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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmnlmxl2 24601 The minimal elements of a preset are the maximal elements of the converse preset. (Contributed by FL, 19-Sep-2011.)
 |-  ( R  e. PresetRel  ->  ( mnl `  R )  =  ( mxl `  `' R ) )
 
Theoremmxlmnl2 24602 The maximal elements of a preset are the minimal elements of the converse preset. (Contributed by FL, 19-Sep-2011.)
 |-  ( R  e. PresetRel  ->  ( mxl `  R )  =  ( mnl `  `' R ) )
 
Theoremdefge3 24603* The greatest element of a poset is an element, when it exists, that is greater than the other elements of the poset. Use the idiom  ( ge `  R )  e.  X when you mean the greatest element of  X exists. (Contributed by FL, 30-Dec-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  ( ge `  R )  e.  X )  ->  A. x  e.  X  x R ( ge `  R ) )
 
Theoremdefse3 24604* The least element of a poset is an element, when it exists, that is less than the other elements of the poset. Use the idiom  (leR `  R
)  e.  X when you mean the least element of  X exists. (Contributed by FL, 30-Dec-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  (leR `  R )  e.  X )  ->  A. x  e.  X  (leR `  R ) R x )
 
Theoremsupaub 24605 If it exists, a supremum of  A is an upper bound for  A. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  ->  ( R  sup w  A )  e.  ( R  ub  A ) )
 
Theoremsupwlub 24606* If it exists, a supremum of  A is the least upper bound for  A. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  ->  A. x  e.  ( R  ub  A ) ( R  sup w  A ) R x )
 
Theoremgeme2 24607 The greatest element of  X is a maximal element. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  ( R  sup w  X )  e.  X )  ->  ( R  sup w  X )  e.  ( mxl `  R ) )
 
Theoreminposetlem 24608* Lemma for inposet 24610. Definition of inclusion of sets using a class. (Contributed by FL, 22-Sep-2008.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |-  ( A C B  <->  A 
 C_  B )
 
Theoreminpc 24609* Inclusion is a proper class. (Contributed by FL, 22-Sep-2008.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |- 
 -.  C  e.  _V
 
Theoreminposet 24610* Inclusion partially orders any set. (Contributed by FL, 22-Sep-2008.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |-  ( A  e.  B  ->  ( C  i^i  ( A  X.  A ) )  e.  PosetRel )
 
Theoremdefinc 24611* Definition of the inclusion. (Contributed by FL, 6-Sep-2009.)
 |-  G  e.  X   &    |-  F  e.  Y   &    |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |-  ( G ( C  i^i  ( A  X.  B ) ) F  <-> 
 ( G  e.  A  /\  F  e.  B  /\  G  C_  F ) )
 
Theoremdominc 24612* The domain of the inclusion relation is  _V. (Contributed by FL, 6-Sep-2009.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |- 
 dom  C  =  _V
 
Theoremrninc 24613* The range of the inclusion relation is  _V. (Contributed by FL, 6-Sep-2009.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |- 
 ran  C  =  _V
 
Theoremdomncnt 24614* Domain of the intersection of the inclusion with a square cross product. (Contributed by FL, 6-Sep-2009.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |- 
 dom  (  C  i^i  ( A  X.  A ) )  =  A
 
Theoremranncnt 24615* Range of the intersection of the inclusion with a square cross product. (Contributed by FL, 6-Sep-2009.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |- 
 ran  (  C  i^i  ( A  X.  A ) )  =  A
 
Theoremsupwval 24616 Value of an infimum under a weak ordering. (Contributed by FL, 19-Sep-2011.)
 |-  (
 ( Rel  R  /\  R  e.  U  /\  A  e.  W )  ->  ( R  sup w  A )  =  ( `' R  inf w  A ) )
 
Theoremnfwpr4c 24617 Infimum of an unordered pair of comparable elements. (Contributed by FL, 19-Sep-2011.)
 |-  (
 ( R  e.  PosetRel  /\  A R B )  ->  ( R  inf w  { A ,  B }
 )  =  A )
 
