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Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiccleub2 24301 An element of a closed interval is more than or equal to its lower bound. (Contributed by FL, 29-May-2014.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  A  <_  C )
 
Theoremiccleub3 24302 An element of a closed interval is less than or equal to its upper bound. (Contributed by FL, 29-May-2014.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  C  e.  RR* )
 
Theoremxrre3 24303 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  <  +oo )
 )  ->  A  e.  RR )
 
Theoreminabs2 24304 Absorption law for intersection. (Contributed by FL, 30-May-2014.)
 |-  ( B  i^i  ( A  u.  B ) )  =  B
 
Theoreminttpemp 24305 Two ways of saying that two triples have no common element. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
 |-  (
 ( A  e.  G  /\  B  e.  H  /\  C  e.  I )  ->  ( ( { A ,  B ,  C }  i^i  { D ,  E ,  F } )  =  (/) 
 <->  ( ( A  =/=  D 
 /\  A  =/=  E  /\  A  =/=  F ) 
 /\  ( B  =/=  D 
 /\  B  =/=  E  /\  B  =/=  F ) 
 /\  ( C  =/=  D 
 /\  C  =/=  E  /\  C  =/=  F ) ) ) )
 
Theoremmapex2 24306* Two ways to express a subset of mappings. (Contributed by FL, 17-Nov-2014.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  { f  |  ( f : A --> B  /\  ph ) }  =  {
 f  e.  ( B 
 ^m  A )  | 
 ph } )
 
Theoremsssu 24307 Equality of a class difference and it subtrahend. (For my private use only. Don't use.) (Contributed by FL, 16-Sep-2016.)
 |-  (
 ( B  \  A )  =  A  <->  ( A  =  (/)  /\  B  =  (/) ) )
 
16.11.6  The "maps to" notation
 
Theoremcmpfun 24308 Functionality of a class given by a "maps to" notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |- 
 Fun  F
 
Theoremcmpdom 24309* Domain of a class given by the "maps to" notation. (Contributed by FL, 17-Feb-2008.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( A. x  e.  A  B  e.  _V  <->  dom  F  =  A )
 
Theoremcmpdom2 24310* Domain of a class given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)
 |-  F  =  ( x  e.  A  |->  ( B G C ) )   =>    |- 
 dom  F  =  A
 
Theoremfopab2g 24311* Functionality of an ordered-pair class abstraction given by the "maps to" notation. (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 31-May-2014.)
 |-  ( F  =  ( x  e.  A  |->  C )  ->  ( A. x  e.  A  C  e.  B  <->  F : A --> B ) )
 
Theoremcrimmt1 24312* Composition of a restricted identity and a mapping (using the maps to notation). See fcoi1 5272. (Contributed by FL, 25-Apr-2012.)
 |-  ( F : A --> B  ->  ( F  o.  ( x  e.  A  |->  x ) )  =  F )
 
Theoremcrimmt2 24313* Composition of a restricted identity and a mapping (using the maps to notation). See fcoi2 5273. (Contributed by FL, 25-Apr-2012.)
 |-  ( F : A --> B  ->  ( ( x  e.  B  |->  x )  o.  F )  =  F )
 
Theoremmapmapmap 24314* Function returning a composite. (Contributed by FL, 19-Nov-2011.)
 |-  F1  =  ( f  e.  ( B  ^m  A )  |->  ( ( E  o.  f
 )  o.  G ) )   =>    |-  ( ( E : B
 --> B1  /\  G : A1 --> A  /\  ( ( A  e.  _V  /\  B  e.  _V )  /\  ( A1  e.  _V  /\  B1  e.  _V ) ) )  ->  F1 : ( B  ^m  A ) --> ( B1  ^m  A1 ) )
 
Theoreminjsurinj 24315* If  E is an injection and  G a surjection  ( f  |->  ( ( E  o.  f )  o.  G
) ) is an injection. Bourbaki E.II.31 prop. 2. (Contributed by FL, 20-Nov-2011.)
 |-  F1  =  ( f  e.  ( B  ^m  A )  |->  ( ( E  o.  f
 )  o.  G ) )   =>    |-  ( ( E : B -1-1-> B1  /\  G : A1
 -onto-> A  /\  ( ( A  e.  _V  /\  B  e.  _V )  /\  ( A1  e.  _V  /\  B1  e.  _V ) ) )  ->  F1 : ( B  ^m  A )
 -1-1-> ( B1  ^m  A1 )
 )
 
16.11.7  Cartesian Products

In what follows I will call nuple an element of a cartesian product.

