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Theorem List for Metamath Proof Explorer - 14201-14300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlatj4 14201 Rearrangement of lattice join of 4 classes. (chj4 22106 analog.) (Contributed by NM, 14-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B )
 )  ->  ( ( X  .\/  Y )  .\/  ( Z  .\/  W ) )  =  ( ( X  .\/  Z )  .\/  ( Y  .\/  W ) ) )
 
Theoremlatj4rot 14202 Rotate lattice join of 4 classes. (Contributed by NM, 11-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B )
 )  ->  ( ( X  .\/  Y )  .\/  ( Z  .\/  W ) )  =  ( ( W  .\/  X )  .\/  ( Y  .\/  Z ) ) )
 
Theoremlatjjdi 14203 Lattice join distributes over itself. (Contributed by NM, 30-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( X  .\/  ( Y 
 .\/  Z ) )  =  ( ( X  .\/  Y )  .\/  ( X  .\/  Z ) ) )
 
Theoremlatjjdir 14204 Lattice join distributes over itself. (Contributed by NM, 2-Aug-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .\/  Y )  .\/  Z )  =  ( ( X  .\/  Z )  .\/  ( Y  .\/  Z ) ) )
 
Theoremmod1ile 14205 The weak direction of the modular law (e.g. pmod1i 29304, atmod1i1 29313) that holds in any lattice. (Contributed by NM, 11-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  .<_  Z  ->  ( X  .\/  ( Y  ./\  Z ) )  .<_  ( ( X 
 .\/  Y )  ./\  Z ) ) )
 
Theoremmod2ile 14206 The weak direction of the modular law (e.g. pmod2iN 29305) that holds in any lattice. (Contributed by NM, 11-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( Z  .<_  X  ->  ( ( X  ./\  Y )  .\/  Z )  .<_  ( X  ./\  ( Y  .\/  Z ) ) ) )
 
Syntaxccla 14207 Extend class notation with complete lattices.
 class  CLat
 
Definitiondf-clat 14208* Define the class of all complete lattices. (Contributed by NM, 18-Oct-2012.)
 |- 
 CLat  =  { p  e.  Poset  |  A. s
 ( s  C_  ( Base `  p )  ->  ( ( ( lub `  p ) `  s
 )  e.  ( Base `  p )  /\  (
 ( glb `  p ) `  s )  e.  ( Base `  p ) ) ) }
 
Theoremisclat 14209* The predicate "is an complete lattice." (Contributed by NM, 18-Oct-2012.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  G  =  ( glb `  K )   =>    |-  ( K  e.  CLat  <->  ( K  e.  Poset  /\  A. s ( s  C_  B  ->  ( ( U `
  s )  e.  B  /\  ( G `
  s )  e.  B ) ) ) )
 
Theoremclatlem 14210 Lemma for properties of a complete lattice. (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  G  =  ( glb `  K )   =>    |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( ( U `  S )  e.  B  /\  ( G `  S )  e.  B )
 )
 
Theoremclatlubcl 14211 LUB always exists in a complete lattice. (chsupcl 21911 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( U `  S )  e.  B )
 
Theoremclatglbcl 14212 GLB always exists in a complete lattice. (chintcl 21903 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( G `  S )  e.  B )
 
Theoremisclati 14213* Properties that determine a complete lattice. (Contributed by NM, 12-Sep-2011.)
 |-  K  e.  Poset   &    |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  G  =  ( glb `  K )   &    |-  ( s  C_  B  ->  ( U `  s
 )  e.  B )   &    |-  ( s  C_  B  ->  ( G `  s )  e.  B )   =>    |-  K  e.  CLat
 
Theoremclatl 14214 A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  CLat  ->  K  e.  Lat )
 