Theoremtolat 24618 A totally ordered set is a lattice. (Contributed by FL, 19-Sep-2011.)
 |-  TosetRel  C_  LatRel
 
Theoremdispos 24619 A restriction of the identity is a poset. (Contributed by FL, 2-Aug-2009.)
 |-  ( A  e.  V  ->  (  _I  |`  A )  e. 
 PosetRel )
 
Theoremderef 24620 An idiom to "dereflexivate" a relation. (Contributed by FL, 30-Dec-2010.)
 |-  ( A  e.  B  ->  -.  A ( C  \  _I  ) A )
 
Theoremdfps2 24621 Alternate definition of a poset. Bourbaki E.III.2 prop. 1. (Contributed by FL, 30-Dec-2010.)
 |-  PosetRel  =  {
 r  |  ( Rel  r  /\  ( r  o.  r )  =  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r ) ) }
 
Theoremtoplat 24622* A topology when ordered by the inclusion is a lattice. This fact leads to the idea of pointless topology, that is a lattice looked at with the eyes of a topologist. (Contributed by FL, 6-Sep-2009.)
 |-  C  =  { <. u ,  v >.  |  u  C_  v }   =>    |-  ( J  e.  Top  ->  ( C  i^i  ( J  X.  J ) )  e.  LatRel )
 
Theoremdfdir2 24623* A directed set (also called a set filtering on the right by Bourbaki) is a preordered set whose every pair of elements has an upper bound. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 21-Nov-2013.)
 |-  DirRel  =  (PresetRel  i^i  { r  |  A. x  e.  U. U. r A. y  e.  U. U. r E. z  e.  U. U. r z  e.  (
 r  ub  { x ,  y } ) }
 )
 
Theoremisdir2 24624* Alternate definition of a direction. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  D   =>    |-  ( D  e.  DirRel  <->  ( D  e. PresetRel  /\  A. x  e.  X  A. y  e.  X  E. z  e.  X  z  e.  ( D  ub  { x ,  y } ) ) )
 
Theoremdirpre 24625 A direction is a preset. (Contributed by FL, 19-Sep-2011.)
 |-  ( D  e.  DirRel  ->  D  e. PresetRel )
 
Theoremdirub 24626* In a direction, every pair of elements has an upper bound. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  D   =>    |-  ( D  e.  DirRel  ->  A. x  e.  X  A. y  e.  X  E. z  e.  X  z  e.  ( D  ub  { x ,  y }
 ) )
 
Theoremlatdir 24627 A lattice is a direction. (Contributed by FL, 19-Sep-2011.)
 |-  LatRel  C_  DirRel
 
Syntaxclbl 24628 Extend class notation to include bound lattice.
 class  BndLat
 
Definitiondf-bnlat 24629 A bound lattice is a lattice that has a greatest and a least element. (Contributed by FL, 21-May-2012.)
 |-  BndLat  =  {
 r  e.  LatRel  |  ( (leR `  r )  e.  dom  r  /\  ( ge `  r )  e. 
 dom  r ) }
 
16.12.14  Finite composites ( i. e. finite sums, products ... )
 
Syntaxcprd 24630 Extend class notation to include finite products/sums.
 class  prod_ k  e.  A G B
 
