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Theorem List for Metamath Proof Explorer - 15701-15800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdlatjmdi 15701 In a distributive lattice, joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e. DLat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  .\/  ( Y  ./\  Z ) )  =  ( ( X  .\/  Y )  ./\  ( X  .\/  Z ) ) )
 
9.2.6  Posets and lattices as relations
 
Syntaxcps 15702 Extend class notation with the class of all posets.
 class  PosetRel
 
Syntaxctsr 15703 Extend class notation with the class of all totally ordered sets.
 class  TosetRel
 
Definitiondf-ps 15704 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 15705 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
 ) }
 
Theoremisps 15706 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 15707 A poset is a relation. (Contributed by NM, 12-May-2008.)
 |-  ( A  e.  PosetRel  ->  Rel 
 A )
 
Theorempsref2 15708 A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)
 |-  ( R  e.  PosetRel  ->  ( R  i^i  `' R )  =  (  _I  |` 
 U. U. R ) )
 
Theorempstr2 15709 A poset is transitive. (Contributed by FL, 3-Aug-2009.)
 |-  ( R  e.  PosetRel  ->  ( R  o.  R ) 
 C_  R )
 
Theorempslem 15710 Lemma for psref 15712 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 15711 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 15712 A poset is reflexive. (Contributed by NM, 13-May-2008.)
 |-  X  =  dom  R   =>    |-  (
 ( R  e.  PosetRel  /\  A  e.  X )  ->  A R A )
 
Theorempsrn 15713 The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.)
 |-  X  =  dom  R   =>    |-  ( R  e.  PosetRel  ->  X  =  ran  R )
 
Theorempsasym 15714 A poset is antisymmetric. (Contributed by NM, 12-May-2008.)
 |-  ( ( R  e.  PosetRel  /\  A R B  /\  B R A )  ->  A  =  B )
 
Theorempstr 15715 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 15716 The converse of a poset is a poset. In the general case  ( `' R  e.  PosetRel  ->  R  e.  PosetRel ) is not true. See cnvpsb 15717 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 15717 The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)
 |-  ( Rel  R  ->  ( R  e.  PosetRel  <->  `' R  e.  PosetRel ) )
 
Theorempsss 15718 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 15719 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 15720 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 15721 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 15722* 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 15723 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 15724 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 15725 A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  ( R  e.  TosetRel  ->  R  e.  PosetRel )
 
Theoremcnvtsr 15726 The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  ( R  e.  TosetRel  ->  `' R  e.  TosetRel  )
 
Theoremtsrss 15727 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  )
 
Theoremledm 15728 domain of  <_ is  RR*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  RR*  =  dom  <_
 
Theoremlern 15729 The range of  <_ is  RR*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  RR*  =  ran  <_
 
Theoremlefld 15730 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 15731 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 15732 Extend class notation with the class of all directed sets.
 class  DirRel
 
Syntaxctail 15733 Extend class notation with the tail function.
 class  tail
 
Definitiondf-dir 15734 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 15735* 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 15736 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 15737 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 15738 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 15739 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 15740 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 15741* 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 15742 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  Magmas

According to Wikipedia ("Magma (algebra)", 08-Jan-2020, https://en.wikipedia.org/wiki/magma_(algebra)) "In abstract algebra, a magma [...] is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation. The binary operation must be closed by definition but no other properties are imposed.".

Since the concept of a "binary operation" is used in different variants, these differences are explained in more detail in the following:

With df-mpt2 6300, binary operations are defined by a rule, and with df-ov 6298, the value of a binary operation applied to two operands can be expressed. In both cases, the two operands can belong to different sets, and the result can be an element of a third set. However, according to Wikipedia "Binary operation", see https://en.wikipedia.org/wiki/Binary_operation (19-Jan-2020), "... a binary operation on a set  S is a mapping of the elements of the Cartesian product  S  X.  S to S:  f : ( S  X.  S
--> S ). Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure).". To distinguish this more restrictive definition (in Wikipedia and most of the literature) from the general case, binary operations mapping the elements of the Cartesian product  S  X.  S are more precisely called internal binary operations. If, in addition, the result is also contained in the set  S, the operation should be called closed internal binary operation. Therefore, a "binary operation on a set  S" according to Wikipedia is a "closed internal binary operation" in a more precise terminology. If the sets are different, the operation is explicitly called external binary operation (see Wikipedia https://en.wikipedia.org/wiki/Binary_operation#External_binary_operations ).