Theoremisglbd 14215* Properties that determine the greatest lower bound of a complete lattice. (Contributed by Mario Carneiro, 19-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   &    |-  (
 ( ph  /\  y  e.  S )  ->  H  .<_  y )   &    |-  ( ( ph  /\  x  e.  B  /\  A. y  e.  S  x  .<_  y )  ->  x  .<_  H )   &    |-  ( ph  ->  K  e.  CLat )   &    |-  ( ph  ->  S 
 C_  B )   &    |-  ( ph  ->  H  e.  B )   =>    |-  ( ph  ->  ( G `  S )  =  H )
 
Theoremlublem 14216* Lemma for least upper bound properties in a complete lattice. (Contributed by NM, 19-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( A. y  e.  S  y  .<_  ( U `
  S )  /\  A. z  e.  B  (
 A. y  e.  S  y  .<_  z  ->  ( U `  S )  .<_  z ) ) )
 
Theoremlubub 14217 The LUB of a complete lattice subset is an upper bound. (Contributed by NM, 19-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B  /\  X  e.  S )  ->  X  .<_  ( U `  S ) )
 
Theoremlubl 14218* The LUB of a complete lattice subset is a least bound. (Contributed by NM, 19-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B  /\  X  e.  B )  ->  ( A. y  e.  S  y  .<_  X  ->  ( U `  S ) 
 .<_  X ) )
 
Theoremlubss 14219 Subset law for least upper bounds. (chsupss 21913 analog.) (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( U `  S ) 
 .<_  ( U `  T ) )
 
Theoremlubel 14220 An element of a set is less than or equal to the least upper bound of the set. (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  X  e.  S  /\  S  C_  B )  ->  X  .<_  ( U `  S ) )
 
Theoremlubun 14221 The LUB of a union. (Contributed by NM, 5-Mar-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  U  =  ( lub `  K )   =>    |-  ( ( K  e.  CLat  /\  S  C_  B  /\  T  C_  B )  ->  ( U `  ( S  u.  T ) )  =  ( ( U `
  S )  .\/  ( U `  T ) ) )
 
Theoremclatglb 14222* Properties of greatest lower bound of a complete lattice. (Contributed by NM, 5-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( A. y  e.  S  ( G `  S )  .<_  y  /\  A. z  e.  B  (
 A. y  e.  S  z  .<_  y  ->  z  .<_  ( G `  S ) ) ) )
 
Theoremclatglble 14223 A greatest lower bound is a least element. (Contributed by NM, 5-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B  /\  X  e.  S )  ->  ( G `  S )  .<_  X )
 
Theoremclatleglb 14224* Two ways of expressing "less than or equal to the greatest lower bound." (Contributed by NM, 5-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  X  e.  B  /\  S  C_  B )  ->  ( X  .<_  ( G `
  S )  <->  A. y  e.  S  X  .<_  y ) )
 
Theoremclatglbss 14225 Subset law for greatest lower bound. (Contributed by Mario Carneiro, 16-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( G `  T ) 
 .<_  ( G `  S ) )
 
9.2.3  The dual of an ordered set
 
Syntaxcodu 14226 Class function defining dual orders.
 class ODual
 
Definitiondf-odu 14227 Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 14231, oduleval 14229, and oduleg 14230 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 14285. (Contributed by Stefan O'Rear, 29-Jan-2015.)

 |- ODual  =  ( w  e.  _V  |->  ( w sSet  <. ( le ` 
 ndx ) ,  `' ( le `  w )
 >. ) )
 
Theoremoduval 14228 Value of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  .<_  =  ( le `  O )   =>    |-  D  =  ( O sSet  <. ( le `  ndx ) ,  `'  .<_  >. )
 
Theoremoduleval 14229 Value of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  .<_  =  ( le `  O )   =>    |-  `'  .<_  =  ( le `  D )
 
Theoremoduleg 14230 Truth of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  .<_  =  ( le `  O )   &    |-  G  =  ( le `  D )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A G B 
 <->  B  .<_  A ) )
 
Theoremodubas 14231 Base set of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  B  =  (
 Base `  O )   =>    |-  B  =  (
 Base `  D )
 
