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Theorem List for Metamath Proof Explorer - 14401-14500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremjoinval2 14401* Value of join for a poset with GLB expanded. (Contributed by NM, 16-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X 
 .\/  Y )  =  (
 iota_ x  e.  B ( ( X  .<_  x 
 /\  Y  .<_  x ) 
 /\  A. z  e.  B  ( ( X  .<_  z 
 /\  Y  .<_  z ) 
 ->  x  .<_  z ) ) ) )
 
Theoremjoinlem 14402* Lemma for join properties. (Contributed by NM, 16-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .\/  Y )  e.  B )  ->  ( ( X  .<_  ( X  .\/  Y )  /\  Y  .<_  ( X  .\/  Y ) )  /\  A. z  e.  B  (
 ( X  .<_  z  /\  Y  .<_  z )  ->  ( X  .\/  Y ) 
 .<_  z ) ) )
 
Theoremlejoin1 14403 A join's first argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .\/  Y )  e.  B )  ->  X  .<_  ( X  .\/  Y ) )
 
Theoremlejoin2 14404 A join's second argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  .\/  Y )  e.  B )  ->  Y  .<_  ( X  .\/  Y ) )
 
Theoremjoinle 14405 A join is less than or equal to a third value iff each argument is less than or equal to the third value. (Contributed by NM, 16-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  .\/  Y )  e.  B ) 
 ->  ( ( X  .<_  Z 
 /\  Y  .<_  Z )  <-> 
 ( X  .\/  Y )  .<_  Z ) )
 
Theoremmeetfval 14406* Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( K  e.  A  ->  ./\ 
 =  ( x  e.  B ,  y  e.  B  |->  ( G `  { x ,  y }
 ) ) )
 
Theoremmeetval 14407 Value of meet for a poset. (Contributed by NM, 12-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y )  =  ( G `  { X ,  Y }
 ) )
 
Theoremmeetval2 14408* Value of meet for a poset with GLB expanded. (Contributed by NM, 19-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X 
 ./\  Y )  =  (
 iota_ x  e.  B ( ( x  .<_  X 
 /\  x  .<_  Y ) 
 /\  A. z  e.  B  ( ( z  .<_  X 
 /\  z  .<_  Y ) 
 ->  z  .<_  x ) ) ) )
 
Theoremmeetlem 14409* Lemma for meet properties. (Contributed by NM, 19-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  e.  B )  ->  ( ( ( X 
 ./\  Y )  .<_  X  /\  ( X  ./\  Y ) 
 .<_  Y )  /\  A. z  e.  B  (
 ( z  .<_  X  /\  z  .<_  Y )  ->  z  .<_  ( X  ./\  Y ) ) ) )
 
Theoremlemeet1 14410 A meet's first argument is greater than or equal to the meet. (Contributed by NM, 19-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  e.  B )  ->  ( X  ./\  Y ) 
 .<_  X )
 
Theoremlemeet2 14411 A meet's second argument is greater than or equal to the meet. (Contributed by NM, 19-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y )  e.  B )  ->  ( X  ./\  Y ) 
 .<_  Y )
 
Theoremmeetle 14412 A meet is greater than or equal to a third value iff each argument is greater than or equal to the third value. (Contributed by NM, 19-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  ./\  Y )  e.  B ) 
 ->  ( Z  .<_  ( X 
 ./\  Y )  <->  ( Z  .<_  X 
 /\  Z  .<_  Y ) ) )
 
TheoremjoincomALT 14413 The join of a poset commutes. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 16-Sep-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X 
 .\/  Y )  =  ( Y  .\/  X )
 )
 
Theoremjoincom 14414 The join of a poset commutes. (The antecedent  ( X  .\/  Y )  e.  B  /\  ( Y  .\/  X )  e.  B i.e. "the joins exist" could be omitted as an artifact of our particular join definition, but other definitions may require it.) (Contributed by NM, 16-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  ( ( X  .\/  Y )  e.  B  /\  ( Y 
 .\/  X )  e.  B ) )  ->  ( X 
 .\/  Y )  =  ( Y  .\/  X )
 )
 
TheoremmeetcomALT 14415 The meet of a poset commutes. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 17-Sep-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X 
 ./\  Y )  =  ( Y  ./\  X )
 )
 
Theoremmeetcom 14416 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 14417 Extend class notation with the class of all tosets.
 class Toset
 
Definitiondf-toset 14418* 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 14419* 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 14420 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 14421 Extend class notation with poset zero.
 class  0.
 
Syntaxcp1 14422 Extend class notation with poset unit.
 class  1.
 