If  X is a cartesian product,  N a nuple of  X,  I an indice,  ( ( X  pr  I ) `  N ) is the  I th coordinate of the nuple  N.

Suppose the set of indices is 
{ 1 ,  2 } and  X is the cartesian product  { { <. 1 ,  a >. , 
<. 2 ,  u >. } ,  { <. 1 ,  a >. , 
<. 2 ,  v
>. } } then  ( ( X  pr  1 ) `
 { <. 1 ,  a >. ,  <. 2 ,  u >. } )  =  a and  ( ( X  pr  2 ) `
 { <. 1 ,  a >. ,  <. 2 ,  u >. } )  =  u.

 
Syntaxcpro 24316 Extend class notation to include the projection mapping.
 class  pr
 
Syntaxcproj 24317 Extend class notation to include the projection mapping.
 class  prj
 
Definitiondf-pro 24318* Definition of a projection (also called a coordinate mapping). Meaninful if  x is a cartesian product and  y is an index. (Contributed by FL, 19-Jun-2011.)
 |-  pr  =  ( x  e.  _V ,  y  e.  _V  |->  ( f  e.  x  |->  ( f `  y
 ) ) )
 
Definitiondf-prj 24319* Definition of a projection. Meaninful if  x is a cartesian product and  y is a set of indices. Suppose  X  =  { { <. 1 ,  a
>. ,  <. 2 ,  u >. } ,  { <. 1 ,  a
>. ,  <. 2 ,  v >. } } then  ( X  prj  1 )  =  {
a } and  ( X  prj  2 )  =  {
u ,  v }. (Contributed by FL, 19-Jun-2011.)
 |-  prj  =  ( x  e.  _V ,  y  e.  _V  |->  ( f  e.  x  |->  ( f  |`  y ) ) )
 
Theoremelixp2b 24320* The base class of the elements of a nuple. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2016.)
 |-  ( F  e.  X_ x  e.  A  B  ->  A. x  e.  A  ( F `  x )  e.  B )
 
Theorembclelnu 24321* The base class of an element of a nuple. (Contributed by FL, 19-Jun-2011.)
 |-  ( x  =  I  ->  B  =  C )   =>    |-  ( ( F  e.  X_ x  e.  A  B  /\  I  e.  A )  ->  ( F `  I )  e.  C )
 
Theoremispr1 24322* Definition of the coordinate mapping (or projection ) of index  I.  P is a cartesian product. (Contributed by FL, 19-Jun-2011.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  (
 ( P  e.  V  /\  I  e.  W )  ->  ( P  pr  I )  =  (
 f  e.  P  |->  ( f `  I ) ) )
 
Theoremprmapcp2 24323* A projection is a mapping from a cartesian product to an element of the family implied in the product. Bourbaki E.II.34 cor. 1. (Contributed by FL, 19-Jun-2011.)
 |-  P  =  X_ x  e.  A  B   &    |-  ( x  =  I 
 ->  B  =  C )   =>    |-  ( ( P  e.  V  /\  I  e.  A )  ->  ( P  pr  I ) : P --> C )
 
Theoremvalpr 24324 The  I th coordinate of the nuple  F. (Contributed by FL, 19-Jun-2011.)
 |-  (
 ( P  e.  V  /\  I  e.  W  /\  F  e.  P ) 
 ->  ( ( P  pr  I ) `  F )  =  ( F `  I ) )
 
