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Theorem List for Metamath Proof Explorer - 14401-14500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxclub 14401 Extend class notation with poset least upper bound.
 class  lub
 
Syntaxcglb 14402 Extend class notation with poset greatest lower bound.
 class  glb
 
Syntaxcjn 14403 Extend class notation with poset join.
 class  join
 
Syntaxcmee 14404 Extend class notation with poset meet.
 class  meet
 
Definitiondf-poset 14405* Define the class of posets. Definition of poset in [Crawley] p. 1. Note that Crawley-Dilworth require that a poset base set be nonempty, but we follow the convention of most authors who don't make this a requirement.

The quantifiers  E. b E. r provide a notational shorthand to allow us to refer to the base and ordering relation as  b and  r the definition rather than having to repeat  (
Base `  f ) and  ( le `  f ) throughout. These quantifiers can be eliminated with ceqsex2v 2995 and related theorems. (Contributed by NM, 18-Oct-2012.)

 |- 
 Poset  =  { f  |  E. b E. r
 ( b  =  (
 Base `  f )  /\  r  =  ( le `  f )  /\  A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y r x )  ->  x  =  y )  /\  (
 ( x r y 
 /\  y r z )  ->  x r
 z ) ) ) }
 
Theoremispos 14406* The predicate "is a poset." (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 4-Nov-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( K  e.  Poset  <->  ( K  e.  _V  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .<_  x 
 /\  ( ( x 
 .<_  y  /\  y  .<_  x )  ->  x  =  y )  /\  ( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) ) ) )
 
Theoremispos2 14407* A poset is an antisymmetric preset.

EDITORIAL: could become the definition of poset. (Contributed by Stefan O'Rear, 1-Feb-2015.)

 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( K  e.  Poset  <->  ( K  e.  Preset  /\  A. x  e.  B  A. y  e.  B  ( ( x 
 .<_  y  /\  y  .<_  x )  ->  x  =  y ) ) )
 
Theoremposprs 14408 A poset is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( K  e.  Poset  ->  K  e.  Preset  )
 
Theoremposi 14409 Lemma for poset properties. (Contributed by NM, 11-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y )  /\  (
 ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) )
 
Theoremposref 14410 A poset ordering is reflexive. (Contributed by NM, 11-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Poset  /\  X  e.  B ) 
 ->  X  .<_  X )
 
Theoremposasymb 14411 A poset ordering is asymetric. (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y 
 /\  Y  .<_  X )  <->  X  =  Y )
 )
 
Theorempostr 14412 A poset ordering is transitive. (Contributed by NM, 11-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) )
 
Theorem0pos 14413 Technical lemma to simplify the statement of ipopos 14588. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 13507) and any relation partially orders an empty set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  (/)  e.  Poset
 
Theoremisposd 14414* Properties that determine a poset (implicit structure version). (Contributed by Mario Carneiro, 29-Apr-2014.)
 |-  ( ph  ->  K  e.  _V )   &    |-  ( ph  ->  B  =  ( Base `  K ) )   &    |-  ( ph  ->  .<_  =  ( le `  K ) )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  x  .<_  x )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  (
 ( x  .<_  y  /\  y  .<_  x )  ->  x  =  y )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B )
 )  ->  ( ( x  .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) )   =>    |-  ( ph  ->  K  e.  Poset )
 
Theoremisposi 14415* Properties that determine a poset (implicit structure version). (Contributed by NM, 11-Sep-2011.)
 |-  K  e.  _V   &    |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ( x  e.  B  ->  x  .<_  x )   &    |-  (
 ( x  e.  B  /\  y  e.  B )  ->  ( ( x 
 .<_  y  /\  y  .<_  x )  ->  x  =  y ) )   &    |-  (
 ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  ( ( x 
 .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) )   =>    |-  K  e.  Poset
 
Theoremisposix 14416* Properties that determine a poset (explicit structure version). Note that the numeric indices of the structure components are not mentioned explicitly in either the theorem or its proof. (Contributed by NM, 9-Nov-2012.)
 |-  B  e.  _V   &    |-  .<_  e. 
 _V   &    |-  K  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( le ` 
 ndx ) ,  .<_  >. }   &    |-  ( x  e.  B  ->  x  .<_  x )   &    |-  (
 ( x  e.  B  /\  y  e.  B )  ->  ( ( x 
 .<_  y  /\  y  .<_  x )  ->  x  =  y ) )   &    |-  (
 ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  ( ( x 
 .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) )   =>    |-  K  e.  Poset
 
Definitiondf-plt 14417 Define less-than ordering for posets and related structures. Unlike df-base 13476 and df-ple 13551, this is a derived component extractor and not an extensible structure component extractor that defines the poset. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.)
 |- 
 lt  =  ( p  e.  _V  |->  ( ( le `  p ) 
 \  _I  ) )
 
Theorempltfval 14418 Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |- 
 .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   =>    |-  ( K  e.  A  -> 
 .<  =  (  .<_  \  _I  ) )
 