Definitiondf-prod 24631* Define the composite for the law  G of a finite sequence of elements whose values are defined by the expression  B and whose set of indices is  A.  A may be empty. It may be thougt as a product (if  G is a multiplication), a sum (if  G is an addition) or whatever. The variable  k is normally a free variable in  B ( i.e.  B can be thought of as  B ( k )). The definition is meaningful when  A is a finite set of sequential integers and  G is an internal operation. Our definition corresponds to the first part of the definition of df-sum 12089. The operation  + has been replaced by the generic operation  G. The reference to the concept of limit has been removed because one wants to use the product in contexts where limits are irrelevant. I could be still more generic and replace  ( m ... n ) by a finite totally ordered set. I would then get the definition given by Bourbaki in the first chapter of the algebra book of his treatise ( A I.3 def.4 ). I don't because the present definition is easier to deal with and because there exists an order isomorphism between any finite totally ordered set and any finite sets of integers. I don't specify anything about  G because nothing is required of  g in the definition of  seq. I hope it will be ok. Otherwise one could add  G  e.  Magma. (Contributed by FL, 5-Sep-2010.)
 |-  prod_ k  e.  A G B  =  if ( A  =  (/)
 ,  (GId `  G ) ,  { x  |  E. m E. n  e.  ( ZZ>= `  m )
 ( A  =  ( m ... n ) 
 /\  x  e.  (  seq  m ( G ,  ( k  e.  _V  |->  B ) ) `  n ) ) }
 )
 
Syntaxcprd2 24632 Extend class notation to include finite supports products/sums.
 class prod2 k  e.  A G B
 
Syntaxcprd3 24633 Extend class notation to include finite supports products/sums.
 class prod3 k  e.  A G B
 
Definitiondf-prod2 24634* Definition of a sum or product operator to be used with generic structures defined by extensible structures.  A is the set of indices,  G the operation,  B an expression,  k is normally a free variable in  B.  A may be any extensible structure with a base set. Its base set may be infinite provided that the "support" is finite. The support is the set:  { k  e.  ( Base `  A
)  |  B  =/=  ( 0g `  G
) }. The base set of  A may be empty.  G must be an extensible structure with a law commutative, associative with a neutral element. (Contributed by FL, 17-Oct-2011.)
 |- prod2 k  e.  A G B  =  if ( ( Base `  G )  =  (/) ,  ( 0g `  G ) ,  ( iota x E. m  e.  NN  E. f
 ( f : ( 1 ... m ) -1-1-onto-> { k  e.  ( Base `  A )  |  B  =/=  ( 0g `  G ) }  /\  x  =  (  seq  1 ( ( +g  `  G ) ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) ) ) )
 
Definitiondf-prod3 24635* Definition of a sum or product operator to be used with generic structures defined by extensible structures.  A is the set of indices,  G the operation,  B an expression,  k is normally a free variable in  B.  A must be a total order. Its base set may be infinite provided that the "support" is finite. The support is the set:  { k  e.  ( Base `  A
)  |  B  =/=  ( 0g `  G
) }. The base set of  A may be empty.  G must be an associative law with a neutral element. (Contributed by FL, 17-Oct-2011.)
 |- prod3 k  e.  A G B  =  if ( ( Base `  G )  =  (/) ,  ( 0g `  G ) ,  ( iota x E. m  e.  NN  E. f
 ( f  e.  (
 ( 1 ... m )  OrIso  { k  e.  ( Base `  A )  |  B  =/=  ( 0g `  G ) }
 )  /\  x  =  (  seq  1 ( (
 +g  `  G ) ,  ( n  e.  NN  |->  [_ ( f `  n )  /  k ]_ B ) ) `  m ) ) ) )
 
Theoremprodex 24636 A finite composite is a set. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  prod_ k  e.  A G B  e.  _V
 
Theoremprod0 24637 The value of  prod_ k  e.  (/) G B. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  prod_ k  e.  (/) G B  =  (GId `  G )
 
Theoremprodeq1 24638 Equality theorem for a composite. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  ( A  =  B  ->  prod_
 k  e.  A G C  =  prod_ k  e.  B G C )
 
Theoremprodeq2 24639 Equality theorem for a composite. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  ( G  =  H  ->  prod_
 k  e.  A G C  =  prod_ k  e.  A H C )
 
Theoremprodeq3ii 24640* Equality theorem for a composite. (Contributed by Mario Carneiro, 26-May-2014.)
 |-  ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C )  ->  prod_ k  e.  A G B  =  prod_ k  e.  A G C )
 
