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Theorem List for Metamath Proof Explorer - 14001-14100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremyonedalem3b 14001* Lemma for yoneda 14005. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  ( SetCat `  U )   &    |-  T  =  (
 SetCat `  V )   &    |-  Q  =  ( O FuncCat  S )   &    |-  H  =  (HomF `  Q )   &    |-  R  =  ( ( Q  X.c  O ) FuncCat  T )   &    |-  E  =  ( O evalF 
 S )   &    |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
 ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  V  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U )  C_  V )   &    |-  ( ph  ->  F  e.  ( O  Func  S ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  G  e.  ( O  Func  S ) )   &    |-  ( ph  ->  P  e.  B )   &    |-  ( ph  ->  A  e.  ( F ( O Nat  S ) G ) )   &    |-  ( ph  ->  K  e.  ( P (  Hom  `  C ) X ) )   &    |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y ) `  x ) ( O Nat  S ) f )  |->  ( ( a `
  x ) `  (  .1.  `  x )
 ) ) )   =>    |-  ( ph  ->  ( ( G M P ) ( <. ( F ( 1st `  Z ) X ) ,  ( G ( 1st `  Z ) P ) >. (comp `  T ) ( G ( 1st `  E ) P ) ) ( A ( <. F ,  X >. ( 2nd `  Z ) <. G ,  P >. ) K ) )  =  ( ( A ( <. F ,  X >. ( 2nd `  E ) <. G ,  P >. ) K ) (
 <. ( F ( 1st `  Z ) X ) ,  ( F ( 1st `  E ) X ) >. (comp `  T ) ( G ( 1st `  E ) P ) ) ( F M X ) ) )
 
Theoremyonedalem3 14002* Lemma for yoneda 14005. (Contributed by Mario Carneiro, 28-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  ( SetCat `  U )   &    |-  T  =  (
 SetCat `  V )   &    |-  Q  =  ( O FuncCat  S )   &    |-  H  =  (HomF `  Q )   &    |-  R  =  ( ( Q  X.c  O ) FuncCat  T )   &    |-  E  =  ( O evalF 
 S )   &    |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
 ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  V  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U )  C_  V )   &    |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y ) `  x ) ( O Nat  S ) f )  |->  ( ( a `
  x ) `  (  .1.  `  x )
 ) ) )   =>    |-  ( ph  ->  M  e.  ( Z ( ( Q  X.c  O ) Nat  T ) E ) )
 
Theoremyonedainv 14003* The Yoneda Lemma with explicit inverse. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  ( SetCat `  U )   &    |-  T  =  (
 SetCat `  V )   &    |-  Q  =  ( O FuncCat  S )   &    |-  H  =  (HomF `  Q )   &    |-  R  =  ( ( Q  X.c  O ) FuncCat  T )   &    |-  E  =  ( O evalF 
 S )   &    |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
 ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  V  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U )  C_  V )   &    |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y ) `  x ) ( O Nat  S ) f )  |->  ( ( a `
  x ) `  (  .1.  `  x )
 ) ) )   &    |-  I  =  (Inv `  R )   &    |-  N  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( u  e.  ( ( 1st `  f ) `  x )  |->  ( y  e.  B  |->  ( g  e.  ( y ( 
 Hom  `  C ) x )  |->  ( ( ( x ( 2nd `  f ) y ) `
  g ) `  u ) ) ) ) )   =>    |-  ( ph  ->  M ( Z I E ) N )
 