Theoremnpincppr 24325* A set of nuples is included in the cartesian product of the projections of the nuples. Bourbaki E.II.32. (Contributed by FL, 20-Jun-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)
 |-  P  =  X_ x  e.  A  B   =>    |-  ( ( F  C_  P  /\  P  e.  Q )  ->  F  C_  X_ x  e.  A  ( ( P  pr  x ) " F ) )
 
Theoremrepfuntw 24326 Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.)
 |-  I  e.  A   &    |-  J  e.  B   =>    |-  ( I  =/=  J  ->  ( F  Fn  { I ,  J }  <->  F  =  { <. I ,  ( F `
  I ) >. , 
 <. J ,  ( F `
  J ) >. } ) )
 
Theoremrepcpwti 24327* A representation of a cartesian product with two indices. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
 |-  A  =  { I ,  J }   &    |-  B  =  if ( x  =  I ,  M ,  N )   &    |-  I  e.  C   &    |-  J  e.  D   =>    |-  ( I  =/=  J  ->  X_ x  e.  A  B  =  {
 f  |  E. a  e.  M  E. b  e.  N  f  =  { <. I ,  a >. , 
 <. J ,  b >. } } )
 
Theoremcbcpcp 24328* The canonical bijection between a cross product and a cartesian product (whose set of indices is composed of two different elements). Bourbaki E.II.33 . (Contributed by FL, 26-Jun-2011.)
 |-  A  =  { I ,  J }   &    |-  B  =  if ( x  =  I ,  M ,  N )   &    |-  F  =  ( a  e.  M ,  b  e.  N  |->  {
 <. I ,  a >. , 
 <. J ,  b >. } )   &    |-  I  e.  C   &    |-  J  e.  D   =>    |-  ( I  =/=  J  ->  F : ( M  X.  N ) -1-1-onto-> X_ x  e.  A  B )
 
Theoremisprj1 24329* Definition of a projection.  I is a set of indices.  P is a cartesian product. (Contributed by FL, 19-Jun-2011.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  (
 ( P  e.  V  /\  I  e.  W )  ->  ( P  prj  I )  =  ( f  e.  P  |->  ( f  |`  I ) ) )
 
Theoremisprj2 24330* Definition of a projection.  I is a set of indices.  P is a cartesian product. (Contributed by FL, 19-Jun-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)
 |-  P  =  X_ x  e.  A  B   =>    |-  ( ( I  e.  V  /\  A. x  e.  A  B  e.  D )  ->  ( P  prj  I )  =  ( f  e.  P  |->  ( f  |`  I ) ) )
 
Theoremprjmapcp 24331* A projection is a mapping from a cartesian product to one of its restriction. (Contributed by FL, 19-Jun-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)
 |-  (
 ( I  C_  A  /\  A  e.  C  /\  A. x  e.  A  B  e.  D )  ->  ( X_ x  e.  A  B  prj  I ) : X_ x  e.  A  B --> X_ x  e.  I  B )
 
Theoremcbicp 24332* Canonical bijection between a cartesian product indexed by a singleton and the base class of the elements of the 1-uple. Bourbaki E II.32 (Contributed by FL, 6-Jun-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)
 |-  ( x  =  A  ->  B  =  C )   =>    |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( X_ x  e.  { A } B  pr  A ) : X_ x  e.  { A } B -1-1-onto-> C )
 
Theoremprl 24333* Existence of a "prolongement" of a cartesian product. Bourbaki E.II.34 prop. 6. (Contributed by FL, 7-Nov-2011.)
 |-  A  e.  D   =>    |-  ( ( A. x  e.  A  C  =/=  (/)  /\  B  C_  A  /\  G  e.  X_ x  e.  B  C )  ->  E. f  e.  X_  x  e.  A  C G  C_  f )
 
Theoremprl1 24334* Existence of a "prolongement" of a cartesian product. Bourbaki E.II.34 prop. 6. (Contributed by FL, 20-Nov-2011.)
 |-  A  e.  D   =>    |-  ( ( A. x  e.  A  C  =/=  (/)  /\  B  C_  A )  ->  A. g  e.  X_  x  e.  B  C E. f  e.  X_  x  e.  A  C g  C_  f )
 
