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Theorem List for Metamath Proof Explorer - 14101-14200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempltval3 14101 Alternate expression for less-than relation. (dfpss3 3262 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .<  =  ( lt `  K )   =>    |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <-> 
 ( X  .<_  Y  /\  -.  Y  .<_  X ) ) )
 
Theorempltnlt 14102 The less-than relation implies the negation of its inverse. (Contributed by NM, 18-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   =>    |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  -.  Y  .<  X )
 
Theorempltn2lp 14103 The less-than relation has no 2-cycle loops. (pssn2lp 3277 analog.) (Contributed by NM, 2-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   =>    |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  -.  ( X  .<  Y  /\  Y  .<  X ) )
 
Theoremplttr 14104 The less-than relation is transitive. (psstr 3280 analog.) (Contributed by NM, 2-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  X  .<  Z ) )
 
Theorempltletr 14105 Transitive law for chained less-than and less-than-or-equal. (psssstr 3282 analog.) (Contributed by NM, 2-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .<  =  ( lt `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .<  Y  /\  Y  .<_  Z )  ->  X  .<  Z ) )
 
Theoremplelttr 14106 Transitive law for chained less-than-or-equal and less-than. (sspsstr 3281 analog.) (Contributed by NM, 2-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .<  =  ( lt `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .<_  Y  /\  Y  .<  Z )  ->  X  .<  Z ) )
 
Theorempospo 14107 Write a poset structure in terms of the proper-class poset predicate (strict less than version). (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .<  =  ( lt `  K )   =>    |-  ( K  e.  V  ->  ( K  e.  Poset  <->  ( 
 .<  Po  B  /\  (  _I  |`  B )  C_  .<_  ) ) )
 
Definitiondf-lub 14108* Define poset least upper bound. If it doesn't exist, an undefined value not in the base set is returned. (Contributed by NM, 12-Sep-2011.)
 |- 
 lub  =  ( p  e.  _V  |->  ( s  e. 
 ~P ( Base `  p )  |->  ( iota_ x  e.  ( Base `  p )
 ( A. y  e.  s  y ( le `  p ) x  /\  A. z  e.  ( Base `  p )
 ( A. y  e.  s  y ( le `  p ) z  ->  x ( le `  p ) z ) ) ) ) )
 
Definitiondf-glb 14109* Define poset greatest lower bound. (Contributed by NM, 19-Jul-2012.)
 |- 
 glb  =  ( p  e.  _V  |->  ( s  e. 
 ~P ( Base `  p )  |->  ( iota_ x  e.  ( Base `  p )
 ( A. y  e.  s  x ( le `  p ) y  /\  A. z  e.  ( Base `  p )
 ( A. y  e.  s  z ( le `  p ) y  ->  z ( le `  p ) x ) ) ) ) )
 
Definitiondf-join 14110* Define poset join. (Contributed by NM, 12-Sep-2011.)
 |- 
 join  =  ( p  e.  _V  |->  ( x  e.  ( Base `  p ) ,  y  e.  ( Base `  p )  |->  ( ( lub `  p ) `  { x ,  y } ) ) )
 
Definitiondf-meet 14111* Define poset meet. (Contributed by NM, 12-Sep-2011.)
 |- 
 meet  =  ( p  e.  _V  |->  ( x  e.  ( Base `  p ) ,  y  e.  ( Base `  p )  |->  ( ( glb `  p ) `  { x ,  y } ) ) )
 
Theoremlubfval 14112* Value of least upper bound function of a poset. (Contributed by NM, 12-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  ( K  e.  A  ->  U  =  ( s  e. 
 ~P B  |->  ( iota_ x  e.  B ( A. y  e.  s  y  .<_  x  /\  A. z  e.  B  ( A. y  e.  s  y  .<_  z 
 ->  x  .<_  z ) ) ) ) )
 
Theoremlubval 14113* Value of least upper bound of a poset. (Contributed by NM, 12-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  A  /\  S  C_  B )  ->  ( U `  S )  =  ( iota_ x  e.  B ( A. y  e.  S  y  .<_  x  /\  A. z  e.  B  (
 A. y  e.  S  y  .<_  z  ->  x  .<_  z ) ) ) )
 
Theoremlubprop 14114* Properties of greatest lower bound of a poset. (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  A  /\  S  C_  B  /\  ( U `  S )  e.  B )  ->  ( A. y  e.  S  y  .<_  ( U `  S )  /\  A. z  e.  B  ( A. y  e.  S  y  .<_  z  ->  ( U `  S ) 
 .<_  z ) ) )
 
Theoremluble 14115 A greatest lower bound is a least element. (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( ( K  e.  A  /\  S  C_  B )  /\  ( ( U `
  S )  e.  B  /\  X  e.  S ) )  ->  X  .<_  ( U `  S ) )
 
Theoremlubid 14116* The LUB of elements less than or equal to a fixed value equals that value. (Contributed by NM, 19-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  Poset  /\  X  e.  B ) 
 ->  ( U `  { y  e.  B  |  y  .<_  X } )  =  X )
 
Theoremglbfval 14117* Value of least upper bound function of a poset. (Contributed by NM, 19-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   =>    |-  ( K  e.  A  ->  G  =  ( s  e. 
 ~P B  |->  ( iota_ x  e.  B ( A. y  e.  s  x  .<_  y  /\  A. z  e.  B  ( A. y  e.  s  z  .<_  y 
 ->  z  .<_  x ) ) ) ) )
 