Theoremyonffthlem 14004* Lemma for yonffth 14006. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  ( SetCat `  U )   &    |-  T  =  (
 SetCat `  V )   &    |-  Q  =  ( O FuncCat  S )   &    |-  H  =  (HomF `  Q )   &    |-  R  =  ( ( Q  X.c  O ) FuncCat  T )   &    |-  E  =  ( O evalF 
 S )   &    |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
 ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  V  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U )  C_  V )   &    |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y ) `  x ) ( O Nat  S ) f )  |->  ( ( a `
  x ) `  (  .1.  `  x )
 ) ) )   &    |-  I  =  (Inv `  R )   &    |-  N  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( u  e.  ( ( 1st `  f ) `  x )  |->  ( y  e.  B  |->  ( g  e.  ( y ( 
 Hom  `  C ) x )  |->  ( ( ( x ( 2nd `  f ) y ) `
  g ) `  u ) ) ) ) )   =>    |-  ( ph  ->  Y  e.  ( ( C Full  Q )  i^i  ( C Faith  Q ) ) )
 
Theoremyoneda 14005* The Yoneda Lemma. There is a natural isomorphism between the functors  Z and  E, where  Z ( F ,  X ) is the natural transformations from Yon ( X )  =  Hom  (  -  ,  X ) to  F, and  E ( F ,  X )  =  F ( X ) is the evaluation functor. Here we need two universes to state the claim: the smaller universe  U is used for forming the functor category  Q  =  C op  ->  SetCat ( U ), which itself does not (necessarily) live in  U but instead is an element of the larger universe  V. (If  U is a Grothendieck universe, then it will be closed under this "presheaf" operation, and so we can set  U  =  V in this case.) (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  B  =  (
 Base `  C )   &    |-  .1.  =  ( Id `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  ( SetCat `  U )   &    |-  T  =  (
 SetCat `  V )   &    |-  Q  =  ( O FuncCat  S )   &    |-  H  =  (HomF `  Q )   &    |-  R  =  ( ( Q  X.c  O ) FuncCat  T )   &    |-  E  =  ( O evalF 
 S )   &    |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
 ) )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  V  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U )  C_  V )   &    |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y ) `  x ) ( O Nat  S ) f )  |->  ( ( a `
  x ) `  (  .1.  `  x )
 ) ) )   &    |-  I  =  (  Iso  `  R )   =>    |-  ( ph  ->  M  e.  ( Z I E ) )
 
Theoremyonffth 14006 The Yoneda Lemma. The Yoneda embedding, the curried Hom functor, is full and faithful, and hence is a representation of the category  C as a full subcategory of the category  Q of presheaves on  C. (Contributed by Mario Carneiro, 29-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  (
 SetCat `  U )   &    |-  Q  =  ( O FuncCat  S )   &    |-  ( ph  ->  C  e.  Cat )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   =>    |-  ( ph  ->  Y  e.  ( ( C Full  Q )  i^i  ( C Faith  Q ) ) )
 
Theoremyoniso 14007* If the codomain is recoverable from a hom-set, then the Yoneda embedding is injective on objects, and hence is an isomorphism from  C into a full subcategory of a presheaf category. (Contributed by Mario Carneiro, 30-Jan-2017.)
 |-  Y  =  (Yon `  C )   &    |-  O  =  (oppCat `  C )   &    |-  S  =  (
 SetCat `  U )   &    |-  D  =  (CatCat `  V )   &    |-  B  =  ( Base `  D )   &    |-  I  =  (  Iso  `  D )   &    |-  Q  =  ( O FuncCat  S )   &    |-  E  =  ( Qs 
 ran  ( 1st `  Y ) )   &    |-  ( ph  ->  V  e.  X )   &    |-  ( ph  ->  C  e.  B )   &    |-  ( ph  ->  U  e.  W )   &    |-  ( ph  ->  ran  (  Homf  `  C )  C_  U )   &    |-  ( ph  ->  E  e.  B )   &    |-  ( ( ph  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) ) 
 ->  ( F `  ( x (  Hom  `  C ) y ) )  =  y )   =>    |-  ( ph  ->  Y  e.  ( C I E ) )
 
PART 9  BASIC ORDER THEORY
 
9.1  Presets and directed sets using extensible structures
 
Syntaxcpreset 14008 Extend class notation with the class of all presets.
 class  Preset
 
Syntaxcdrs 14009 Extend class notation with the class of all directed sets.
 class Dirset
 
Definitiondf-preset 14010* Define the class of preordered sets (presets). A preset is a set equipped with a transitive and reflexive relation.