The definition of magmas (Mgm, see df-mgm 15746) concentrates on the closure property of the associated operation, and poses no additional restrictions on it. In this way, it is most general and flexible.

 
Syntaxcplusf 15743 Extend class notation with group addition as a function.
 class  +f
 
Syntaxcmgm 15744 Extend class notation with class of all magmas.
 class Mgm
 
Definitiondf-plusf 15745* 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 for any magma is that it is a guaranteed function (mgmplusf 15755), while  +g only has closure (mgmcl 15749). (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 +f  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( +g  `  g ) y ) ) )
 
Definitiondf-mgm 15746* A magma is a set equipped with an everywhere defined internal operation. Definition of a magma in [BourbakiAlg1] p. 1, or of a groupoid in [Bruck] p. 1. Note: The term "groupoid" is now widely used to refer to other objects: (small) categories all of whose morphisms are invertible, or groups with a partial function replacing the binary operation. Therefore, we will only use the term "magma" for the present notion in set.mm. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
 |- Mgm 
 =  { g  | 
 [. ( Base `  g
 )  /  b ]. [. ( +g  `  g
 )  /  o ]. A. x  e.  b  A. y  e.  b  ( x o y )  e.  b }
 
Theoremismgm 15747* The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  ( M  e.  V  ->  ( M  e. Mgm  <->  A. x  e.  B  A. y  e.  B  ( x  .o.  y )  e.  B ) )
 
Theoremismgmn0 15748* The predicate "is a magma" for a structure with a nonempty base set. (Contributed by AV, 29-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  ( A  e.  B  ->  ( M  e. Mgm  <->  A. x  e.  B  A. y  e.  B  ( x  .o.  y )  e.  B ) )
 
Theoremmgmcl 15749 Closure of the operation of a magma. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 13-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  (
 ( M  e. Mgm  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .o.  Y )  e.  B )
 
Theoremisnmgm 15750 A condition for a structure not to be a magma. (Contributed by AV, 30-Jan-2020.) (Proof shortened by NM, 5-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  (
 ( X  e.  B  /\  Y  e.  B  /\  ( X  .o.  Y ) 
 e/  B )  ->  M  e/ Mgm  )
 
Theoremplusffval 15751* 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 15752 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 15753 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 15754 The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+^  =  ( +f `  G )   =>    |-  .+^ 
 Fn  ( B  X.  B )
 
Theoremmgmplusf 15755 The group addition function of a magma is a function into its base set. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revisd by AV, 28-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .+^  =  ( +f `  M )   =>    |-  ( M  e. Mgm  ->  .+^  : ( B  X.  B ) --> B )
 
Theoremintopsn 15756 The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 17793. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 23-Jan-2020.)
 |-  ( (  .o.  :
 ( B  X.  B )
 --> B  /\  Z  e.  B )  ->  ( B  =  { Z }  <->  .o. 
 =  { <. <. Z ,  Z >. ,  Z >. } ) )
 
Theoremmgmb1mgm1 15757 The only magma with a base set consisting of one element is the trivial magma (at least if its operation is an internal binary operation). (Contributed by AV, 23-Jan-2020.) (Revised by AV, 7-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   =>    |-  ( ( M  e. Mgm  /\  Z  e.  B  /\  .+  Fn  ( B  X.  B ) ) 
 ->  ( B  =  { Z }  <->  .+  =  { <. <. Z ,  Z >. ,  Z >. } ) )
 
Theoremmgm1 15758 The structure with one element and the only closed internal operation for a singleton is a magma. (Contributed by AV, 10-Feb-2020.)
 |-  M  =  { <. (
 Base `  ndx ) ,  { I } >. , 
 <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
 >. }   =>    |-  ( I  e.  V  ->  M  e. Mgm  )
 
Theoremopifismgm 15759* A structure with a group addition operation expressed by a conditional operator is a magma if both values of the conditional operator are contained in the base set. (Contributed by AV, 9-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  ( +g  `  M )  =  ( x  e.  B ,  y  e.  B  |->  if ( ps ,  C ,  D )
 )   &    |-  ( ph  ->  B  =/= 
 (/) )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  C  e.  B )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  D  e.  B )   =>    |-  ( ph  ->  M  e. Mgm  )
 
10.1.2  Identity elements

According to Wikipedia ("Identity element", 7-Feb-2020, https://en.wikipedia.org/wiki/Identity_element): "In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it.". Or in more detail "... an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity." We concentrate on two-sided identities in the following. The existence of an identity (an identity is unique if it exists, see mgmidmo 15760) is an important property of monoids (see mndid 15806), and therefore also for groups (see grpid 15957), but also for magmas not required to be associative. Non-associative magmas having an identity element are called "unital magmas" (see definition 2 in [BourbakiAlg1] p. 12) or, if the magmas are cancellative, "loops" (see definition in [Bruck] p. 15).

