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Theorem List for Metamath Proof Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmeetcom 14101 The meet of a poset commutes. (The antecedent  ( X  ./\  Y )  e.  B  /\  ( Y  ./\  X )  e.  B i.e. "the meets exist" could be omitted as an artifact of our particular join definition, but other definitions may require it.) (Contributed by NM, 17-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  ( ( X  ./\  Y )  e.  B  /\  ( Y 
 ./\  X )  e.  B ) )  ->  ( X 
 ./\  Y )  =  ( Y  ./\  X )
 )
 
Syntaxctos 14102 Extend class notation with the class of all tosets.
 class Toset
 
Definitiondf-toset 14103* Define the class of totally ordered sets (tosets). (Contributed by FL, 17-Nov-2014.)
 |- Toset  =  { f  e.  Poset  | 
 [. ( Base `  f
 )  /  b ]. [. ( le `  f
 )  /  r ]. A. x  e.  b  A. y  e.  b  ( x r y  \/  y r x ) }
 
Theoremistos 14104* The predicate "is a toset." (Contributed by FL, 17-Nov-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( K  e. Toset  <->  ( K  e.  Poset  /\  A. x  e.  B  A. y  e.  B  ( x  .<_  y  \/  y  .<_  x ) ) )
 
Theoremtosso 14105 Write the totally ordered set structure predicate in terms of the proper class strict order predicate. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .<  =  ( lt `  K )   =>    |-  ( K  e.  V  ->  ( K  e. Toset  <->  (  .<  Or  B  /\  (  _I  |`  B )  C_  .<_  ) ) )
 
Syntaxcp0 14106 Extend class notation with poset zero.
 class  0.
 
Syntaxcp1 14107 Extend class notation with poset unit.
 class  1.
 
Definitiondf-p0 14108 Define poset zero. (Contributed by NM, 12-Oct-2011.)
 |- 
 0.  =  ( p  e.  _V  |->  ( ( glb `  p ) `  ( Base `  p )
 ) )
 
Definitiondf-p1 14109 Define poset unit. (Contributed by NM, 22-Oct-2011.)
 |- 
 1.  =  ( p  e.  _V  |->  ( ( lub `  p ) `  ( Base `  p )
 ) )
 
Theoremp0val 14110 Value of poset zero. (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( K  e.  V  ->  .0.  =  ( G `
  B ) )
 
Theoremp1val 14111 Value of poset zero. (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( K  e.  V  ->  .1.  =  ( U `
  B ) )
 
Theoremp0le 14112 Poset zero (if defined) is less than any element. (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  Poset  /\  .0.  e.  B  /\  X  e.  B )  ->  .0.  .<_  X )
 
Theoremple1 14113 Any element is less than or equal to poset one (if defined). (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  Poset  /\  X  e.  B  /\  .1.  e.  B )  ->  X  .<_  .1.  )
 
9.2.2  Lattices
 
Syntaxclat 14114 Extend class notation with the class of all lattices.
 class  Lat
 
Definitiondf-lat 14115* Define the class of all lattices. A lattice is a poset in which the join and meet of any two elements always exists. (Contributed by NM, 18-Oct-2012.)
 |- 
 Lat  =  { p  e.  Poset  |  A. x  e.  ( Base `  p ) A. y  e.  ( Base `  p ) ( ( x ( join `  p ) y )  e.  ( Base `  p )  /\  ( x (
 meet `  p ) y )  e.  ( Base `  p ) ) }
 
Theoremislat 14116* The predicate "is a lattice." (Contributed by NM, 18-Oct-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( K  e.  Lat  <->  ( K  e.  Poset  /\  A. x  e.  B  A. y  e.  B  ( ( x  .\/  y
 )  e.  B  /\  ( x  ./\  y )  e.  B ) ) )
 
Theoremlatlem 14117 Lemma for lattice properties. (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .\/  Y )  e.  B  /\  ( X  ./\  Y )  e.  B ) )
 
Theoremlatpos 14118 A lattice is a poset. (Contributed by NM, 17-Sep-2011.)
 |-  ( K  e.  Lat  ->  K  e.  Poset )
 
Theoremlatjcl 14119 Closure of join operation in a lattice. (chjcom 22046 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y )  e.  B )
 
Theoremlatmcl 14120 Closure of meet operation in a lattice. (incom 3336 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y )  e.  B )
 
Theoremislati 14121* Properties that determine a lattice. (Contributed by NM, 12-Sep-2011.)
 |-  K  e.  Poset   &    |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  (
 ( x  e.  B  /\  y  e.  B )  ->  ( x  .\/  y )  e.  B )   &    |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  ./\  y )  e.  B )   =>    |-  K  e.  Lat
 
Theoremlatref 14122 A lattice ordering is reflexive. (ssid 3172 analog.) (Contributed by NM, 8-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B ) 
 ->  X  .<_  X )
 
Theoremlatasymb 14123 A lattice ordering is asymetric. (eqss 3169 analog.) (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y 
 /\  Y  .<_  X )  <->  X  =  Y )
 )
 
Theoremlatasym 14124 A lattice ordering is asymetric. (eqss 3169 analog.) (Contributed by NM, 8-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y 
 /\  Y  .<_  X ) 
 ->  X  =  Y ) )
 
Theoremlattr 14125 A lattice ordering is transitive. (sstr 3162 analog.) (Contributed by NM, 17-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) )
 