Theoremglbval 14118* Value of greatest lower bound of a poset. (Contributed by NM, 19-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  A  /\  S  C_  B )  ->  ( G `  S )  =  ( iota_ x  e.  B ( A. y  e.  S  x  .<_  y  /\  A. z  e.  B  (
 A. y  e.  S  z  .<_  y  ->  z  .<_  x ) ) ) )
 
Theoremglbprop 14119* Properties of greatest lower bound of a poset. (Contributed by NM, 19-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  A  /\  S  C_  B  /\  ( G `  S )  e.  B )  ->  ( A. y  e.  S  ( G `  S ) 
 .<_  y  /\  A. z  e.  B  ( A. y  e.  S  z  .<_  y  ->  z  .<_  ( G `  S ) ) ) )
 
Theoremglble 14120 A greatest lower bound is a least element. (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( ( K  e.  A  /\  S  C_  B )  /\  ( ( G `
  S )  e.  B  /\  X  e.  S ) )  ->  ( G `  S ) 
 .<_  X )
 
Theoremjoinfval 14121* Value of join function for a poset. (Contributed by NM, 12-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( K  e.  A  ->  .\/ 
 =  ( x  e.  B ,  y  e.  B  |->  ( U `  { x ,  y }
 ) ) )
 
Theoremjoinval 14122 Value of join for a poset. (Contributed by NM, 12-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  (
 ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y )  =  ( U ` 
 { X ,  Y } ) )
 
Theoremjoinval2 14123* 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 14124* 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 14125 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 14126 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 14127 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 14128* 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 14129 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 14130* 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 14131* 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 14132 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 14133 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 14134 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 14135 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 14136 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 14137 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 14138 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 14139 Extend class notation with the class of all tosets.
 class Toset
 
Definitiondf-toset 14140* 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 14141* 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 14142 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 14143 Extend class notation with poset zero.
 class  0.
 
Syntaxcp1 14144 Extend class notation with poset unit.
 class  1.
 
Definitiondf-p0 14145 Define poset zero. (Contributed by NM, 12-Oct-2011.)
 |- 
 0.  =  ( p  e.  _V  |->  ( ( glb `  p ) `  ( Base `  p )
 ) )
 
Definitiondf-p1 14146 Define poset unit. (Contributed by NM, 22-Oct-2011.)
 |- 
 1.  =  ( p  e.  _V  |->  ( ( lub `  p ) `  ( Base `  p )
 ) )
 
Theoremp0val 14147 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 14148 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 14149 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 14150 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 14151 Extend class notation with the class of all lattices.
 class  Lat
 
Definitiondf-lat 14152* 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 14153* 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 14154 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 14155 A lattice is a poset. (Contributed by NM, 17-Sep-2011.)
 |-  ( K  e.  Lat  ->  K  e.  Poset )
 
Theoremlatjcl 14156 Closure of join operation in a lattice. (chjcom 22085 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 14157 Closure of meet operation in a lattice. (incom 3361 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 14158* 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 14159 A lattice ordering is reflexive. (ssid 3197 analog.) (Contributed by NM, 8-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B ) 
 ->  X  .<_  X )
 
Theoremlatasymb 14160 A lattice ordering is asymetric. (eqss 3194 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 14161 A lattice ordering is asymetric. (eqss 3194 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 14162 A lattice ordering is transitive. (sstr 3187 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 14163 Deduce equality from lattice ordering. (eqssd 3196 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 14164 A lattice ordering is transitive. Deduction version of lattr 14162. (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 14165 The join of a lattice commutes. (chjcom 22085 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 14166 A join's first argument is less than or equal to the join. (chub1 22086 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 14167 A join's second argument is less than or equal to the join. (chub2 22087 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 14168 A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 22088 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 14169 Less-than-or-equal-to in terms of join. (chlejb1 22091 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 14170 Less-than-or-equal-to in terms of join. (chlejb2 22092 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 14171 Add join to both sides of a lattice ordering. (chlej1i 22052 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 14172 Add join to both sides of a lattice ordering. (chlej2i 22053 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 14173 Add join to both sides of a lattice ordering. (chlej12i 22054 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 14174 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 14175 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 14176 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 14177 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 14178 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 14179 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 14180 Lattice join is idempotent. (chjidm 22099 analog.) (Contributed by NM, 8-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X 
 .\/  X )  =  X )
 
Theoremlatmcom 14181 The join of a lattice commutes. (incom 3361 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 14182 A meet is less than or equal to its first argument. (inss1 3389 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 14183 A meet is less than or equal to its second argument. (inss2 3390 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 14184 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 3391 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 14185 Less-than-or-equal-to in terms of meet. (df-ss 3166 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 14186 Less-than-or-equal-to in terms of meet. (sseqin2 3388 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 14187 Add meet to both sides of a lattice ordering. (ssrin 3394 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 14188 Add meet to both sides of a lattice ordering. (sslin 3395 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 14189 Add join to both sides of a lattice ordering. (ss2in 3396 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 14190 Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3401 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 14191 Equivalent expressions for "not less than" in a lattice. (chnle 22093 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 14192 Lattice join is idempotent. (inidm 3378 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X 
 ./\  X )  =  X )
 
Theoremlatabs1 14193 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 22095 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 14194 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 22096 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 14195 An ortholattice is distributive in one ordering direction. (ledi 22119 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 14196 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 14197 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 14198 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 14199 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 14200 The least upper bound of a singleton. (chsupsn 21992 analog.) (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  Lat  /\  X  e.  B ) 
 ->  ( U `  { X } )  =  X )
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