Theoremprl2 24335* Existence of a "prolongement" of a cartesian product. Bourbaki E.II.34 prop. 6. (Contributed by FL, 20-Nov-2011.)
 |-  A  e.  D   =>    |-  ( ( A. x  e.  A  C  =/=  (/)  /\  B  C_  A )  ->  A. g  e.  X_  x  e.  B  C E. f  e.  X_  x  e.  A  C g  =  ( f  |`  B ) )
 
Theoremprjmapcp2 24336* A projection is a mapping from a cartesian product onto one of its restriction. Bourbaki E.II.33 prop. 5. (Contributed by FL, 20-Nov-2011.) (Revised by Mario Carneiro, 31-May-2014.)
 |-  (
 ( I  C_  A  /\  A  e.  C  /\  ( A. x  e.  A  B  e.  D  /\  A. x  e.  A  B  =/= 
 (/) ) )  ->  ( X_ x  e.  A  B  prj  I ) :
 X_ x  e.  A  B -onto-> X_ x  e.  I  B )
 
Theoremdstr 24337* Distribution of union over intersection. Bourbaki E.II.35 prop. 8. (Contributed by FL, 18-Jun-2011.)
 |-  (
 y  =  ( f `
  x )  ->  C  =  D )   &    |-  X  =  X_ x  e.  A  B   &    |-  A  e.  E   =>    |-  U_ x  e.  A  |^|_
 y  e.  B  C  =  |^|_ f  e.  X  U_ x  e.  A  D
 
16.11.8  Operations on subsets and functions
 
Syntaxccst 24338 Extend class notation with an operator that derives an operation on subsets of a set from an operation on the elements of this set.
 class  cset
 
Definitiondf-cst 24339* Define an operation on the subsets derived from an operation  g on the elements. Meaningful if 
g is a binary internal operation. (Contributed by FL, 18-Apr-2010.)
 |-  cset  =  ( g  e.  _V  |->  ( x  e.  ~P dom  dom  g ,  y  e.  ~P dom  dom  g  |-> 
 ran  (  u  e.  x ,  v  e.  y  |->  ( u g v ) ) ) )
 
Theoremiscst1 24340* An operation on the subsets derived from an operation on the elements. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  X  =  dom  dom  G   &    |-  H  =  (
 cset `  G )   =>    |-  ( G  e.  A  ->  H  =  ( x  e.  ~P X ,  y  e.  ~P X  |->  ran  (  u  e.  x ,  v  e.  y  |->  ( u G v ) ) ) )
 
Theoremiscst2 24341* The value of the couple  <. A ,  B >. through the derived operation. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  X  =  dom  dom  G   &    |-  H  =  (
 cset `  G )   =>    |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  ( A H B )  =  ran  (  u  e.  A ,  v  e.  B  |->  ( u G v ) ) )
 
Theoremiscst3 24342* Property equivalent to the fact of belonging to the value of a pair through the derived operation. (Contributed by FL, 18-Apr-2010.)
 |-  X  =  dom  dom  G   &    |-  H  =  (
 cset `  G )   =>    |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  ( C  e.  ( A H B )  <->  E. u  e.  A  E. v  e.  B  C  =  ( u G v ) ) )
 
Theoremiscst4 24343* The value of the couple  <. A ,  B >. through the derived operation  H (expressed with a union). (Contributed by FL, 31-Dec-2010.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  X  =  dom  dom  G   &    |-  H  =  (
 cset `  G )   =>    |-  ( ( G  e.  M  /\  A  e.  ~P X  /\  B  e.  ~P X )  ->  ( A H B )  =  U_ x  e.  B  ( A H { x } ) )
 
16.11.9  Arithmetic
 
Theorem3timesi 24344 Three times a number. (Contributed by FL, 17-Oct-2010.)
 |-  A  e.  CC   =>    |-  ( 3  x.  A )  =  ( A  +  ( A  +  A ) )
 