Definitiondf-p0 14423 Define poset zero. (Contributed by NM, 12-Oct-2011.)
 |- 
 0.  =  ( p  e.  _V  |->  ( ( glb `  p ) `  ( Base `  p )
 ) )
 
Definitiondf-p1 14424 Define poset unit. (Contributed by NM, 22-Oct-2011.)
 |- 
 1.  =  ( p  e.  _V  |->  ( ( lub `  p ) `  ( Base `  p )
 ) )
 
Theoremp0val 14425 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 14426 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 14427 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 14428 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 14429 Extend class notation with the class of all lattices.
 class  Lat
 
Definitiondf-lat 14430* 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 14431* 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 14432 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 14433 A lattice is a poset. (Contributed by NM, 17-Sep-2011.)
 |-  ( K  e.  Lat  ->  K  e.  Poset )
 
Theoremlatjcl 14434 Closure of join operation in a lattice. (chjcom 22961 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 14435 Closure of meet operation in a lattice. (incom 3493 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 14436* 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 14437 A lattice ordering is reflexive. (ssid 3327 analog.) (Contributed by NM, 8-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B ) 
 ->  X  .<_  X )
 
Theoremlatasymb 14438 A lattice ordering is asymetric. (eqss 3323 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 14439 A lattice ordering is asymetric. (eqss 3323 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 14440 A lattice ordering is transitive. (sstr 3316 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 14441 Deduce equality from lattice ordering. (eqssd 3325 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 14442 A lattice ordering is transitive. Deduction version of lattr 14440. (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 14443 The join of a lattice commutes. (chjcom 22961 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 14444 A join's first argument is less than or equal to the join. (chub1 22962 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 14445 A join's second argument is less than or equal to the join. (chub2 22963 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 14446 A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 22964 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 14447 Less-than-or-equal-to in terms of join. (chlejb1 22967 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 14448 Less-than-or-equal-to in terms of join. (chlejb2 22968 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 14449 Add join to both sides of a lattice ordering. (chlej1i 22928 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 14450 Add join to both sides of a lattice ordering. (chlej2i 22929 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 14451 Add join to both sides of a lattice ordering. (chlej12i 22930 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 14452 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 14453 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 14454 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 14455 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 14456 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 14457 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 14458 Lattice join is idempotent. (chjidm 22975 analog.) (Contributed by NM, 8-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X 
 .\/  X )  =  X )
 
Theoremlatmcom 14459 The join of a lattice commutes. (incom 3493 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 14460 A meet is less than or equal to its first argument. (inss1 3521 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 14461 A meet is less than or equal to its second argument. (inss2 3522 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 14462 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 3523 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 14463 Less-than-or-equal-to in terms of meet. (df-ss 3294 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 14464 Less-than-or-equal-to in terms of meet. (sseqin2 3520 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 14465 Add meet to both sides of a lattice ordering. (ssrin 3526 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 14466 Add meet to both sides of a lattice ordering. (sslin 3527 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 14467 Add join to both sides of a lattice ordering. (ss2in 3528 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 14468 Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3533 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 14469 Equivalent expressions for "not less than" in a lattice. (chnle 22969 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 14470 Lattice join is idempotent. (inidm 3510 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X 
 ./\  X )  =  X )
 
Theoremlatabs1 14471 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 22971 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 14472 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 22972 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 14473 An ortholattice is distributive in one ordering direction. (ledi 22995 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 14474 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 14475 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 14476 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 14477 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 14478 The least upper bound of a singleton. (chsupsn 22868 analog.) (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  Lat  /\  X  e.  B ) 
 ->  ( U `  { X } )  =  X )
 
Theoremlatjass 14479 Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 22988 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 14480 Swap 1st and 2nd members of lattice join. (chj12 22989 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 14481 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 14482 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 14483 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 14484 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 14485 Rearrangement of lattice join of 4 classes. (chj4 22990 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 14486 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 14487 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 14488 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 14489 The weak direction of the modular law (e.g. pmod1i 30330, atmod1i1 30339) 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 14490 The weak direction of the modular law (e.g. pmod2iN 30331) 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 14491 Extend class notation with complete lattices.
 class  CLat
 
Definitiondf-clat 14492* 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 14493* The predicate "is a 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 14494 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 14495 LUB always exists in a complete lattice. (chsupcl 22795 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( U `  S )  e.  B )
 
Theoremclatglbcl 14496 GLB always exists in a complete lattice. (chintcl 22787 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( G `  S )  e.  B )
 
Theoremisclati 14497* 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 14498 A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  CLat  ->  K  e.  Lat )
 
Theoremisglbd 14499* 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 14500* Lemma for the 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 ) ) )
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