Theoremprodeq3 24641* Equality theorem for a composite. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2014.)
 |-  ( A. k  e.  A  B  =  C  ->  prod_
 k  e.  A G B  =  prod_ k  e.  A G C )
 
Theoremnfprod1 24642* Bound-variable hypothesis builder for  prod_. (Contributed by FL, 14-Sep-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ k A   &    |-  F/_ k G   =>    |-  F/_ k prod_ k  e.  A G B
 
Theoremnfprod 24643* Bound-variable hypothesis builder for  prod_. If  x is (effectively) not free in  A,  G and  B, it is not free in  prod_ k  e.  A G B. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ x A   &    |-  F/_ x G   &    |-  F/_ x B   =>    |-  F/_ x prod_ k  e.  A G B
 
Theoremcbvprodi 24644 Change bound variable in a finite composite. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  F/_ k B   &    |-  F/_ j C   &    |-  ( j  =  k  ->  B  =  C )   =>    |- 
 prod_ j  e.  A G B  =  prod_ k  e.  A G C
 
Theoremprodeq1d 24645 Conditions for two composites to be equal. (Contributed by FL, 5-Sep-2010.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  prod_ k  e.  A G C  =  prod_ k  e.  B G C )
 
Theoremprodeq2d 24646 Conditions for two composites to be equal. (Contributed by FL, 5-Sep-2010.)
 |-  ( ph  ->  G  =  H )   =>    |-  ( ph  ->  prod_ k  e.  A G C  =  prod_ k  e.  A H C )
 
Theoremprodeq3d 24647* Conditions for two composites to be equal. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  ( ph  ->  A. k  e.  A  C  =  D )   =>    |-  ( ph  ->  prod_ k  e.  A G C  =  prod_ k  e.  A G D )
 
Theoremprodeq123d 24648* Conditions for two composites to be equal. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  G  =  H )   &    |-  ( ph  ->  A. k  e.  A  C  =  D )   =>    |-  ( ph  ->  prod_ k  e.  A G C  =  prod_ k  e.  B H D )
 
Theoremprodeq123i 24649* Conditions for two composites to be equal. (Contributed by FL, 6-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  A  =  B   &    |-  G  =  H   &    |-  (
 k  e.  A  ->  C  =  D )   =>    |-  prod_ k  e.  A G C  =  prod_ k  e.  B H D
 
Theoremprodeqfv 24650* Convert a composite of function values to a composite of classes  A ( k ). (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  F  =  ( k  e.  B  |->  A )   =>    |-  ( C  C_  B  -> 
 prod_ m  e.  C G ( F `  m )  =  prod_ k  e.  C G A )
 
Theoremdffprod 24651 Special case of composite over a finite index set. (Contributed by FL, 5-Sep-2010.) (Proof shortened by Mario Carneiro, 26-May-2014.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  prod_ k  e.  ( M ... N ) G A  =  ( 
 seq  M ( G ,  ( k  e.  _V  |->  A ) ) `  N ) )
 
Theoremfprodser 24652* A finite composite expressed in terms of a partial composite of an infinite series. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  prod_ k  e.  ( M ... N ) G ( F `  k )  =  (  seq  M ( G ,  F ) `  N ) )
 
Theoremfprodserf 24653 Version of fprodser 24652 with a bound-variable hypothesis instead of a distinct variable condition. (Contributed by FL, 5-Sep-2010.)
 |-  F/_ k F   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  prod_ k  e.  ( M
 ... N ) G ( F `  k
 )  =  (  seq  M ( G ,  F ) `  N ) )
 
Theoremfprod1i 24654* The finite composite of  A ( k ) from  k  =  M to  M (i.e. a composite with only one term) is  B i.e.  A ( M ). (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  (
 k  =  M  ->  A  =  B )   =>    |-  ( ( M  e.  ZZ  /\  B  e.  V )  ->  prod_ k  e.  ( M ... M ) G A  =  B )
 