Theorempospropd 14232* Posethood is determined only by structure components and only by the value of the relation within the base set. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  ( ph  ->  K  e.  V )   &    |-  ( ph  ->  L  e.  W )   &    |-  ( ph  ->  B  =  (
 Base `  K ) )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( le `  K ) y  <->  x ( le `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  Poset 
 <->  L  e.  Poset ) )
 
Theoremodupos 14233 Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   =>    |-  ( O  e.  Poset  ->  D  e.  Poset )
 
Theoremoduposb 14234 Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   =>    |-  ( O  e.  V  ->  ( O  e.  Poset  <->  D  e.  Poset ) )
 
Theoremmeet0 14235 Lemma for odujoin 14240. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  ( meet `  (/) )  =  (/)
 
Theoremjoin0 14236 Lemma for odumeet 14238. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  ( join `  (/) )  =  (/)
 
Theoremoduglb 14237 Greatest lower bounds in a dual order are least upper bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  U  =  ( lub `  O )   =>    |-  ( O  e.  V  ->  U  =  ( glb `  D ) )
 
Theoremodumeet 14238 Meets in a dual order are joins in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  .\/  =  ( join `  O )   =>    |-  .\/  =  ( meet `  D )
 
Theoremodulub 14239 Least upper bounds in a dual order are greatest lower bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  L  =  ( glb `  O )   =>    |-  ( O  e.  V  ->  L  =  ( lub `  D ) )
 
Theoremodujoin 14240 Joins in a dual order are meets in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  ./\  =  ( meet `  O )   =>    |-  ./\  =  ( join `  D )
 
Theoremodulatb 14241 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   =>    |-  ( O  e.  V  ->  ( O  e.  Lat  <->  D  e.  Lat ) )
 
Theoremoduclatb 14242 Being a complete lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   =>    |-  ( O  e.  CLat  <->  D  e.  CLat )
 
Theoremodulat 14243 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   =>    |-  ( O  e.  Lat  ->  D  e.  Lat )
 
Theoremposlubmo 14244* Least upper bounds in a poset are unique if they exist. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by NM, 16-Jun-2017.)
 |- 
 .<_  =  ( le `  K )   &    |-  B  =  ( Base `  K )   =>    |-  ( ( K  e.  Poset  /\  S  C_  B )  ->  E* x  e.  B ( A. y  e.  S  y  .<_  x  /\  A. z  e.  B  ( A. y  e.  S  y  .<_  z  ->  x  .<_  z ) ) )
 
Theoremposlubd 14245* Properties which determine a least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |- 
 .<_  =  ( le `  K )   &    |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  ( ph  ->  K  e.  Poset )   &    |-  ( ph  ->  S  C_  B )   &    |-  ( ph  ->  T  e.  B )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  x  .<_  T )   &    |-  ( ( ph  /\  y  e.  B  /\  A. x  e.  S  x  .<_  y ) 
 ->  T  .<_  y )   =>    |-  ( ph  ->  ( U `  S )  =  T )
 
Theoremposlubdg 14246* Properties which determine a least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |- 
 .<_  =  ( le `  K )   &    |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  U  =  ( lub `  K ) )   &    |-  ( ph  ->  K  e.  Poset )   &    |-  ( ph  ->  S 
 C_  B )   &    |-  ( ph  ->  T  e.  B )   &    |-  ( ( ph  /\  x  e.  S )  ->  x  .<_  T )   &    |-  ( ( ph  /\  y  e.  B  /\  A. x  e.  S  x  .<_  y )  ->  T  .<_  y )   =>    |-  ( ph  ->  ( U `  S )  =  T )
 