Preorders are a natural generalization of order for sets where there is a well-defined ordering, but it in some sense "fails to capture the whole story", in that there may be pairs of elements which are indistinguishable under the order. Two elements which are not equal but are less-or-equal to each other behave the same under all order operations and may be thought of as "tied".

A preorder can naturally be strengthened by requiring that there are no ties, resulting in a partial order, or by stating that all comparable pairs of elements are tied, resulting in an equivalence relation. Every preorder naturally factors into these two types; the tied relation on a preorder is an equivalence relation and the quotient under that relation is a partial order. (Contributed by FL, 17-Nov-2014.) (Revised by Stefan O'Rear, 31-Jan-2015.)

 |-  Preset  =  { f  | 
 [. ( Base `  f
 )  /  b ]. [. ( le `  f
 )  /  r ]. A. x  e.  b  A. y  e.  b  A. z  e.  b  ( x r x  /\  ( ( x r y  /\  y r z )  ->  x r z ) ) }
 
Definitiondf-drs 14011* Define the class of directed sets. A directed set is a nonempty preordered set where every pair of elements have some upper bound. Note that it is not required that there exist a least upper bound.

There is no consensus in the literature over whether directed sets are allowed to be empty. It is slightly more convenient for us if they are not. (Contributed by Stefan O'Rear, 1-Feb-2015.)

 |- Dirset  =  { f  e.  Preset  | 
 [. ( Base `  f
 )  /  b ]. [. ( le `  f
 )  /  r ]. ( b  =/=  (/)  /\  A. x  e.  b  A. y  e.  b  E. z  e.  b  ( x r z  /\  y r z ) ) }
 
Theoremisprs 14012* Property of being a preordered set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( K  e.  Preset  <->  ( K  e.  _V  /\  A. x  e.  B  A. y  e.  B  A. z  e.  B  ( x  .<_  x 
 /\  ( ( x 
 .<_  y  /\  y  .<_  z )  ->  x  .<_  z ) ) ) )
 
Theoremprslem 14013 Lemma for prsref 14014 and prstr 14015. (Contributed by Mario Carneiro, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  .<_  X  /\  ( ( X  .<_  Y  /\  Y  .<_  Z )  ->  X  .<_  Z ) ) )
 
Theoremprsref 14014 Less-or-equal is reflexive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Preset  /\  X  e.  B ) 
 ->  X  .<_  X )
 
Theoremprstr 14015 Less-or-equal is transitive in a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Preset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  .<_  Y  /\  Y  .<_  Z ) ) 
 ->  X  .<_  Z )
 
Theoremisdrs 14016* Property of being a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( K  e. Dirset  <->  ( K  e.  Preset  /\  B  =/=  (/)  /\  A. x  e.  B  A. y  e.  B  E. z  e.  B  ( x  .<_  z 
 /\  y  .<_  z ) ) )
 
Theoremdrsdir 14017* Direction of a directed set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e. Dirset  /\  X  e.  B  /\  Y  e.  B )  ->  E. z  e.  B  ( X  .<_  z  /\  Y  .<_  z ) )
 
Theoremdrsprs 14018 A directed set is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( K  e. Dirset  ->  K  e.  Preset  )
 
Theoremdrsbn0 14019 The base of a directed set is not empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   =>    |-  ( K  e. Dirset  ->  B  =/=  (/) )
 
Theoremdrsdirfi 14020* Any finite number of elements in a directed set have a common upper bound. Here is where the non-emptiness constraint in df-drs 14011 first comes into play; without it we would need an additional constraint that  X not be empty. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e. Dirset  /\  X  C_  B  /\  X  e.  Fin )  ->  E. y  e.  B  A. z  e.  X  z 
 .<_  y )
 