In the context of extensible structures, the identity element (of any magma  M) is defined as "group identity element"  ( 0g `  M
), see df-0g 14714. Related theorems which are already valid for magmas are provided in the following.

 
Theoremmgmidmo 15760* 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 )
 
Theoremgrpidval 15761* 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 ) ) )
 
Theoremgrpidpropd 15762* 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, 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 ) )
 
Theoremfn0g 15763 The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |- 
 0g  Fn  _V
 
Theorem0g0 15764 The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)
 |-  (/)  =  ( 0g `  (/) )
 
Theoremismgmid 15765* 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 15766* 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 15767* 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 15768* 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.  )
 
Theoremgrpidd 15769* Deduce the identity element of a magma 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 ) )
 
Theoremgsumvallem1 15770* 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.  } )
 
10.1.3  Ordered sums in a magma

Usually, the symbol  gsumg is used in the context of (abelian) groups. Therefore it is called "group sum". It can be used, however, also for magmas, that's why the related theorems are provided in the following. If the magma is either not commutative or not associative or has no identity, special care has to be taken. E.g. the order of the single additions could be important, see remark 2. in the comment for df-gsum 14715.

 
Theoremgsumvalx 15771* Expand out the substitutions in df-gsum 14715. (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 15772* Expand out the substitutions in df-gsum 14715. (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 15773 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 15819 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 ) )
 
Theoremgsumpropd2lem 15774* Lemma for gsumpropd2 15775 (Contributed by Thierry Arnoux, 28-Jun-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  ( Base `  G )  =  ( Base `  H )
 )   &    |-  ( ( ph  /\  (
 s  e.  ( Base `  G )  /\  t  e.  ( Base `  G )
 ) )  ->  (
 s ( +g  `  G ) t )  e.  ( Base `  G )
 )   &    |-  ( ( ph  /\  (
 s  e.  ( Base `  G )  /\  t  e.  ( Base `  G )
 ) )  ->  (
 s ( +g  `  G ) t )  =  ( s ( +g  `  H ) t ) )   &    |-  ( ph  ->  Fun 
 F )   &    |-  ( ph  ->  ran 
 F  C_  ( Base `  G ) )   &    |-  A  =  ( `' F "
 ( _V  \  {
 s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G ) t )  =  t  /\  ( t ( +g  `  G ) s )  =  t ) } )
 )   &    |-  B  =  ( `' F " ( _V  \  { s  e.  ( Base `  H )  | 
 A. t  e.  ( Base `  H ) ( ( s ( +g  `  H ) t )  =  t  /\  (
 t ( +g  `  H ) s )  =  t ) } )
 )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsumpropd2 15775* A stronger version of gsumpropd 15773, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 15776. (Contributed by Thierry Arnoux, 28-Jun-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  ( Base `  G )  =  ( Base `  H )
 )   &    |-  ( ( ph  /\  (
 s  e.  ( Base `  G )  /\  t  e.  ( Base `  G )
 ) )  ->  (
 s ( +g  `  G ) t )  e.  ( Base `  G )
 )   &    |-  ( ( ph  /\  (
 s  e.  ( Base `  G )  /\  t  e.  ( Base `  G )
 ) )  ->  (
 s ( +g  `  G ) t )  =  ( s ( +g  `  H ) t ) )   &    |-  ( ph  ->  Fun 
 F )   &    |-  ( ph  ->  ran 
 F  C_  ( Base `  G ) )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsummgmpropd 15776* A stronger version of gsumpropd 15773 if at least one of the involved structures is a magma, see gsumpropd2 15775. (Contributed by AV, 31-Jan-2020.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  ( Base `  G )  =  ( Base `  H )
 )   &    |-  ( ph  ->  G  e. Mgm  )   &    |-  ( ( ph  /\  ( s  e.  ( Base `  G )  /\  t  e.  ( Base `  G ) ) ) 
 ->  ( s ( +g  `  G ) t )  =  ( s (
 +g  `  H )
 t ) )   &    |-  ( ph  ->  Fun  F )   &    |-  ( ph  ->  ran  F  C_  ( Base `  G ) )   =>    |-  ( ph  ->  ( G  gsumg  F )  =  ( H  gsumg  F ) )
 
Theoremgsumress 15777* 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 ) )
 
Theoremgsumval1 15778* 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 15779 Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( G  gsumg  (/) )  =  .0.
 
