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Theorem List for Metamath Proof Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlatnlej 14101 An idiom to express that a lattice element differs from two others. (Contributed by NM, 28-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  -.  X  .<_  ( Y  .\/  Z ) )  ->  ( X  =/=  Y  /\  X  =/=  Z ) )
 
Theoremlatnlej1l 14102 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  -.  X  .<_  ( Y  .\/  Z ) )  ->  X  =/=  Y )
 
Theoremlatnlej1r 14103 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  -.  X  .<_  ( Y  .\/  Z ) )  ->  X  =/=  Z )
 
Theoremlatnlej2 14104 An idiom to express that a lattice element differs from two others. (Contributed by NM, 10-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  -.  X  .<_  ( Y  .\/  Z ) )  ->  ( -.  X  .<_  Y  /\  -.  X  .<_  Z ) )
 
Theoremlatnlej2l 14105 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  -.  X  .<_  ( Y  .\/  Z ) )  ->  -.  X  .<_  Y )
 
Theoremlatnlej2r 14106 An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  -.  X  .<_  ( Y  .\/  Z ) )  ->  -.  X  .<_  Z )
 
Theoremlatjidm 14107 Lattice join is idempotent. (chjidm 22024 analog.) (Contributed by NM, 8-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X 
 .\/  X )  =  X )
 
Theoremlatmcom 14108 The join of a lattice commutes. (incom 3303 analog.) (Contributed by NM, 6-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y )  =  ( Y 
 ./\  X ) )
 
Theoremlatmle1 14109 A meet is less than or equal to its first argument. (inss1 3331 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y )  .<_  X )
 
Theoremlatmle2 14110 A meet is less than or equal to its second argument. (inss2 3332 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y )  .<_  Y )
 
Theoremlatlem12 14111 An element is less than or equal to a meet iff the element is less than or equal to each argument of the meet. (ssin 3333 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .<_  Y 
 /\  X  .<_  Z )  <->  X  .<_  ( Y  ./\  Z ) ) )
 
Theoremlatleeqm1 14112 Less-than-or-equal-to in terms of meet. (df-ss 3108 analog.) (Contributed by NM, 7-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <-> 
 ( X  ./\  Y )  =  X ) )
 
Theoremlatleeqm2 14113 Less-than-or-equal-to in terms of meet. (sseqin2 3330 analog.) (Contributed by NM, 7-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <-> 
 ( Y  ./\  X )  =  X ) )
 
Theoremlatmlem1 14114 Add meet to both sides of a lattice ordering. (ssrin 3336 analog.) (Contributed by NM, 10-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( X  .<_  Y  ->  ( X  ./\  Z )  .<_  ( Y  ./\  Z ) ) )
 
Theoremlatmlem2 14115 Add meet to both sides of a lattice ordering. (sslin 3337 analog.) (Contributed by NM, 10-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( X  .<_  Y  ->  ( Z  ./\  X )  .<_  ( Z  ./\  Y ) ) )
 
Theoremlatmlem12 14116 Add join to both sides of a lattice ordering. (ss2in 3338 analog.) (Contributed by NM, 10-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B )
 )  ->  ( ( X  .<_  Y  /\  Z  .<_  W )  ->  ( X  ./\  Z )  .<_  ( Y  ./\  W )
 ) )
 
Theoremlatnlemlt 14117 Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3343 analog.) (Contributed by NM, 5-Feb-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .<  =  ( lt `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  X  .<_  Y  <-> 
 ( X  ./\  Y ) 
 .<  X ) )
 
Theoremlatnle 14118 Equivalent expressions for "not less than" in a lattice. (chnle 22018 analog.) (Contributed by NM, 16-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .<  =  ( lt `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  (
 ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( -.  Y  .<_  X  <->  X  .<  ( X  .\/  Y ) ) )
 
Theoremlatmidm 14119 Lattice join is idempotent. (inidm 3320 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X 
 ./\  X )  =  X )
 
Theoremlatabs1 14120 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 22020 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  ( X  ./\  Y ) )  =  X )
 
Theoremlatabs2 14121 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 22021 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  ( X  .\/  Y ) )  =  X )
 
Theoremlatledi 14122 An ortholattice is distributive in one ordering direction. (ledi 22044 analog.) (Contributed by NM, 7-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  ./\  Y )  .\/  ( X  ./\  Z ) )  .<_  ( X  ./\  ( Y  .\/  Z ) ) )
 