Theoremfprodp1 24655* The composite of the next term in a finite composite of  A
( k ) is the previous term composed with  A ( N  + 
1 )  =  B. (Contributed by Mario Carneiro, 26-May-2014.)
 |-  (
 k  =  ( N  +  1 )  ->  A  =  B )   =>    |-  (
 ( N  e.  ( ZZ>=
 `  M )  /\  B  e.  V )  -> 
 prod_ k  e.  ( M ... ( N  +  1 ) ) G A  =  ( prod_
 k  e.  ( M
 ... N ) G A G B ) )
 
Theoremfprodp1i 24656* The composite of the next term in a finite composite of  A
( k ) is the previous term composed with  A ( N  + 
1 )  =  B. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  (
 k  =  ( N  +  1 )  ->  A  =  B )   &    |-  B  e.  _V   =>    |-  ( N  e.  ( ZZ>=
 `  M )  ->  prod_ k  e.  ( M
 ... ( N  +  1 ) ) G A  =  ( prod_
 k  e.  ( M
 ... N ) G A G B ) )
 
Theoremfprod1s 24657 The finite composite of a sequence 
A ( k ) from 
M to  M (i.e. a composite with only one term) is  A ( M ). (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  (
 ( M  e.  ZZ  /\  [_ M  /  k ]_ A  e.  V )  ->  prod_ k  e.  ( M ... M ) G A  =  [_ M  /  k ]_ A )
 
Theoremfprod1fi 24658* The finite composite of a term  A ( k ) from 
M to  M (i.e. a composite with only one term) is  A ( M )  =  B, where  k is effectively not free in  B. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  F/_ k B   &    |-  ( k  =  M  ->  A  =  B )   =>    |-  ( ( M  e.  ZZ  /\  B  e.  V )  ->  prod_ k  e.  ( M ... M ) G A  =  B )
 
Theoremfprodp1s 24659 The composite of the next term in a finite sum of  A ( k ) is the previous term composed with 
A ( N  + 
1 ). (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  (
 ( N  e.  ( ZZ>=
 `  M )  /\  [_ ( N  +  1 )  /  k ]_ A  e.  V )  -> 
 prod_ k  e.  ( M ... ( N  +  1 ) ) G A  =  ( prod_
 k  e.  ( M
 ... N ) G A G [_ ( N  +  1 )  /  k ]_ A ) )
 
Theoremfprodp1fi 24660* The composite of the next term in a finite composite of  A
( k ) is the previous term composed with  A ( N  + 
1 )  =  B. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  B  e.  _V   &    |-  F/_ k B   &    |-  (
 k  =  ( N  +  1 )  ->  A  =  B )   =>    |-  ( N  e.  ( ZZ>= `  M )  ->  prod_ k  e.  ( M ... ( N  +  1 )
 ) G A  =  ( prod_ k  e.  ( M ... N ) G A G B ) )
 
Theoremfsumprd 24661* Relation between  sum_ and  prod_. (Contributed by FL, 5-Sep-2010.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  ( A. k  e.  ( M ... N ) A  e.  CC  ->  sum_ k  e.  ( M ... N ) A  =  prod_ k  e.  ( M ... N )  +  A )
 
Theoremfprod2 24662* The finite composite of  A ( k ) from  k  =  M to  ( M  +  1 ) (i.e. a composite with two terms) is  ( B1 G B 2
) i.e.  ( A ( M ) G A ( M  +  1 ) ). (Contributed by FL, 22-Nov-2010.) (Revised by Mario Carneiro, 26-May-2014.)
 |-  (
 k  =  M  ->  A  =  B1 )   &    |-  ( k  =  ( M  +  1 )  ->  A  =  B 2 )   =>    |-  ( ( M  e.  ZZ  /\  B1  e.  V  /\  B 2  e.  W )  ->  prod_ k  e.  ( M ... ( M  +  1 ) ) G A  =  ( B1 G B 2 ) )
 
16.12.15  Operation properties
 
Syntaxccm1 24663 Extend class notation with a class that adds commutativity to semi-groups, monoids and so on.
 class  Com1
 