Theoremlatasymd 14126 Deduce equality from lattice ordering. (eqssd 3171 analog.) (Contributed by NM, 18-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ( ph  ->  K  e.  Lat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  X 
 .<_  Y )   &    |-  ( ph  ->  Y 
 .<_  X )   =>    |-  ( ph  ->  X  =  Y )
 
Theoremlattrd 14127 A lattice ordering is transitive. Deduction version of lattr 14125. (Contributed by NM, 3-Sep-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ( ph  ->  K  e.  Lat )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  X  .<_  Y )   &    |-  ( ph  ->  Y  .<_  Z )   =>    |-  ( ph  ->  X  .<_  Z )
 
Theoremlatjcom 14128 The join of a lattice commutes. (chjcom 22046 analog.) (Contributed by NM, 16-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y )  =  ( Y 
 .\/  X ) )
 
Theoremlatlej1 14129 A join's first argument is less than or equal to the join. (chub1 22047 analog.) (Contributed by NM, 17-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  ( X 
 .\/  Y ) )
 
Theoremlatlej2 14130 A join's second argument is less than or equal to the join. (chub2 22048 analog.) (Contributed by NM, 17-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  Y  .<_  ( X 
 .\/  Y ) )
 
Theoremlatjle12 14131 A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 22049 analog.) (Contributed by NM, 17-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .<_  Z 
 /\  Y  .<_  Z )  <-> 
 ( X  .\/  Y )  .<_  Z ) )
 
Theoremlatleeqj1 14132 Less-than-or-equal-to in terms of join. (chlejb1 22052 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <-> 
 ( X  .\/  Y )  =  Y )
 )
 
Theoremlatleeqj2 14133 Less-than-or-equal-to in terms of join. (chlejb2 22053 analog.) (Contributed by NM, 14-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <-> 
 ( Y  .\/  X )  =  Y )
 )
 
Theoremlatjlej1 14134 Add join to both sides of a lattice ordering. (chlej1i 22013 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( X  .<_  Y  ->  ( X  .\/  Z )  .<_  ( Y  .\/  Z ) ) )
 
Theoremlatjlej2 14135 Add join to both sides of a lattice ordering. (chlej2i 22014 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( X  .<_  Y  ->  ( Z  .\/  X )  .<_  ( Z  .\/  Y ) ) )
 
Theoremlatjlej12 14136 Add join to both sides of a lattice ordering. (chlej12i 22015 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  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 )
 ) )
 
Theoremlatnlej 14137 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 14138 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 14139 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 14140 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 14141 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 14142 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 14143 Lattice join is idempotent. (chjidm 22060 analog.) (Contributed by NM, 8-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X 
 .\/  X )  =  X )
 
Theoremlatmcom 14144 The join of a lattice commutes. (incom 3336 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 14145 A meet is less than or equal to its first argument. (inss1 3364 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 14146 A meet is less than or equal to its second argument. (inss2 3365 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 14147 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 3366 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 14148 Less-than-or-equal-to in terms of meet. (df-ss 3141 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 14149 Less-than-or-equal-to in terms of meet. (sseqin2 3363 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 14150 Add meet to both sides of a lattice ordering. (ssrin 3369 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 14151 Add meet to both sides of a lattice ordering. (sslin 3370 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 14152 Add join to both sides of a lattice ordering. (ss2in 3371 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 14153 Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3376 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 14154 Equivalent expressions for "not less than" in a lattice. (chnle 22054 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 14155 Lattice join is idempotent. (inidm 3353 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X 
 ./\  X )  =  X )
 
Theoremlatabs1 14156 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 22056 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 14157 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 22057 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 14158 An ortholattice is distributive in one ordering direction. (ledi 22080 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 14159 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 14160 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 14161 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 14162 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 14163 The least upper bound of a singleton. (chsupsn 21953 analog.) (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  Lat  /\  X  e.  B ) 
 ->  ( U `  { X } )  =  X )
 
Theoremlatjass 14164 Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 22073 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 14165 Swap 1st and 2nd members of lattice join. (chj12 22074 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 14166 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 14167 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 14168 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 14169 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 14170 Rearrangement of lattice join of 4 classes. (chj4 22075 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 14171 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 14172 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 14173 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 14174 The weak direction of the modular law (e.g. pmod1i 29287, atmod1i1 29296) 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 14175 The weak direction of the modular law (e.g. pmod2iN 29288) 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 14176 Extend class notation with complete lattices.
 class  CLat
 
Definitiondf-clat 14177* 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 14178* 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 14179 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 14180 LUB always exists in a complete lattice. (chsupcl 21880 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( U `  S )  e.  B )
 
Theoremclatglbcl 14181 GLB always exists in a complete lattice. (chintcl 21872 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( G `  S )  e.  B )
 
Theoremisclati 14182* 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 14183 A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  CLat  ->  K  e.  Lat )
 
Theoremisglbd 14184* 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 14185* 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 14186 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 14187* 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 14188 Subset law for least upper bounds. (chsupss 21882 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 14189 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 14190 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 14191* 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 14192 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 14193* 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 14194 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 14195 Class function defining dual orders.
 class ODual
 
Definitiondf-odu 14196 Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 14200, oduleval 14198, and oduleg 14199 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 14254. (Contributed by Stefan O'Rear, 29-Jan-2015.)

 |- ODual  =  ( w  e.  _V  |->  ( w sSet  <. ( le ` 
 ndx ) ,  `' ( le `  w )
 >. ) )
 
Theoremoduval 14197 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 14198 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 14199 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 14200 Base set of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  B  =  (
 Base `  O )   =>    |-  B  =  (
 Base `  D )
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