Theoremposglbd 14247* Properties which determine a greatest lower bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |- 
 .<_  =  ( le `  K )   &    |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  G  =  ( glb `  K ) )   &    |-  ( ph  ->  K  e.  Poset )   &    |-  ( ph  ->  S 
 C_  B )   &    |-  ( ph  ->  T  e.  B )   &    |-  ( ( ph  /\  x  e.  S )  ->  T  .<_  x )   &    |-  ( ( ph  /\  y  e.  B  /\  A. x  e.  S  y 
 .<_  x )  ->  y  .<_  T )   =>    |-  ( ph  ->  ( G `  S )  =  T )
 
9.2.4  Subset order structures
 
Syntaxcipo 14248 Class function defining inclusion posets.
 class toInc
 
Definitiondf-ipo 14249* For any family of sets, define the poset of that family ordered by inclusion. See ipobas 14252, ipolerval 14253, and ipole 14255 for its contract.

EDITORIAL: I'm not thrilled with the name. Any suggestions? (Contributed by Stefan O'Rear, 30-Jan-2015.) (New usage is discouraged.)

 |- toInc  =  ( f  e.  _V  |->  [_
 { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  /  o ]_ ( { <. ( Base `  ndx ) ,  f >. , 
 <. (TopSet `  ndx ) ,  (ordTop `  o ) >. }  u.  { <. ( le `  ndx ) ,  o >. ,  <. ( oc
 `  ndx ) ,  ( x  e.  f  |->  U. { y  e.  f  |  ( y  i^i  x )  =  (/) } ) >. } ) )
 
Theoremipostr 14250 The structure of df-ipo 14249 is a structure defining indexes up to 11. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |-  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. (TopSet `  ndx ) ,  J >. }  u.  { <. ( le `  ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ._|_  >. } ) Struct  <. 1 , ; 1 1 >.
 
Theoremipoval 14251* Value of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  I  =  (toInc `  F )   &    |-  .<_  =  { <. x ,  y >.  |  ( { x ,  y }  C_  F  /\  x  C_  y ) }   =>    |-  ( F  e.  V  ->  I  =  ( { <. ( Base `  ndx ) ,  F >. , 
 <. (TopSet `  ndx ) ,  (ordTop `  .<_  ) >. }  u.  { <. ( le ` 
 ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ( x  e.  F  |->  U. {
 y  e.  F  |  ( y  i^i  x )  =  (/) } ) >. } ) )
 
Theoremipobas 14252 Base set of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Revised by Mario Carneiro, 25-Oct-2015.)
 |-  I  =  (toInc `  F )   =>    |-  ( F  e.  V  ->  F  =  ( Base `  I ) )
 
Theoremipolerval 14253* Relation of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  I  =  (toInc `  F )   =>    |-  ( F  e.  V  ->  { <. x ,  y >.  |  ( { x ,  y }  C_  F  /\  x  C_  y ) }  =  ( le `  I ) )
 
Theoremipotset 14254 Topology of the inclusion poset. (Contributed by Mario Carneiro, 24-Oct-2015.)
 |-  I  =  (toInc `  F )   &    |-  .<_  =  ( le `  I )   =>    |-  ( F  e.  V  ->  (ordTop `  .<_  )  =  (TopSet `  I )
 )
 
Theoremipole 14255 Weak order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  I  =  (toInc `  F )   &    |-  .<_  =  ( le `  I )   =>    |-  ( ( F  e.  V  /\  X  e.  F  /\  Y  e.  F ) 
 ->  ( X  .<_  Y  <->  X  C_  Y ) )
 
Theoremipolt 14256 Strict order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  I  =  (toInc `  F )   &    |-  .<  =  ( lt `  I )   =>    |-  ( ( F  e.  V  /\  X  e.  F  /\  Y  e.  F )  ->  ( X 
 .<  Y  <->  X  C.  Y ) )
 
Theoremipopos 14257 The inclusion poset on a family of sets is actually a poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  I  =  (toInc `  F )   =>    |-  I  e.  Poset
 
Theoremisipodrs 14258* Condition for a family of sets to be directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( (toInc `  A )  e. Dirset  <->  ( A  e.  _V 
 /\  A  =/=  (/)  /\  A. x  e.  A  A. y  e.  A  E. z  e.  A  ( x  u.  y )  C_  z ) )
 
