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Theorem List for Metamath Proof Explorer - 25201-25300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdomrancur1clem 25201 Lemma for domrancur1c 25202. (Contributed by FL, 17-May-2010.)
 |-  (
 ( F  Fn  ( A  X.  B )  /\  ( A  e.  C  /\  B  e.  D ) )  ->  ( F  o.  `' ( 2nd  |`  M ) )  e.  _V )
 
Theoremdomrancur1c 25202* 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 25203 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 25204 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 25205* 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 25206 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 ) )
 
18.13.12  Order theory (Extensible Structure Builder)
 
Syntaxcorhom 25207 Extend class notation with the class of all decreasing functions.
 class  OrHom
 
Syntaxcoriso 25208 Extend class notation with the class of all the order isomorphisms.
 class  OrIso
 
Definitiondf-orhom 25209* 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 25210* 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 25211* 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 25212* 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 25213* 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 25214 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 ) ) )
 
18.13.13  Order theory
 
Syntaxcpresetrel 25215 Extend class notation with the class of all the presets.
 class PresetRel
 
Syntaxcmxl 25216 Extend class notation with a function that returns the maximal elements of a preset.
 class  mxl
 
Syntaxcmnl 25217 Extend class notation with a function that returns the minimal elements of a preset.
 class  mnl
 
Syntaxcub 25218 Extend class notation with a function that returns the upper bounds of a part of a preset.
 class  ub
 
Syntaxclb 25219 Extend class notation with a function that returns the lower bounds of a part of a preset.
 class  lb
 
Syntaxcge 25220 Extend class notation with a function that returns the greatest element of a poset.
 class  ge
 
Syntaxcse 25221 Extend class notation with a function that returns the smallest element of a poset.
 class leR
 
Syntaxcantidir 25222 Extend class notation with the class of all the anti-directions.
 class  AntiDir
 
Definitiondf-prs 25223 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 25224 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 25225 A preset is a relation. (Contributed by FL, 18-May-2011.)
 |-  ( R  e. PresetRel  ->  Rel  R )
 
Theorempreodom2 25226 The domain of a preset equals its field. (Contributed by FL, 22-May-2011.)
 |-  ( R  e. PresetRel  ->  dom  R  =  U.
 U. R )
 
Theoremppldrels 25227 The field of a preset is a set. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  X  e.  _V )
 
Theorempreoref12 25228 A preset is reflexive. (Contributed by FL, 18-May-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  (  _I  |`  X )  C_  R )
 
Theorempreoref22 25229 A preset is reflexive. (Contributed by FL, 22-May-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e. PresetRel  /\  A  e.  X )  ->  A R A )
 
Theorempreoran2 25230 The range of a preset equals its field. (Contributed by FL, 22-May-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  ran  R  =  X )
 
Theorempre1befi2 25231 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 25232 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 25233 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 25234 A preset is transitive. (Contributed by FL, 22-May-2011.)
 |-  ( R  e. PresetRel  ->  ( R  o.  R )  C_  R )
 
Theorempreotr2 25235 A preset is transitive. (Contributed by FL, 23-May-2011.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  (
 ( R  e. PresetRel  /\  ( A R B  /\  B R C ) )  ->  A R C )
 
Theoremaltprs2 25236 The composite of a preset with itself. (Contributed by FL, 13-May-2011.)
 |-  ( R  e. PresetRel  ->  ( R  o.  R )  =  R )
 
Theoremint2pre 25237 The intersection of two presets is a preset. (Contributed by FL, 28-Dec-2011.)
 |-  (
 ( R  e. PresetRel  /\  S  e. PresetRel )  ->  ( R  i^i  S )  e. PresetRel )
 
Theoremsqpre 25238 A square product is a preset. (Contributed by FL, 28-Dec-2011.)
 |-  ( A  e.  V  ->  ( A  X.  A )  e. PresetRel )
 
Theoremindpre 25239 The relation induced by a preset on a part of its field is a preset. (Contributed by FL, 28-Dec-2011.)
 |-  (
 ( R  e. PresetRel  /\  A  e.  B )  ->  ( R  i^i  ( A  X.  A ) )  e. PresetRel )
 
Theoremposprsr 25240 A partial order is a preset. (Contributed by FL, 1-May-2011.)
 |-  PosetRel  C_ PresetRel
 
Theoremposispre 25241 A poset is a preset. (Contributed by FL, 19-Sep-2011.)
 |-  ( A  e.  PosetRel  ->  A  e. PresetRel )
 
Theoremempos 25242 The empty set is a poset. (Contributed by FL, 6-Oct-2008.)
 |-  (/)  e.  PosetRel
 
Theoremdupre1 25243 The converse of a preset is a preset. The case  ( `' R  e. PresetRel  ->  R  e. PresetRel ) is true only if 
R is a relation. See dupre2 25244. (Contributed by FL, 5-Jan-2009.)
 |-  ( R  e. PresetRel  ->  `' R  e. PresetRel )
 
Theoremdupre2 25244 The converse of a preset is a preset. (Contributed by FL, 19-Sep-2011.)
 |-  ( Rel  R  ->  ( R  e. PresetRel  <->  `' R  e. PresetRel ) )
 