Theorem2eq3m1 24345  2 equals  3 minus  1. (Contributed by FL, 17-Oct-2010.)
 |-  2  =  ( 3  -  1
 )
 
TheoremnZdef 24346* Two ways to define  n ZZ. In the first way I multiply the set  { N } by the set  ZZ ( I think this is this sort of multiplication that is at the origin of the denotation  n ZZ). In the second way I multiply the integer  N by an element of  ZZ. (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
 |-  ( N  e.  ZZ  ->  ( { N }  ( cset `  (  x.  |`  ( ZZ 
 X.  ZZ ) ) ) ZZ )  =  { x  |  E. y  e.  ZZ  x  =  ( N  x.  y ) } )
 
16.11.10  Lattice (algebraic definition)
 
Syntaxclatalg 24347 Extend class notation to include the class of lattices.
 class  LatAlg
 
Definitiondf-latalg 24348* Algebraic definition of a lattice.  j is called the join of the lattice (and is denoted by 
\/ in textbooks) ,  m is called the meet (and is denoted by 
/\ in textbooks). (Contributed by FL, 11-Dec-2009.)
 |-  LatAlg  =  { <. j ,  m >.  | 
 E. t ( j : ( t  X.  t ) --> t  /\  m : ( t  X.  t ) --> t  /\  A. x  e.  t  A. y  e.  t  (
 ( x j y )  =  ( y j x )  /\  ( x m y )  =  ( y m x )  /\  (
 ( x m ( x j y ) )  =  x  /\  ( x j ( x m y ) )  =  x  /\  A. z  e.  t  (
 ( x m ( y m z ) )  =  ( ( x m y ) m z )  /\  ( x j ( y j z ) )  =  ( ( x j y ) j z ) ) ) ) ) }
 
Theoremislatalg 24349* The predicate "is a lattice". (Contributed by FL, 11-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B )  ->  ( <. J ,  M >.  e.  LatAlg  <->  ( J :
 ( X  X.  X )
 --> X  /\  M :
 ( X  X.  X )
 --> X  /\  A. x  e.  X  A. y  e.  X  ( ( x J y )  =  ( y J x )  /\  ( x M y )  =  ( y M x )  /\  ( ( x M ( x J y ) )  =  x  /\  ( x J ( x M y ) )  =  x  /\  A. z  e.  X  ( ( x M ( y M z ) )  =  ( ( x M y ) M z )  /\  ( x J ( y J z ) )  =  ( ( x J y ) J z ) ) ) ) ) ) )
 
Theoremjop 24350 Join is a binary internal operation. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  J : ( X  X.  X ) --> X )
 
Theoremmop 24351 Meet is a binary internal operation. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  M : ( X  X.  X ) --> X )
 
Theoremcljo 24352 Closure of join. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  (
 ( P  e.  X  /\  Q  e.  X ) 
 ->  ( P J Q )  e.  X )
 )
 
Theoremclme 24353 Closure of meet. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  (
 ( P  e.  X  /\  Q  e.  X ) 
 ->  ( P M Q )  e.  X )
 )
 
Theoremlabs1 24354* Absorption law.  ( x  /\  ( x  \/  y
) )  =  x. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  A. x  e.  X  A. y  e.  X  ( x M ( x J y ) )  =  x )
 
Theoremlabss1 24355 Absorption law.  ( P  /\  ( P  \/  Q
) )  =  P. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  (
 ( P  e.  X  /\  Q  e.  X ) 
 ->  ( P M ( P J Q ) )  =  P ) )
 
Theoremlabs2 24356* Absorption law.  ( x  \/  ( x  /\  y
) )  =  x. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  A. x  e.  X  A. y  e.  X  ( x J ( x M y ) )  =  x )
 
Theoremlabss2 24357 Absorption law.  ( P  \/  ( P  /\  Q ) )  =  P. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  (
 ( P  e.  X  /\  Q  e.  X ) 
 ->  ( P J ( P M Q ) )  =  P ) )
 