Theoremlatmlej11 14123 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  ./\ 
 Y )  .<_  ( X 
 .\/  Z ) )
 
Theoremlatmlej12 14124 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  ./\ 
 Y )  .<_  ( Z 
 .\/  X ) )
 
Theoremlatmlej21 14125 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( Y  ./\ 
 X )  .<_  ( X 
 .\/  Z ) )
 
Theoremlatmlej22 14126 Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( Y  ./\ 
 X )  .<_  ( Z 
 .\/  X ) )
 
Theoremlubsn 14127 The least upper bound of a singleton. (chsupsn 21917 analog.) (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  Lat  /\  X  e.  B ) 
 ->  ( U `  { X } )  =  X )
 
Theoremlatjass 14128 Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 22037 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 14129 Swap 1st and 2nd members of lattice join. (chj12 22038 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 14130 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 14131 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 14132 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 14133 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 14134 Rearrangement of lattice join of 4 classes. (chj4 22039 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 14135 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 14136 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 14137 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 14138 The weak direction of the modular law (e.g. pmod1i 29167, atmod1i1 29176) 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 14139 The weak direction of the modular law (e.g. pmod2iN 29168) 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 14140 Extend class notation with complete lattices.
 class  CLat
 
Definitiondf-clat 14141* 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 14142* 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 14143 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 14144 LUB always exists in a complete lattice. (chsupcl 21844 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( U `  S )  e.  B )
 
Theoremclatglbcl 14145 GLB always exists in a complete lattice. (chintcl 21836 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( G `  S )  e.  B )
 
Theoremisclati 14146* 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 14147 A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  CLat  ->  K  e.  Lat )
 
Theoremisglbd 14148* 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 14149* 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 14150 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 14151* 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 14152 Subset law for least upper bounds. (chsupss 21846 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 14153 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 14154 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 14155* 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 14156 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 14157* 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 14158 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 14159 Class function defining dual orders.
 class ODual
 
Definitiondf-odu 14160 Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 14164, oduleval 14162, and oduleg 14163 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 14218. (Contributed by Stefan O'Rear, 29-Jan-2015.)

 |- ODual  =  ( w  e.  _V  |->  ( w sSet  <. ( le ` 
 ndx ) ,  `' ( le `  w )
 >. ) )
 
Theoremoduval 14161 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 14162 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 14163 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 14164 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 14165* 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 14166 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 14167 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 14168 Lemma for odujoin 14173. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  ( meet `  (/) )  =  (/)
 
Theoremjoin0 14169 Lemma for odumeet 14171. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  ( join `  (/) )  =  (/)
 
Theoremoduglb 14170 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 14171 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 14172 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 14173 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 14174 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 14175 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 14176 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   =>    |-  ( O  e.  Lat  ->  D  e.  Lat )
 
Theoremposlubmo 14177* Least upper bounds in a poset are unique if they exist. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |- 
 .<_  =  ( le `  K )   &    |-  B  =  ( Base `  K )   =>    |-  ( ( K  e.  Poset  /\  S  C_  B )  ->  E* x ( x  e.  B  /\  ( A. y  e.  S  y  .<_  x  /\  A. z  e.  B  ( A. y  e.  S  y  .<_  z  ->  x  .<_  z ) ) ) )
 
Theoremposlubd 14178* 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 14179* 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 14180* 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 14181 Class function defining inclusion posets.
 class toInc
 
Definitiondf-ipo 14182* For any family of sets, define the poset of that family ordered by inclusion. See ipobas 14185, ipolerval 14186, and ipole 14188 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 14183 The structure of df-ipo 14182 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 14184* 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 14185 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 14186* 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 14187 Topology of the inclusion poset. (Contributed by Mario Carneiro, 24-Oct-2015.)
 |-  I  =  (toInc `  F )   &    |-  .<_  =  ( le `  I )   =>    |-  ( F  e.  V  ->  (ordTop `  .<_  )  =  (TopSet `  I )
 )
 
Theoremipole 14188 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 14189 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 14190 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 14191* 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 14192 Direction by inclusion as used here implies sethood. (Contributed by Stefan O'Rear, 2-Apr-2015.)
 |-  ( (toInc `  A )  e. Dirset  ->  A  e.  _V )
 
Theoremipodrsfi 14193* 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 14194 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 14195* 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 14196* An algebraic closure system satisfies isacs3 14204. (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 14197 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 14198* 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 14199* 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 14200 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 ) )
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