Theoremnfwval 25245 An infimum is the supremum of the converse relation. (Contributed by FL, 6-Sep-2009.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  (
 ( R  e.  U  /\  A  e.  W ) 
 ->  ( R  inf w  A )  =  ( `' R  sup w  A ) )
 
Definitiondf-mxl 25246* Define the maximal elements of a set. I.e. the elements of the set that are not smaller than the other elements. Meaningful if  r is at least a preset. Read  ( mxl `  R
) as the maximal elements of the set  U. U. R preordered by  R. Bourbaki E III 8. Experimental. (Contributed by FL, 16-May-2011.)
 |-  mxl  =  ( r  e.  _V  |->  { a  e.  U. U. r  |  A. b  e. 
 U. U. r ( a r b  ->  a  =  b ) } )
 
Definitiondf-mnl 25247* Define the minimal elements of a set. I.e. the elements of the set that are not greater than the other elements. Meaningful is  r is at least a preset. Read  ( mnl `  R
) as the minimal elements of the set  U. U. R preordered by  R. Bourbaki E III 8. Experimental. (Contributed by FL, 19-Sep-2011.)
 |-  mnl  =  ( r  e.  _V  |->  { a  e.  U. U. r  |  A. b  e. 
 U. U. r ( b r a  ->  b  =  a ) } )
 
Definitiondf-ge 25248 Define the greatest element of a poset. I.e. the element of the poset that is larger than the other elements. Meaningful is  r is at least a poset (otherwise there could be more than one supremum due to cycles). Bourbaki E III 10. Experimental. (Contributed by FL, 19-Sep-2011.)
 |-  ge  =  ( r  e.  _V  |->  ( r  sup w  dom  r ) )
 
Definitiondf-ler 25249 Define the least element of a poset. I.e. the element of the poset that is smaller than the other elements. Meaningful is  r is at least a poset. Experimental. (Contributed by FL, 19-Sep-2011.)
 |- leR  =  ( r  e.  _V  |->  ( r  inf w  dom  r ) )
 
Theoremgepsup 25250 The greatest element of a poset is the supremum of the poset. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e.  A  ->  ( ge `  R )  =  ( R  sup w  X ) )
 
Theoremseinf 25251 The least element of a poset is the infimum of the poset. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e.  A  ->  (leR `  R )  =  ( R  inf w  X ) )
 
Theoremsege 25252 The least element of a poset is the greatest element of the converse poset. (Contributed by FL, 30-Dec-2011.)
 |-  ( R  e.  PosetRel  ->  (leR `  R )  =  ( ge `  `' R ) )
 
Definitiondf-ub 25253* Define the upper bounds of a set  x. Meaningful if  r is at least a preset, and  x a subset of the field of 
r. Bourbaki E.III.9 def. 5. Experimental. (Contributed by FL, 16-May-2011.)
 |-  ub  =  ( r  e.  _V ,  x  e.  _V  |->  { a  e.  U. U. r  |  A. b  e.  x  b r a } )
 
Definitiondf-lb 25254* Define the lower bounds of a set  x. Meaningful if  r is at least a preset, and  x a subset of the field of 
r. Experimental. (Contributed by FL, 16-May-2011.)
 |-  lb  =  ( r  e.  _V ,  x  e.  _V  |->  { a  e.  U. U. r  |  A. b  e.  x  a r b } )
 
Definitiondf-antidir 25255* An antidirected set (also called a set filtering on the left by Bourbaki) is a preset whose every pair of elements has a lower bound. (Contributed by FL, 17-Oct-2011.)
 |-  AntiDir  =  (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 lb { x ,  y } ) }
 )
 
Theoremubos 25256* The upper bounds of  A. (Contributed by FL, 16-May-2011.)
 |-  X  =  U. U. R   =>    |-  ( ( R  e.  S  /\  A  e.  B )  ->  ( R  ub  A )  =  { a  e.  X  |  A. b  e.  A  b R a } )
 
Theoremubos2 25257* The upper bounds of  A. (Contributed by FL, 18-Sep-2011.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  X  =  dom  R   =>    |-  ( ( R  e. PresetRel  /\  A  e.  B )  ->  ( R  ub  A )  =  { a  e.  X  |  A. b  e.  A  b R a } )
 
Theorempuub2 25258* The predicate " U is an upper bound of  A." (Contributed by FL, 16-May-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e. PresetRel  /\  A  e.  B )  ->  ( U  e.  ( R  ub  A )  <->  ( U  e.  X  /\  A. b  e.  A  b R U ) ) )
 
Theorempuub 25259* The predicate " U is an upper bound of  A." (Contributed by FL, 16-May-2011.)
 |-  X  =  U. U. R   =>    |-  ( ( R  e.  S  /\  A  e.  B )  ->  ( U  e.  ( R  ub  A )  <->  ( U  e.  X  /\  A. b  e.  A  b R U ) ) )
 
