ( A R B  /\  B R C ) )  ->  A R C )
 
Theoremdirge 14359* For any two elements of a directed set, there exists a third element greater than or equal to both. (Note that this does not say that the two elements have a least upper bound.) (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  DirRel  /\  A  e.  X  /\  B  e.  X )  ->  E. x  e.  X  ( A R x  /\  B R x ) )
 
Theoremtsrdir 14360 A totally ordered set is a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
 |-  ( A  e.  TosetRel  ->  A  e.  DirRel )
 
PART 10  BASIC ALGEBRAIC STRUCTURES
 
10.1  Monoids
 
10.1.1  Definition and basic properties
 
Syntaxcmnd 14361 Extend class notation with class of all monoids.
 class  Mnd
 
Syntaxcgrp 14362 Extend class notation with class of all groups.
 class  Grp
 
Syntaxcminusg 14363 Extend class notation with inverse of group element.
 class  inv g
 
Syntaxcplusf 14364 Extend class notation with group addition as a function.
 class  + f
 
Syntaxcsg 14365 Extend class notation with group subtraction (or division) operation.
 class  -g
 
Syntaxcmg 14366 Extend class notation with a function mapping a group operation to the power operation for the group.
 class .g
 
Definitiondf-mnd 14367* Definition of a monoid. A monoid is a set equipped with an everywhere defined internal operation (so, a magma, see mndcl 14372), whose operation is associative (so, a semigroup, see mndass 14373) and has a two-sided neutral element (see mndid 14374). (Contributed by Mario Carneiro, 6-Jan-2015.)
 |- 
 Mnd  =  { 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 ) ) }
 
Definitiondf-plusf 14368* Define group addition function. Usually we will use  +g directly instead of  + f, and they have the same behavior in most cases. The main advantage of  + f is that it is a guaranteed function (mndplusf 14383), while  +g only has closure (mndcl 14372). (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 + f  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( +g  `  g ) y ) ) )
 
Theoremismnd 14369* The predicate "is a monoid." (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e.  Mnd  <->  (
 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 ) ) )
 
Theoremmgmidmo 14370* A two-sided identity element is unique (if it exists) in any magma. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by NM, 17-Jun-2017.)
 |- 
 E* u  e.  B A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x )
 
Theoremmndlem1 14371 Lemma for monoid properties. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .+  Y )  e.  B  /\  ( ( X  .+  Y )  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) ) )
 
Theoremmndcl 14372 Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
 
Theoremmndass 14373 A monoid operation is associative. (Contributed by NM, 14-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .+  Y )  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
 
Theoremmndid 14374* 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 14375* 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 14376 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 14377 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 14378 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 ) ) )
 
Theoremplusffval 14379* The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .+^  =  ( + f `  G )   =>    |-  .+^  =  ( x  e.  B ,  y  e.  B  |->  ( x  .+  y ) )
 
Theoremplusfval 14380 The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .+^  =  ( + f `  G )   =>    |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .+^  Y )  =  ( X 
 .+  Y ) )
 
Theoremplusfeq 14381 If the addition operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .+^  =  ( + f `  G )   =>    |-  (  .+  Fn  ( B  X.  B )  ->  .+^ 
 =  .+  )
 
Theoremplusffn 14382 The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+^  =  ( + f `  G )   =>    |-  .+^ 
 Fn  ( B  X.  B )
 
Theoremmndplusf 14383 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+^  =  ( + f `  G )   =>    |-  ( G  e.  Mnd  ->  .+^ 
 : ( B  X.  B ) --> B )
 
Theoremgrpidval 14384* The value of the identity element of a group. (Contributed by NM, 20-Aug-2011.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |- 
 .0.  =  ( iota e ( e  e.  B  /\  A. x  e.  B  ( ( e 
 .+  x )  =  x  /\  ( x 
 .+  e )  =  x ) ) )
 
Theoremfn0g 14385 The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |- 
 0g  Fn  _V
 
Theorem0g0 14386 The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)
 |-  (/)  =  ( 0g `  (/) )
 
Theoremismgmid 14387* 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 14388* 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 14389* 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 14390* 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 14391 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 14392 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 14393 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 14394 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 14395* 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 14396* 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 14397 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 14398* 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 14399* 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 14400 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 )
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