Theoremjidd 24358 Join is idempotent.  ( P  \/  P )  =  P. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  ( P  e.  X  ->  ( P J P )  =  P ) )
 
Theoremmidd 24359 Meet is idempotent.  ( P  /\  P )  =  P. (Contributed by FL, 12-Dec-2009.)
 |-  X  =  dom  dom  J   =>    |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  ( P  e.  X  ->  ( P M P )  =  P ) )
 
16.11.11  Currying and Partial Mappings
 
Syntaxccur1 24360 Extend class notation with the definition of currying.
 class  cur1
 
Syntaxccur2 24361 Extend class notation with the definition of currying.
 class  cur2
 
Definitiondf-cur1 24362* Definition of currying (1st sort). Currying the operation  f consists in creating a mapping that returns for every value  x of  dom  dom  f the partial application of  f to  x. (Contributed by FL, 24-Jan-2010.)
 |-  cur1  =  { <. f ,  g >.  |  ( ( Fun  f  /\  Rel  dom  f )  /\  g  =  ( x  e.  dom  dom  f  |->  ( f  o.  `' ( 2nd  |`  ( { x }  X.  _V )
 ) ) ) ) }
 
Definitiondf-cur2 24363* Definition of currying (2nd sort). Currying the operation  f consists in creating a mapping that returns for every value  x of  ran  dom  f the partial application of  f to  x. (Contributed by FL, 24-Jan-2010.)
 |-  cur2  =  { <. f ,  g >.  |  ( Fun  f  /\  Rel  dom  f  /\  g  =  ( x  e.  ran  dom  f  |->  ( f  o.  `' ( 1st  |`  ( _V  X.  { x } ) ) ) ) ) }
 
Theoremcur1val 24364* The value of a curried operation. (Contributed by FL, 24-Jan-2010.)
 |-  (
 ( F  e.  A  /\  Fun  F  /\  Rel  dom 
 F )  ->  ( cur1 `  F )  =  ( x  e.  dom  dom 
 F  |->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V )
 ) ) ) )
 
Theoremcur1vald 24365* The value of a curried operation. (Contributed by FL, 17-May-2010.)
 |-  (
 ( ( F  Fn  ( A  X.  B ) 
 /\  B  =/=  (/) )  /\  ( A  e.  C  /\  B  e.  D ) )  ->  ( cur1 `  F )  =  ( x  e.  A  |->  ( F  o.  `' ( 2nd  |`  ( { x }  X.  _V ) ) ) ) )
 
Theoremdomrancur1b 24366* The currying of a mapping  F whose domain is  ( A  X.  B
) is a mapping whose domain is  A and the range, the class of all the functions from  B to  ran  F. (Contributed by FL, 28-Apr-2010.)
 |-  A  e.  C   &    |-  B  e.  D   &    |-  B  =/= 
 (/)   &    |-  F  Fn  ( A  X.  B )   =>    |-  ( cur1 `  F ) : A --> { f  |  f : B --> ran  F }
 
Theoremdomrancur1clem 24367 Lemma for domrancur1c 24368. (Contributed by FL, 17-May-2010.)
 |-  (
 ( F  Fn  ( A  X.  B )  /\  ( A  e.  C  /\  B  e.  D ) )  ->  ( F  o.  `' ( 2nd  |`  M ) )  e.  _V )
 
Theoremdomrancur1c 24368* The currying of a mapping  F whose domain is  ( A  X.  B
) is a mapping whose domain is  A and the range, the class of all the functions from  B to  ran  F. (Contributed by FL, 17-May-2010.)
 |-  (
 ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B ) ) )  ->  ( cur1 `  F ) : A --> { f  |  f : B --> ran  F }
 )
 
Theoremvalcurfn 24369 The value of a curried function at 
O  e.  A is a mapping. (Contributed by FL, 17-May-2010.)
 |-  (
 ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B ) ) 
 /\  O  e.  A )  ->  ( ( cur1 `  F ) `  O ) : B --> ran  F )
 