Theoremipodrscl 14259 Direction by inclusion as used here implies sethood. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( (toInc `  A )  e. Dirset  ->  A  e.  _V )
 
Theoremipodrsfi 14260* Finite upper bound property for directed collections of sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( ( (toInc `  A )  e. Dirset  /\  X  C_  A  /\  X  e.  Fin )  ->  E. z  e.  A  U. X  C_  z )
 
Theoremfpwipodrs 14261 The finite subsets of any set are directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( A  e.  V  ->  (toInc `  ( ~P A  i^i  Fin ) )  e. Dirset )
 
Theoremipodrsima 14262* The monotone image of a directed set. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( ph  ->  F  Fn  ~P A )   &    |-  (
 ( ph  /\  ( u 
 C_  v  /\  v  C_  A ) )  ->  ( F `  u ) 
 C_  ( F `  v ) )   &    |-  ( ph  ->  (toInc `  B )  e. Dirset )   &    |-  ( ph  ->  B 
 C_  ~P A )   &    |-  ( ph  ->  ( F " B )  e.  V )   =>    |-  ( ph  ->  (toInc `  ( F " B ) )  e. Dirset )
 
Theoremisacs3lem 14263* An algebraic closure system satisfies isacs3 14271. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  ->  ( C  e.  (Moore `  X )  /\  A. s  e. 
 ~P  C ( (toInc `  s )  e. Dirset  ->  U. s  e.  C ) ) )
 
Theoremacsdrsel 14264 An algebraic closure system contains all directed unions of closed sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( ( C  e.  (ACS `  X )  /\  Y  C_  C  /\  (toInc `  Y )  e. Dirset )  ->  U. Y  e.  C )
 
Theoremisacs4lem 14265* In a closure system in which directed unions of closed sets are closed, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C ( (toInc `  s )  e. Dirset  ->  U. s  e.  C ) )  ->  ( C  e.  (Moore `  X )  /\  A. t  e. 
 ~P  ~P X ( (toInc `  t )  e. Dirset  ->  ( F `  U. t
 )  =  U. ( F " t ) ) ) )
 
Theoremisacs5lem 14266* If closure commutes with directed unions, then the closure of a set is the closure of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (Moore `  X )  /\  A. t  e.  ~P  ~P X ( (toInc `  t )  e. Dirset  ->  ( F `  U. t )  =  U. ( F
 " t ) ) )  ->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( F `  s )  =  U. ( F " ( ~P s  i^i  Fin )
 ) ) )
 
Theoremacsdrscl 14267 In an algebraic closure system, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (ACS `  X )  /\  Y  C_  ~P X  /\  (toInc `  Y )  e. Dirset )  ->  ( F `  U. Y )  =  U. ( F " Y ) )
 
Theoremacsficl 14268 A closure in an algebraic closure system is the union of the closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( ( C  e.  (ACS `  X )  /\  S  C_  X )  ->  ( F `  S )  =  U. ( F
 " ( ~P S  i^i  Fin ) ) )
 
Theoremisacs5 14269* A closure system is algebraic iff the closure of a generating set is the union of the closures of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  X ( F `  s )  =  U. ( F
 " ( ~P s  i^i  Fin ) ) ) )
 
Theoremisacs4 14270* A closure system is algebraic iff closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  F  =  (mrCls `  C )   =>    |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  ~P X ( (toInc `  s )  e. Dirset  ->  ( F `  U. s )  =  U. ( F
 " s ) ) ) )
 
Theoremisacs3 14271* A closure system is algebraic iff directed unions of closed sets are closed. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( C  e.  (ACS `  X )  <->  ( C  e.  (Moore `  X )  /\  A. s  e.  ~P  C ( (toInc `  s )  e. Dirset  ->  U. s  e.  C ) ) )
 