Definitiondf-com1 24664* A device to add commutativity to magmas, semi-groups, monoids and so on. A commutative group is composed of 5 properties (internal operation, commutativity, associativity, existence of a neutral element and an inverse). If we switch on or off those four properties we get 32 structures. Instead of giving a name to those 32 structures, I suggest we use intersected classes and speak of  (
Magma  i^i  Com1 ) or  ( Magma  i^i  ExId  ). (Contributed by FL, 5-Sep-2010.)
 |-  Com1  =  { g  |  A. x  e.  dom  dom  g A. y  e.  dom  dom  g ( x g y )  =  ( y g x ) }
 
Theoremiscom 24665* The predicate "is a commutative operation". (Contributed by FL, 5-Sep-2010.)
 |-  X  =  dom  dom  G   =>    |-  ( G  e.  A  ->  ( G  e.  Com1  <->  A. x  e.  X  A. y  e.  X  ( x G y )  =  ( y G x ) ) )
 
Theoremiscomb 24666 The predicate "is a commutative operation". (Contributed by FL, 14-Sep-2010.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  Com1  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
 
16.12.16  Groups and related structures
 
Theoremridlideq 24667* If a magma has a left identity element and a right identity element, they are equal. (Contributed by FL, 25-Sep-2011.)
 |-  (
 ( U  e.  X  /\  V  e.  X ) 
 ->  ( A. x  e.  X  ( ( U G x )  =  x  /\  ( x G V )  =  x )  ->  U  =  V ) )
 
Theoremrzrlzreq 24668* If a magma has a left zero element and a right zero element, they are equal. (Contributed by FL, 25-Dec-2011.)
 |-  (
 ( U  e.  X  /\  V  e.  X ) 
 ->  ( A. x  e.  X  ( ( U G x )  =  U  /\  ( x G V )  =  V )  ->  U  =  V ) )
 
Theoremmgmlion 24669* If a magma has a left identity element, it is onto. (Contributed by FL, 25-Sep-2011.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  Magma  /\  U  e.  X  /\  A. x  e.  X  ( U G x )  =  x )  ->  G : ( X  X.  X ) -onto-> X )
 
Theoremrrisgrp 24670  RR is a group for addition. (Contributed by FL, 22-Dec-2008.)
 |-  (  +  |`  ( RR  X.  RR ) )  e.  GrpOp
 
Theoremdmrngrp 24671 A way to express the domain of a group. (Contributed by FL, 9-Jan-2011.)
 |-  ( G  e.  GrpOp  ->  dom  G  =  ( ran  G  X.  ran  G ) )
 
Theorembsmgrli 24672 The base set of an operation with a right and left identity element is not empty. (Contributed by FL, 18-Feb-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   =>    |-  ( G  e.  ( Magma  i^i  ExId  )  ->  X  =/=  (/) )
 
Theoremsmgrpass2 24673 A semi-group is associative. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  SemiGrp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
 
Theoremablocomgrp 24674 An abelian group is a commutative group. (Contributed by FL, 14-Sep-2010.)
 |-  ( G  e.  AbelOp  ->  G  e.  ( GrpOp  i^i  Com1 )
 )
 
Theoremreacomsmgrp1 24675 Rearrangement of three terms in a commutative semi-group. (Contributed by FL, 14-Sep-2010.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  ( SemiGrp  i^i  Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A G ( B G C ) )  =  ( B G ( A G C ) ) )
 
Theoremreacomsmgrp2 24676 Rearrangement of three terms in a commutative semi-group. (Contributed by FL, 14-Sep-2010.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  ( SemiGrp  i^i  Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A G ( B G C ) )  =  ( C G ( B G A ) ) )
 
Theoremreacomsmgrp3 24677 Rearrangement of three terms in a commutative semi-group. (Contributed by FL, 14-Sep-2010.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  ( SemiGrp  i^i  Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( ( A G C ) G B ) )
 