Theoremgsumval2a 15780* 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 15781 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 ) )
 
Theoremgsumprval 15782 Value of the group sum operation over a pair of sequential integers. (Contributed by AV, 14-Dec-2018.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  =  ( M  +  1 ) )   &    |-  ( ph  ->  F : { M ,  N } --> B )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( ( F `  M )  .+  ( F `  N ) ) )
 
Theoremgsumpr12val 15783 Value of the group sum operation over the pair  { 1 ,  2 }. (Contributed by AV, 14-Dec-2018.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  F : { 1 ,  2 } --> B )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( ( F `  1
 )  .+  ( F `  2 ) ) )
 
10.1.4  Semigroups

The definition of semigroups (SGrp, see df-sgrp 15785) is according to Wikipedia ("Semigroup", 8-Jan-2020, https://en.wikipedia.org/wiki/Semigroup) "In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. ... Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses.".

 
Syntaxcsgrp 15784 Extend class notation with class of all semigroups.
 class SGrp
 
Definitiondf-sgrp 15785* A semigroup is a set equipped with an everywhere defined internal operation (so, a magma, see df-mgm 15746), whose operation is associative. Definition of a semigroup in [Bruck] p. 23, or of an "associative magma" in [BourbakiAlg1] p. 4 . (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
 |- SGrp  =  { g  e. Mgm  |  [. ( Base `  g )  /  b ]. [. ( +g  `  g )  /  o ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( ( x o y ) o z )  =  ( x o ( y o z ) ) }
 
Theoremissgrp 15786* The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by by AV, 6-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  ( M  e. SGrp  <->  ( M  e. Mgm  /\ 
 A. x  e.  B  A. y  e.  B  A. z  e.  B  (
 ( x  .o.  y
 )  .o.  z )  =  ( x  .o.  (
 y  .o.  z )
 ) ) )
 
Theoremissgrpv 15787* The predicate "is a semigroup" for a structure which is a set. (Contributed by AV, 1-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  ( M  e.  V  ->  ( M  e. SGrp  <->  A. x  e.  B  A. y  e.  B  ( ( x  .o.  y
 )  e.  B  /\  A. z  e.  B  ( ( x  .o.  y
 )  .o.  z )  =  ( x  .o.  (
 y  .o.  z )
 ) ) ) )
 
Theoremissgrpn0 15788* The predicate "is a semigroup" for a structure with a nonempty base set. (Contributed by AV, 1-Feb-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  ( A  e.  B  ->  ( M  e. SGrp  <->  A. x  e.  B  A. y  e.  B  ( ( x  .o.  y
 )  e.  B  /\  A. z  e.  B  ( ( x  .o.  y
 )  .o.  z )  =  ( x  .o.  (
 y  .o.  z )
 ) ) ) )
 
Theoremisnsgrp 15789 A condition for a structure not to be a semigroup. (Contributed by AV, 30-Jan-2020.)
 |-  B  =  ( Base `  M )   &    |-  .o.  =  (
 +g  `  M )   =>    |-  (
 ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  ->  ( ( ( X  .o.  Y )  .o. 
 Z )  =/=  ( X  .o.  ( Y  .o.  Z ) )  ->  M  e/ SGrp  ) )
 
Theoremsgrpmgm 15790 A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
 |-  ( M  e. SGrp  ->  M  e. Mgm  )
 
Theoremsgrpass 15791 A semigroup operation is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 30-Jan-2020.)
 |-  B  =  ( Base `  G )   &    |-  .o.  =  (
 +g  `  G )   =>    |-  (
 ( G  e. SGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .o.  Y )  .o. 
 Z )  =  ( X  .o.  ( Y  .o.  Z ) ) )
 
Theoremsgrp1 15792 The structure with one element and the only closed internal operation for a singleton is a semigroup. (Contributed by AV, 10-Feb-2020.)
 |-  M  =  { <. (
 Base `  ndx ) ,  { I } >. , 
 <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
 >. }   =>    |-  ( I  e.  V  ->  M  e. SGrp  )
 