Theoremacsficld 14272 In an algebraic closure system, the closure of a set is the union of the closures of its finite subsets. Deduction form of acsficl 14268. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( N `  S )  =  U. ( N
 " ( ~P S  i^i  Fin ) ) )
 
Theoremacsficl2d 14273* In an algebraic closure system, an element is in the closure of a set if and only if it is in the closure of a finite subset. Alternate form of acsficl 14268. Deduction form. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( Y  e.  ( N `
  S )  <->  E. x  e.  ( ~P S  i^i  Fin ) Y  e.  ( N `  x ) ) )
 
Theoremacsfiindd 14274 In an algebraic closure system, a set is independent if and only if all its finite subsets are independent. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S 
 C_  X )   =>    |-  ( ph  ->  ( S  e.  I  <->  ( ~P S  i^i  Fin )  C_  I
 ) )
 
Theoremacsmapd 14275* In an algebraic closure system, if 
T is contained in the closure of  S, there is a map  f from  T into the set of finite subsets of  S such that the closure of  U. ran  f contains  T. This is proven by applying acsficl2d 14273 to each element of  T. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  ( ph  ->  S 
 C_  X )   &    |-  ( ph  ->  T  C_  ( N `  S ) )   =>    |-  ( ph  ->  E. f
 ( f : T --> ( ~P S  i^i  Fin )  /\  T  C_  ( N `  U. ran  f
 ) ) )
 
Theoremacsmap2d 14276* In an algebraic closure system, if 
S and  T have the same closure and  S is independent, then there is a map  f from  T into the set of finite subsets of  S such that  S equals the union of  ran  f. This is proven by taking the map  f from acsmapd 14275 and observing that, since  S and  T have the same closure, the closure of  U. ran  f must contain  S. Since  S is independent, by mrissmrcd 13536,  U. ran  f must equal  S. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  C_  X )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   =>    |-  ( ph  ->  E. f
 ( f : T --> ( ~P S  i^i  Fin )  /\  S  =  U. ran  f ) )
 
Theoremacsinfd 14277 In an algebraic closure system, if 
S and  T have the same closure and  S is infinite independent, then  T is infinite. This follows from applying unirnffid 7142 to the map given in acsmap2d 14276. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  C_  X )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   &    |-  ( ph  ->  -.  S  e.  Fin )   =>    |-  ( ph  ->  -.  T  e.  Fin )
 
Theoremacsdomd 14278 In an algebraic closure system, if 
S and  T have the same closure and  S is infinite independent, then  T dominates  S. This follows from applying acsinfd 14277 and then applying unirnfdomd 8184 to the map given in acsmap2d 14276. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  C_  X )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   &    |-  ( ph  ->  -.  S  e.  Fin )   =>    |-  ( ph  ->  S  ~<_  T )
 
Theoremacsinfdimd 14279 In an algebraic closure system, if two independent sets have equal closure and one is infinite, then they are equinumerous. This is proven by using acsdomd 14278 twice with acsinfd 14277. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  e.  I
 )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   &    |-  ( ph  ->  -.  S  e.  Fin )   =>    |-  ( ph  ->  S  ~~  T )
 
Theoremacsexdimd 14280* In an algebraic closure system whose closure operator has the exchange property, if two independent sets have equal closure, they are equinumerous. See mreexfidimd 13546 for the finite case and acsinfdimd 14279 for the infinite case. This is a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  (ACS `  X )
 )   &    |-  N  =  (mrCls `  A )   &    |-  I  =  (mrInd `  A )   &    |-  ( ph  ->  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y }
 ) )  \  ( N `  s ) ) y  e.  ( N `
  ( s  u. 
 { z } )
 ) )   &    |-  ( ph  ->  S  e.  I )   &    |-  ( ph  ->  T  e.  I
 )   &    |-  ( ph  ->  ( N `  S )  =  ( N `  T ) )   =>    |-  ( ph  ->  S  ~~  T )
 