Theoremvalcurfn1 24370 The value of a curried function at 
O  e.  A is a partial application. (Contributed by FL, 17-May-2010.)
 |-  G  =  ( F  o.  `' ( 2nd  |`  ( { O }  X.  _V ) ) )   =>    |-  ( ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B ) )  /\  O  e.  A )  ->  ( (
 cur1 `  F ) `  O )  =  G )
 
Theoremvalcurfn2 24371* The value of a curried function at 
O  e.  A is a partial application. (Contributed by FL, 17-May-2010.)
 |-  (
 ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B ) ) 
 /\  O  e.  A )  ->  ( ( cur1 `  F ) `  O )  =  ( x  e.  B  |->  ( O F x ) ) )
 
Theoremvalvalcurfn 24372 The value at  P  e.  B of the value of a curried function at  O  e.  A equals  ( O F P ). (Contributed by FL, 17-May-2010.)
 |-  (
 ( ( A  e.  C  /\  B  e.  D )  /\  ( B  =/=  (/)  /\  F  Fn  ( A  X.  B ) ) 
 /\  ( O  e.  A  /\  P  e.  B ) )  ->  ( ( ( cur1 `  F ) `  O ) `  P )  =  ( O F P ) )
 
16.11.12  Order theory (Extensible Structure Builder)
 
Syntaxcorhom 24373 Extend class notation with the class of all decreasing functions.
 class  OrHom
 
Syntaxcoriso 24374 Extend class notation with the class of all the order isomorphisms.
 class  OrIso
 
Definitiondf-orhom 24375* Increasing functions also called "order homomorphisms", "isotone, monotone or order preserving mappings". To have the class of decreasing functions use  ( r  OrHom  `' s ). Experimental. Bourbaki E.III.7 (Contributed by FL, 17-Nov-2014.)
 |-  OrHom  =  ( r  e.  _V ,  s  e.  _V  |->  { f  e.  ( ( Base `  s
 )  ^m  ( Base `  r ) )  | 
 A. a  e.  ( Base `  r ) A. b  e.  ( Base `  r ) ( a ( le `  r
 ) b  ->  (
 f `  a )
 ( le `  s
 ) ( f `  b ) ) }
 )
 
Definitiondf-oriso 24376* Order isomorphisms. Experimental. Bourbaki E.III.5 (Contributed by FL, 17-Nov-2014.)
 |-  OrIso  =  ( r  e.  _V ,  s  e.  _V  |->  { f  |  ( f : (
 Base `  r ) -1-1-onto-> ( Base `  s )  /\  A. a  e.  ( Base `  r ) A. b  e.  ( Base `  r )
 ( a ( le `  r ) b  <->  ( f `  a ) ( le `  s ) ( f `
  b ) ) ) } )
 
Theoremisorhom 24377* The predicate is an order homomorphism. (Contributed by FL, 17-Nov-2014.)
 |-  X  =  ( Base `  A )   &    |-  Y  =  ( Base `  B )   &    |- &lea  =  ( le `  A )   &    |- &leb  =  ( le `  B )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  OrHom  B )  =  { f  e.  ( Y  ^m  X )  |  A. a  e.  X  A. b  e.  X  ( a&lea  b 
 ->  ( f `  a
 )&leb  ( f `  b ) ) }
 )
 
Theoremisoriso 24378* Order isomorphisms. (Contributed by FL, 17-Nov-2014.)
 |-  X  =  ( Base `  A )   &    |-  Y  =  ( Base `  B )   &    |- &lea  =  ( le `  A )   &    |- &leb  =  ( le `  B )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  OrIso  B )  =  { f  |  ( f : X -1-1-onto-> Y  /\  A. a  e.  X  A. b  e.  X  ( a&lea  b  <->  ( f `  a )&leb  ( f `  b ) ) ) } )
 