10.1.5  Definition and basic properties of monoids

According to Wikipedia ("Monoid", https://en.wikipedia.org/wiki/Monoid, 6-Feb-2020,) "In abstract algebra [...] a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are semigroups with identity.". In the following, monoids are defined in the second way (as semigroups with identity), see df-mnd 15795, whereas many authors define magmas in the first way (as algebraic structure with a single associative binary operation and an identity element, i.e. without the need of a definition for/knowledge about semigroups), see ismnd 15797. See, for example, the definition in [Lang] p. 3: "A monoid is a set G, with a law of composition which is associative, and having a unit element".

 
Syntaxcmnd 15793 Extend class notation with class of all monoids.
 class  Mnd
 
SyntaxcmndOLD 15794 Extend class notation with class of all monoids.
 class MndOLD
 
Definitiondf-mnd 15795* A monoid is a semigroup, which has a two-sided neutral element. Definition 2 in [BourbakiAlg1] p. 12. In other words (according to the definition in [Lang] p. 3), a monoid is a set equipped with an everywhere defined internal operation (see mndcl 15802), whose operation is associative (see mndass 15803) and has a two-sided neutral element (see mndid 15806), see also ismnd 15797. (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
 |- 
 Mnd  =  { g  e. SGrp  |  [. ( Base `  g )  /  b ]. [. ( +g  `  g
 )  /  p ]. E. e  e.  b  A. x  e.  b  ( ( e p x )  =  x  /\  ( x p e )  =  x ) }
 
Theoremismnddef 15796* The predicate "is a monoid", corresponding 1-to-1 to the definition. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 1-Feb-2020.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e.  Mnd  <->  ( G  e. SGrp  /\  E. e  e.  B  A. a  e.  B  ( ( e 
 .+  a )  =  a  /\  ( a 
 .+  e )  =  a ) ) )
 
Theoremismnd 15797* The predicate "is a monoid". This is the definig theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl 15802), whose operation is associative (so, a semigroup, see also mndass 15803) and has a two-sided neutral element (see mndid 15806). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e.  Mnd  <->  (
 A. a  e.  B  A. b  e.  B  ( ( a  .+  b
 )  e.  B  /\  A. c  e.  B  ( ( a  .+  b
 )  .+  c )  =  ( a  .+  (
 b  .+  c )
 ) )  /\  E. e  e.  B  A. a  e.  B  ( ( e 
 .+  a )  =  a  /\  ( a 
 .+  e )  =  a ) ) )
 
Definitiondf-mndOLD 15798* Obsolete version of df-mnd 15795 as of 6-Feb-2020. Definition of a monoid. A monoid is a set equipped with an everywhere defined internal operation (so, a magma, see mndcl 15802), whose operation is associative (so, a semigroup, see mndass 15803) and has a two-sided neutral element (see mndid 15806). (Contributed by Mario Carneiro, 6-Jan-2015.) (New usage is discouraged.)
 |- MndOLD  =  { g  |  [. ( Base `  g )  /  b ]. [. ( +g  `  g )  /  p ]. ( A. x  e.  b  A. y  e.  b  A. z  e.  b  ( ( x p y )  e.  b  /\  ( ( x p y ) p z )  =  ( x p ( y p z ) ) )  /\  E. e  e.  b  A. x  e.  b  (
 ( e p x )  =  x  /\  ( x p e )  =  x ) ) }
 
TheoremismndOLD 15799* Obsolete version of ismnd 15797 as of 6-Feb-2020. The predicate "is a monoid." (Contributed by Mario Carneiro, 6-Jan-2015.) (New usage is discouraged.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e. MndOLD  <->  ( A. a  e.  B  A. b  e.  B  A. c  e.  B  (
 ( a  .+  b
 )  e.  B  /\  ( ( a  .+  b )  .+  c )  =  ( a  .+  ( b  .+  c ) ) )  /\  E. e  e.  B  A. a  e.  B  ( ( e 
 .+  a )  =  a  /\  ( a 
 .+  e )  =  a ) ) )
 
Theoremmndsgrp 15800 A monoid is a semigroup. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) (Proof shortened by AV, 6-Feb-2020.)
 |-  ( G  e.  Mnd  ->  G  e. SGrp  )
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