Theoremmrelatglb 14281 Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  I  =  (toInc `  C )   &    |-  G  =  ( glb `  I )   =>    |-  (
 ( C  e.  (Moore `  X )  /\  U  C_  C  /\  U  =/=  (/) )  ->  ( G `  U )  =  |^| U )
 
Theoremmrelatglb0 14282 The empty intersection in a Moore space is realized by the base set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  I  =  (toInc `  C )   &    |-  G  =  ( glb `  I )   =>    |-  ( C  e.  (Moore `  X )  ->  ( G `  (/) )  =  X )
 
Theoremmrelatlub 14283 Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  I  =  (toInc `  C )   &    |-  F  =  (mrCls `  C )   &    |-  L  =  ( lub `  I )   =>    |-  (
 ( C  e.  (Moore `  X )  /\  U  C_  C )  ->  ( L `  U )  =  ( F `  U. U ) )
 
Theoremmreclat 14284 A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  I  =  (toInc `  C )   =>    |-  ( C  e.  (Moore `  X )  ->  I  e.  CLat )
 
9.2.5  Distributive lattices
 
Theoremlatmass 14285 Lattice meet is associative. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  ./\  Y )  ./\  Z )  =  ( X  ./\  ( Y  ./\  Z ) ) )
 
Theoremlatdisdlem 14286* Lemma for latdisd 14287. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( K  e.  Lat  ->  ( A. u  e.  B  A. v  e.  B  A. w  e.  B  ( u  .\/  ( v  ./\  w ) )  =  ( ( u  .\/  v
 )  ./\  ( u  .\/  w ) )  ->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  ./\  ( y  .\/  z ) )  =  ( ( x  ./\  y )  .\/  ( x  ./\  z
 ) ) ) )
 
Theoremlatdisd 14287* In a lattice, joins distribute over meets if and only if meets distribute over joins; the distributive property is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( K  e.  Lat  ->  ( A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .\/  ( y  ./\  z ) )  =  ( ( x  .\/  y )  ./\  ( x 
 .\/  z ) )  <->  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  ./\  ( y  .\/  z ) )  =  ( ( x  ./\  y )  .\/  ( x 
 ./\  z ) ) ) )
 
Syntaxcdlat 14288 The class of distributive lattices.
 class DLat
 
Definitiondf-dlat 14289* A distributive lattice is a lattice in which meets distribute over joins, or equivalently (latdisd 14287) joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |- DLat  =  { k  e.  Lat  | 
 [. ( Base `  k
 )  /  b ]. [. ( join `  k )  /  j ]. [. ( meet `  k )  /  m ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x m ( y j z ) )  =  ( ( x m y ) j ( x m z ) ) }
 
Theoremisdlat 14290* Property of being a distributive lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( K  e. DLat  <->  ( K  e.  Lat  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  ./\  ( y  .\/  z ) )  =  ( ( x  ./\  y )  .\/  ( x 
 ./\  z ) ) ) )
 
Theoremdlatmjdi 14291 In a distributive lattice, meets distribute over joins. (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 ) ) )
 
Theoremdlatl 14292 A distributive lattice is a lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( K  e. DLat  ->  K  e.  Lat )
 
Theoremodudlatb 14293 The dual of a distributive lattice is a distributive lattice and conversely. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  D  =  (ODual `  K )   =>    |-  ( K  e.  V  ->  ( K  e. DLat  <->  D  e. DLat ) )
 
Theoremdlatjmdi 14294 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 14295 Extend class notation with the class of all posets.
 class  PosetRel
 
Syntaxctsr 14296 Extend class notation with the class of all totally ordered sets.
 class  TosetRel
 
Syntaxcspw 14297 Extend class notation with the supremum of an ordered set.
 class  sup w
 
Syntaxcinf 14298 Extend class notation with the infimum of an ordered set.
 class  inf w
 
Syntaxcla 14299 Extend class notation with the class of all lattices.
 class  LatRel
 
Definitiondf-ps 14300 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 ) ) }
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