Theoremprltub 25260 If  R is a preset,  U R V and  U is an upper bound of  A then  V is an upper bound of  A. Bourbaki E.III.9 nb 8. (Contributed by FL, 23-May-2011.) (Proof shortened by Mario Carneiro, 3-May-2015.)
 |-  (
 ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  ->  V  e.  ( R  ub  A ) )
 
Theoremubpar 25261 If  U is an upper bound of  A and  B  C_  A then  U is an upper bound of  B. Bourbaki E.III.9 n 8. (Contributed by FL, 23-May-2011.)
 |-  (
 ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  ->  ( U  e.  ( R  ub  A )  ->  U  e.  ( R  ub  B ) ) )
 
Theoremsupdef 25262* If it exists, a supremum of  A is greater or equal to every element of  A and is the least upper bound of  A. Here the existence of the supremum is expressed by the idiom  ( R  sup w  A
)  e.  X. (Contributed by FL, 23-May-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  W  /\  ( R  sup w  A )  e.  X )  ->  ( A. y  e.  A  y R ( R  sup w  A )  /\  A. y  e.  X  ( A. z  e.  A  z R y 
 ->  ( R  sup w  A ) R y ) ) )
 
Theoremsupdefa 25263 The greatest element of a poset is greater than the other elements of the poset. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( ( R  e.  PosetRel  /\  A  e.  X  /\  ( R  sup w  X )  e.  X )  ->  A R ( R 
 sup w  X )
 )
 
Theoremmxlelt 25264* The maximal elements of the preset 
R. (Contributed by FL, 16-May-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  U. U. R   =>    |-  ( R  e.  S  ->  ( mxl `  R )  =  { a  e.  X  |  A. b  e.  X  ( a R b  ->  a  =  b ) } )
 
Theoremmxlelt2 25265* The maximal elements of the preset 
R. (Contributed by FL, 16-May-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  ( mxl `  R )  =  { a  e.  X  |  A. b  e.  X  ( a R b 
 ->  a  =  b
 ) } )
 
Theoremmnlelt2 25266* The minimal elements of the preset 
R. (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  ( mnl `  R )  =  { a  e.  X  |  A. b  e.  X  ( b R a 
 ->  b  =  a
 ) } )
 
Theoremismxl2 25267* The predicate " A is a maximal element of the preset 
R " . (Contributed by FL, 22-May-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  ( A  e.  ( mxl `  R )  <->  ( A  e.  X  /\  A. b  e.  X  ( A R b  ->  A  =  b ) ) ) )
 
Theoremismnl2 25268* The predicate " A is a minimal element of the preset 
R " . (Contributed by FL, 19-Sep-2011.)
 |-  X  =  dom  R   =>    |-  ( R  e. PresetRel  ->  ( A  e.  ( mnl `  R )  <->  ( A  e.  X  /\  A. b  e.  X  ( b R A  ->  b  =  A ) ) ) )
 
Theoremmnlmxl2 25269 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 25270 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 25271* 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 25272* 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 25273 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 25274* 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 25275 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 25276* Lemma for inposet 25278. 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 25277* Inclusion is a proper class. (Contributed by FL, 22-Sep-2008.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |- 
 -.  C  e.  _V
 
Theoreminposet 25278* 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 25279* 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 25280* The domain of the inclusion relation is  _V. (Contributed by FL, 6-Sep-2009.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |- 
 dom  C  =  _V
 
Theoremrninc 25281* The range of the inclusion relation is  _V. (Contributed by FL, 6-Sep-2009.)
 |-  C  =  { <. x ,  y >.  |  x  C_  y }   =>    |- 
 ran  C  =  _V
 
Theoremdomncnt 25282* 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 25283* 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 25284 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 25285 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 25286 A totally ordered set is a lattice. (Contributed by FL, 19-Sep-2011.)
 |-  TosetRel  C_  LatRel
 
Theoremdispos 25287 A restriction of the identity is a poset. (Contributed by FL, 2-Aug-2009.)
 |-  ( A  e.  V  ->  (  _I  |`  A )  e. 
 PosetRel )
 
Theoremderef 25288 An idiom to "dereflexivate" a relation. (Contributed by FL, 30-Dec-2010.)
 |-  ( A  e.  B  ->  -.  A ( C  \  _I  ) A )
 
Theoremdfps2 25289 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 25290* 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 25291* 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 25292* 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 25293 A direction is a preset. (Contributed by FL, 19-Sep-2011.)
 |-  ( D  e.  DirRel  ->  D  e. PresetRel )
 
Theoremdirub 25294* 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 25295 A lattice is a direction. (Contributed by FL, 19-Sep-2011.)
 |-  LatRel  C_  DirRel
 
Syntaxclbl 25296 Extend class notation to include bound lattice.
 class  BndLat
 
Definitiondf-bnlat 25297 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 ) }
 
18.13.14  Finite composites ( i. e. finite sums, products ... )
 
Syntaxcprd 25298 Extend class notation to include finite products/sums.
 class  prod_ k  e.  A G B
 
Definitiondf-prod 25299* 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 12159. 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 25300 Extend class notation to include finite supports products/sums.
 class prod2 k  e.  A G B
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