Theoremisoriso2 24379* Order isomorphisms. (Contributed by FL, 17-Nov-2014.)
 |-  X  =  ( Base `  A )   &    |-  Y  =  ( Base `  B )   &    |- &lea  =  ( le `  A )   &    |- &leb  =  ( le `  B )   =>    |-  ( ( A  e.  C  /\  B  e.  D  /\  F  e.  E ) 
 ->  ( F  e.  ( A  OrIso  B )  <->  ( F : X
 -1-1-onto-> Y  /\  A. a  e.  X  A. b  e.  X  ( a&lea  b  <-> 
 ( F `  a
 )&leb  ( F `  b ) ) ) ) )
 
Theoremoriso 24380 If  F is an order isomorphism so is  `' F. (Contributed by FL, 11-Nov-2014.)
 |-  X  =  ( Base `  A )   &    |-  Y  =  ( Base `  B )   &    |- &lea  =  ( le `  A )   &    |- &leb  =  ( le `  B )   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( F  e.  ( A  OrIso  B ) 
 ->  `' F  e.  ( B  OrIso  A ) ) )
 
16.11.13  Order theory
 
Syntaxcpresetrel 24381 Extend class notation with the class of all the presets.
 class PresetRel
 
Syntaxcmxl 24382 Extend class notation with a function that returns the maximal elements of a preset.
 class  mxl
 
Syntaxcmnl 24383 Extend class notation with a function that returns the minimal elements of a preset.
 class  mnl
 
Syntaxcub 24384 Extend class notation with a function that returns the upper bounds of a part of a preset.
 class  ub
 
Syntaxclb 24385 Extend class notation with a function that returns the lower bounds of a part of a preset.
 class  lb
 
Syntaxcge 24386 Extend class notation with a function that returns the greatest element of a poset.
 class  ge
 
Syntaxcse 24387 Extend class notation with a function that returns the smallest element of a poset.
 class leR
 
Syntaxcantidir 24388 Extend class notation with the class of all the anti-directions.
 class  AntiDir
 
Definitiondf-prs 24389 Define the class of all presets. A preset is a transitive and reflexive relation. ("preset" is the short for preordered set.) (Contributed by FL, 1-May-2011.)
 |- PresetRel  =  {
 r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  (  _I  |`  U. U. r )  C_  r ) }
 
Theoremisprsr 24390 The predicate "is a preset". (Contributed by FL, 1-May-2011.)
 |-  ( R  e.  A  ->  ( R  e. PresetRel  <->  ( Rel  R  /\  ( R  o.  R )  C_  R  /\  (  _I  |`  U. U. R )  C_  R ) ) )
 
Theorempreorel 24391 A preset is a relation. (Contributed by FL, 18-May-2011.)
 |-  ( R  e. PresetRel  ->  Rel  R )
 
Theorempreodom2 24392 The domain of a preset equals its field. (Contributed by FL, 22-May-2011.)
 |-  ( R  e. PresetRel  ->  dom  R  =  U.
 U. R )
 
Theoremppldrels 24393 The field of a preset is a set. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  X  e.  _V )
 
Theorempreoref12 24394 A preset is reflexive. (Contributed by FL, 18-May-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  (  _I  |`  X )  C_  R )
 
Theorempreoref22 24395 A preset is reflexive. (Contributed by FL, 22-May-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e. PresetRel  /\  A  e.  X )  ->  A R A )
 
Theorempreoran2 24396 The range of a preset equals its field. (Contributed by FL, 22-May-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  ran  R  =  X )
 
Theorempre1befi2 24397 If  A  <_  B then 
A belongs to the field of the preset. (Contributed by FL, 23-May-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e. PresetRel  /\  A R B )  ->  A  e.  X )
 
Theorempre2befi2 24398 If  A  <_  B then 
B belongs to the field of the preset. (Contributed by FL, 23-May-2011.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  X  =  dom  R   =>    |-  ( ( R  e. PresetRel  /\  A R B )  ->  B  e.  X )
 
Theoremdomcnvpre 24399 If  R is a preset, its domain and the domain of its converse are equal. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  X  =  dom  `'  R )
 
Theorempreotr1 24400 A preset is transitive. (Contributed by FL, 22-May-2011.)
 |-  ( R  e. PresetRel  ->  ( R  o.  R )  C_  R )
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