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Theorem List for Metamath Proof Explorer - 14201-14300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlatjass 14201 Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 22112 analog.) (Contributed by NM, 17-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .\/  Y )  .\/  Z )  =  ( X  .\/  ( Y  .\/  Z ) ) )
 
Theoremlatj12 14202 Swap 1st and 2nd members of lattice join. (chj12 22113 analog.) (Contributed by NM, 4-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( X  .\/  ( Y 
 .\/  Z ) )  =  ( Y  .\/  ( X  .\/  Z ) ) )
 
Theoremlatj32 14203 Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 2-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .\/  Y )  .\/  Z )  =  ( ( X  .\/  Z )  .\/  Y )
 )
 
Theoremlatj13 14204 Swap 1sd and 3rd members of lattice join. (Contributed by NM, 4-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( X  .\/  ( Y 
 .\/  Z ) )  =  ( Z  .\/  ( Y  .\/  X ) ) )
 
Theoremlatj31 14205 Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 23-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .\/  Y )  .\/  Z )  =  ( ( Z  .\/  Y )  .\/  X )
 )
 
Theoremlatjrot 14206 Rotate lattice join of 3 classes. (Contributed by NM, 23-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .\/  Y )  .\/  Z )  =  ( ( Z  .\/  X )  .\/  Y )
 )
 
Theoremlatj4 14207 Rearrangement of lattice join of 4 classes. (chj4 22114 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 14208 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 14209 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 14210 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 14211 The weak direction of the modular law (e.g. pmod1i 30037, atmod1i1 30046) 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 14212 The weak direction of the modular law (e.g. pmod2iN 30038) 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 14213 Extend class notation with complete lattices.
 class  CLat
 
Definitiondf-clat 14214* 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 14215* 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 14216 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 14217 LUB always exists in a complete lattice. (chsupcl 21919 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( U `  S )  e.  B )
 
Theoremclatglbcl 14218 GLB always exists in a complete lattice. (chintcl 21911 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( G `  S )  e.  B )
 
Theoremisclati 14219* 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 14220 A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  CLat  ->  K  e.  Lat )
 
Theoremisglbd 14221* 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 14222* 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 14223 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 14224* 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 14225 Subset law for least upper bounds. (chsupss 21921 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 14226 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 14227 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 14228* 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 14229 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 14230* 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 14231 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 14232 Class function defining dual orders.
 class ODual
 
Definitiondf-odu 14233 Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 14237, oduleval 14235, and oduleg 14236 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 14291. (Contributed by Stefan O'Rear, 29-Jan-2015.)

 |- ODual  =  ( w  e.  _V  |->  ( w sSet  <. ( le ` 
 ndx ) ,  `' ( le `  w )
 >. ) )
 
Theoremoduval 14234 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 14235 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 14236 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 14237 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 14238* 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 14239 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 14240 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 14241 Lemma for odujoin 14246. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  ( meet `  (/) )  =  (/)
 
Theoremjoin0 14242 Lemma for odumeet 14244. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  ( join `  (/) )  =  (/)
 
Theoremoduglb 14243 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 14244 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 14245 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 14246 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 14247 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 14248 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 14249 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   =>    |-  ( O  e.  Lat  ->  D  e.  Lat )
 
Theoremposlubmo 14250* 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 14251* 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 14252* 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 14253* 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 14254 Class function defining inclusion posets.
 class toInc
 
Definitiondf-ipo 14255* For any family of sets, define the poset of that family ordered by inclusion. See ipobas 14258, ipolerval 14259, and ipole 14261 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 14256 The structure of df-ipo 14255 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 14257* 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 14258 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 14259* 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 14260 Topology of the inclusion poset. (Contributed by Mario Carneiro, 24-Oct-2015.)
 |-  I  =  (toInc `  F )   &    |-  .<_  =  ( le `  I )   =>    |-  ( F  e.  V  ->  (ordTop `  .<_  )  =  (TopSet `  I )
 )
 
Theoremipole 14261 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 14262 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 14263 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 14264* 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 14265 Direction by inclusion as used here implies sethood. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( (toInc `  A )  e. Dirset  ->  A  e.  _V )
 
Theoremipodrsfi 14266* 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 14267 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 14268* 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 14269* An algebraic closure system satisfies isacs3 14277. (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 14270 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 14271* 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 14272* 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 14273 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 14274 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 14275* 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 14276* 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 14277* 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 14278 In an algebraic closure system, the closure of a set is the union of the closures of its finite subsets. Deduction form of acsficl 14274. (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 14279* 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 14274. 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 14280 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 14281* 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 14279 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 14282* 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 14281 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 13542,  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 14283 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 7147 to the map given in acsmap2d 14282. 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 14284 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 14283 and then applying unirnfdomd 8189 to the map given in acsmap2d 14282. 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 14285 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 14284 twice with acsinfd 14283. 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 14286* In an algebraic closure system whose closure operator has the exchange property, if two independent sets have equal closure, they are equinumerous. See mreexfidimd 13552 for the finite case and acsinfdimd 14285 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 14287 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 14288 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 14289 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 14290 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 14291 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 14292* Lemma for latdisd 14293. (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 14293* 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 14294 The class of distributive lattices.
 class DLat
 
Definitiondf-dlat 14295* A distributive lattice is a lattice in which meets distribute over joins, or equivalently (latdisd 14293) 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 14296* 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 14297 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 14298 A distributive lattice is a lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  ( K  e. DLat  ->  K  e.  Lat )
 
Theoremodudlatb 14299 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 14300 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 ) ) )
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