Theoremreacomsmgrp4 24678 Rearrangement of terms in a commutative semi-group. (Contributed by FL, 18-Sep-2010.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  ( SemiGrp  i^i  Com1 )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A G ( B G C ) )  =  ( ( C G B ) G A ) )
 
Theoremclfsebs 24679* Closure of a finite composite of elements of the base set of an internal operation. (Contributed by FL, 14-Sep-2010.) (Proof shortened by Mario Carneiro, 26-May-2014.)
 |-  X  =  dom  dom  G   =>    |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e.  Magma  /\  A. k  e.  ( M ... N ) A  e.  X )  ->  prod_ k  e.  ( M ... N ) G A  e.  X )
 
Theoremclfsebsg 24680* Closure of a finite composite of elements of the base set of an internal operation. (Closed version.) (Contributed by FL, 14-Sep-2010.)
 |-  (
 ( N  e.  ( ZZ>=
 `  M )  /\  G  e.  Magma  /\  A. k  e.  ( M ... N ) A  e.  dom 
 dom  G )  ->  prod_ k  e.  ( M ... N ) G A  e.  dom  dom 
 G )
 
Theoremclfsebsr 24681* Closure of a finite composite of elements of the base set of an internal operation. (Case of a magma with a right and left identity element.) (Contributed by FL, 14-Sep-2010.)
 |-  X  =  ran  G   =>    |-  ( ( N  e.  ( ZZ>= `  M )  /\  G  e.  ( Magma  i^i  ExId  )  /\  A. k  e.  ( M ... N ) A  e.  X )  ->  prod_ k  e.  ( M ... N ) G A  e.  X )
 
Theoremfincmpzer 24682* Finite composite of identity elements. (Contributed by FL, 14-Sep-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2014.)
 |-  U  =  (GId `  G )   =>    |-  (
 ( N  e.  ( ZZ>=
 `  M )  /\  G  e.  ( Magma  i^i  ExId  ) )  ->  prod_ k  e.  ( M ... N ) G U  =  U )
 
Theoremresgcom 24683 Rearrangement of four terms in a commutative, associative magma. (Contributed by FL, 14-Sep-2010.)
 |-  X  =  dom  dom  G   =>    |-  ( ( G  e.  ( SemiGrp  i^i  Com1 )  /\  ( ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  D  e.  X ) ) )  ->  ( ( A G B ) G ( C G D ) )  =  ( ( A G C ) G ( B G D ) ) )
 
Theoremfprodadd 24684* The composite of two finite composites. (Contributed by FL, 14-Sep-2010.) (Proof shortened by Mario Carneiro, 30-May-2014.)
 |-  X  =  dom  dom  G   =>    |-  ( ( N  e.  ( ZZ>= `  M )  /\  A. k  e.  ( M ... N ) ( A  e.  X  /\  B  e.  X )  /\  G  e.  ( SemiGrp  i^i  Com1 ) )  ->  prod_ k  e.  ( M ... N ) G ( A G B )  =  ( prod_ k  e.  ( M
 ... N ) G A G prod_ k  e.  ( M ... N ) G B ) )
 
Theoremabloinvop 24685 The inverse of the abelian group operation doesn't reverse the arguments. cf grpoinvop 20833. (Contributed by FL, 14-Sep-2010.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  (
 ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  =  ( ( N `
  A ) G ( N `  B ) ) )
 
Theoremisppm 24686 The sequence of partial composites of elements of a magma is a sequence of elements of this magma. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 30-May-2014.)
 |-  X  =  dom  dom  G   &    |-  Z  =  (
 ZZ>= `  M )   =>    |-  ( ( G  e.  Magma  /\  M  e.  ZZ  /\  F : Z --> X )  ->  seq  M ( G ,  F ) : Z --> X )
 