Theorempltval 14419 Less-than relation. (df-pss 3338 analog.) (Contributed by NM, 12-Oct-2011.)
 |- 
 .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  C ) 
 ->  ( X  .<  Y  <->  ( X  .<_  Y 
 /\  X  =/=  Y ) ) )
 
Theorempltle 14420 Less-than implies less-than-or-equal. (pssss 3444 analog.) (Contributed by NM, 4-Dec-2011.)
 |- 
 .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  C ) 
 ->  ( X  .<  Y  ->  X 
 .<_  Y ) )
 
Theorempltne 14421 Less-than relation. (df-pss 3338 analog.) (Contributed by NM, 2-Dec-2011.)
 |- 
 .<  =  ( lt `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  C ) 
 ->  ( X  .<  Y  ->  X  =/=  Y ) )
 
Theorempltirr 14422 The less-than relation is not reflexive. (pssirr 3449 analog.) (Contributed by NM, 7-Feb-2012.)
 |- 
 .<  =  ( lt `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B )  ->  -.  X  .<  X )
 
Theorempleval2i 14423 One direction of pleval2 14424. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .<  =  ( lt `  K )   =>    |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  ( X  .<  Y  \/  X  =  Y ) ) )
 
Theorempleval2 14424 Less-than-or-equal in terms of less-than. (sspss 3448 analog.) (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 8-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .<  =  ( lt `  K )   =>    |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <-> 
 ( X  .<  Y  \/  X  =  Y )
 ) )
 
Theorempltnle 14425 Less-than implies not inverse less-than-or-equal. (Contributed by NM, 18-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .<  =  ( lt `  K )   =>    |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  -.  Y  .<_  X )
 
Theorempltval3 14426 Alternate expression for less-than relation. (dfpss3 3435 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 14427 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 14428 The less-than relation has no 2-cycle loops. (pssn2lp 3450 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 14429 The less-than relation is transitive. (psstr 3453 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 14430 Transitive law for chained less-than and less-than-or-equal. (psssstr 3455 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 14431 Transitive law for chained less-than-or-equal and less-than. (sspsstr 3454 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 14432 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 14433* 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 14434* 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 14435* 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 14436* 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 14437* Value of the 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 14438* Value of the 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 14439* 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 14440 The greatest lower bound is the 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 14441* 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 14442* Value of the 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 14443* 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 14444* 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 14445 The greatest lower bound is the 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 14446* 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 14447 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 14448* 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 14449* 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 14450 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 14451 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 14452 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 14453* 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 14454 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 14455* 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 14456* 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 14457 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 14458 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 14459 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 14460 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 14461 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 14462 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 14463 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 14464 Extend class notation with the class of all tosets.
 class Toset
 
Definitiondf-toset 14465* 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 14466* 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 14467 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 14468 Extend class notation with poset zero.
 class  0.
 
Syntaxcp1 14469 Extend class notation with poset unit.
 class  1.
 
Definitiondf-p0 14470 Define poset zero. (Contributed by NM, 12-Oct-2011.)
 |- 
 0.  =  ( p  e.  _V  |->  ( ( glb `  p ) `  ( Base `  p )
 ) )
 
Definitiondf-p1 14471 Define poset unit. (Contributed by NM, 22-Oct-2011.)
 |- 
 1.  =  ( p  e.  _V  |->  ( ( lub `  p ) `  ( Base `  p )
 ) )
 
Theoremp0val 14472 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 14473 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 14474 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 14475 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 14476 Extend class notation with the class of all lattices.
 class  Lat
 
Definitiondf-lat 14477* 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 14478* 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 14479 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 14480 A lattice is a poset. (Contributed by NM, 17-Sep-2011.)
 |-  ( K  e.  Lat  ->  K  e.  Poset )
 
Theoremlatjcl 14481 Closure of join operation in a lattice. (chjcom 23010 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 14482 Closure of meet operation in a lattice. (incom 3535 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 14483* 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 14484 A lattice ordering is reflexive. (ssid 3369 analog.) (Contributed by NM, 8-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B ) 
 ->  X  .<_  X )
 
Theoremlatasymb 14485 A lattice ordering is asymetric. (eqss 3365 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 14486 A lattice ordering is asymetric. (eqss 3365 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 14487 A lattice ordering is transitive. (sstr 3358 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 14488 Deduce equality from lattice ordering. (eqssd 3367 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 14489 A lattice ordering is transitive. Deduction version of lattr 14487. (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 14490 The join of a lattice commutes. (chjcom 23010 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 14491 A join's first argument is less than or equal to the join. (chub1 23011 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 14492 A join's second argument is less than or equal to the join. (chub2 23012 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 14493 A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 23013 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 14494 Less-than-or-equal-to in terms of join. (chlejb1 23016 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 14495 Less-than-or-equal-to in terms of join. (chlejb2 23017 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 14496 Add join to both sides of a lattice ordering. (chlej1i 22977 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 14497 Add join to both sides of a lattice ordering. (chlej2i 22978 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 14498 Add join to both sides of a lattice ordering. (chlej12i 22979 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 14499 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 14500 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 )
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