Theoremseqzp2 24687 Value of the arbitrary-based recursive sequence builder at a successor value when the operation  G is associative. Compare with seqp1 10992. (Contributed by FL, 24-Jan-2010.)
 |-  X  =  dom  dom  G   &    |-  Z  =  (
 ZZ>= `  M )   &    |-  G  e. 
 SemiGrp   &    |-  F : Z --> X   =>    |-  ( N  e.  Z  ->  (  seq  M ( G ,  F ) `
  ( N  +  1 ) )  =  ( ( F `  M ) G ( 
 seq  ( M  +  1 ) ( G ,  F ) `  ( N  +  1
 ) ) ) )
 
Theoremmndoisass 24688 A monoid is associative. (Contributed by FL, 2-Nov-2009.)
 |-  ( G  e. MndOp  ->  G  e.  Ass )
 
Theoremmndoid 24689* A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   =>    |-  ( G  e. MndOp  ->  E. x  e.  X  A. y  e.  X  (
 ( x G y )  =  y  /\  ( y G x )  =  y ) )
 
Theoremmndoio 24690 A monoid is an internal operation. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   =>    |-  ( G  e. MndOp  ->  G : ( X  X.  X ) --> X )
 
Theoremmndoass 24691* A monoid is associative. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   =>    |-  ( G  e. MndOp  ->  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )
 
Theoremmndoass2 24692 A monoid is associative. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  ran  G   =>    |-  ( ( G  e. MndOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
 
Syntaxclsg 24693 Extend class notation with monoid exponentiation.
 class  ^ md
 
Definitiondf-expsg 24694* Define the exponentiation of an element of a monoid. Experimental. I define exponentiation on a monoid (and not on a semi-group or a magma ) because I need an identity element for the basis hypothesis and associativity for interesting properties such as the composite of two exponentiated elements.  ZZ is used in df-gx 20787 here I used  NN0 because the inverse is not defined in a monoid. (Contributed by FL, 2-Sep-2013.)
 |-  ^ md  =  ( g  e. MndOp  |->  ( x  e.  ran  g ,  y  e.  NN0  |->  if (
 y  =  0 ,  (GId `  g ) ,  (  seq  1 ( g ,  ( NN 
 x.  { x } )
 ) `  y )
 ) ) )
 
Theoremexpmiz 24695 Value of a member of a monoid (or any other structure where GId is defined ) raised to the 0th power. (Contributed by FL, 12-Dec-2009.)
 |-  F  =  ( rec ( ( a  e.  _V  |->  ( a G A ) ) ,  (GId `  G ) )  |`  om )   =>    |-  ( F `  (/) )  =  (GId `  G )
 
Theoremexpm 24696* Exponentiation of a monoid. The value at a successor. What I am calculating is  ( A ( ^ md `  G ) N ). (Contributed by FL, 12-Dec-2009.)
 |-  F  =  ( rec ( ( a  e.  _V  |->  ( a G A ) ) ,  (GId `  G ) )  |`  om )   =>    |-  ( N  e.  om  ->  ( F `  suc  N )  =  ( ( F `  N ) G A ) )
 
Theoremexpus 24697* The exponentiation of a member of a monoid belongs to the underlying set. (Contributed by FL, 12-Dec-2009.)
 |-  F  =  ( rec ( ( a  e.  _V  |->  ( a G A ) ) ,  (GId `  G ) )  |`  om )   &    |-  G  e. MndOp   &    |-  A  e.  X   &    |-  X  =  ran  G   =>    |-  ( x  e.  om  ->  ( F `  x )  e.  X )
 
Theoremmgmrddd 24698 The range of the domain of a magma equals the domain of the domain. (Contributed by FL, 17-May-2010.)
 |-  ( G  e.  Magma  ->  ran  dom  G  =  dom  dom  G )
 
Theoremunsgrp 24699 The underlying set of a group is a set. (Contributed by FL, 17-May-2010.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  X  e.  _V )
 
Theoremsymgfo 24700 The operation of a symetry group is onto. (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  X  =  ( Base `  G )   &    |-  P  =  ( +g  `  G )   =>    |-  ( A  e.  V  ->  P : ( X  X.  X ) -onto-> X )
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