Theoremisdrs2 14021* Directed sets may be defined in terms of finite subsets. Again, without nonemptiness we would need to restrict to nonempty subsets here. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( K  e. Dirset  <->  ( K  e.  Preset  /\ 
 A. x  e.  ( ~P B  i^i  Fin ) E. y  e.  B  A. z  e.  x  z 
 .<_  y ) )
 
9.2  Posets and lattices using extensible structures
 
9.2.1  Posets
 
Syntaxcpo 14022 Extend class notation with the class of posets.
 class  Poset
 
Syntaxcplt 14023 Extend class notation with less-than for posets.
 class  lt
 
Syntaxclub 14024 Extend class notation with poset least upper bound.
 class  lub
 
Syntaxcglb 14025 Extend class notation with poset greatest lower bound.
 class  glb
 
Syntaxcjn 14026 Extend class notation with poset join.
 class  join
 
Syntaxcmee 14027 Extend class notation with poset meet.
 class  meet
 
Definitiondf-poset 14028* 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 2793 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 14029* 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 14030* 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 14031 A poset is a preset. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( K  e.  Poset  ->  K  e.  Preset  )
 
Theoremposi 14032 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 14033 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 14034 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 14035 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 14036 Technical lemma to simplify the statement of ipopos 14211. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 13132) and any relation partially orders an empty set. (Contributed by Stefan O'Rear, 30-Jan-2015.)
 |-  (/)  e.  Poset
 
Theoremisposd 14037* 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 14038* 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 14039* 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 14040 Define less-than ordering for posets and related structures. Unlike df-base 13101 and df-ple 13176, 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 14041 Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |- 
 .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   =>    |-  ( K  e.  A  -> 
 .<  =  (  .<_  \  _I  ) )
 
Theorempltval 14042 Less-than relation. (df-pss 3129 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 14043 Less-than implies less-than-or-equal. (pssss 3232 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 14044 Less-than relation. (df-pss 3129 analog.) (Contributed by NM, 2-Dec-2011.)
 |- 
 .<  =  ( lt `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  C ) 
 ->  ( X  .<  Y  ->  X  =/=  Y ) )
 
Theorempltirr 14045 The less-than relation is not reflexive. (pssirr 3237 analog.) (Contributed by NM, 7-Feb-2012.)
 |- 
 .<  =  ( lt `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B )  ->  -.  X  .<  X )
 
Theorempleval2i 14046 One direction of pleval2 14047. (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 14047 Less-than-or-equal in terms of less-than. (sspss 3236 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 14048 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 14049 Alternate expression for less-than relation. (dfpss3 3223 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 14050 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 14051 The less-than relation has no 2-cycle loops. (pssn2lp 3238 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 14052 The less-than relation is transitive. (psstr 3241 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 14053 Transitive law for chained less-than and less-than-or-equal. (psssstr 3243 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 14054 Transitive law for chained less-than-or-equal and less-than. (sspsstr 3242 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 14055 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 14056* 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 14057* 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 14058* 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 14059* 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 14060* 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 14061* 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 14062* 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 14063 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 14064* 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 14065* 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 14066* 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 14067* 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 14068 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 14069* 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 14070 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 14071* 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 14072* 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 14073 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 14074 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 14075 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 14076* 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 14077 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 14078* 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 14079* 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 14080 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 14081 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 14082 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 14083 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 14084 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 14085 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 14086 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 14087 Extend class notation with the class of all tosets.
 class Toset
 
Definitiondf-toset 14088* 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 14089* 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 14090 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 14091 Extend class notation with poset zero.
 class  0.
 
Syntaxcp1 14092 Extend class notation with poset unit.
 class  1.
 
Definitiondf-p0 14093 Define poset zero. (Contributed by NM, 12-Oct-2011.)
 |- 
 0.  =  ( p  e.  _V  |->  ( ( glb `  p ) `  ( Base `  p )
 ) )
 
Definitiondf-p1 14094 Define poset unit. (Contributed by NM, 22-Oct-2011.)
 |- 
 1.  =  ( p  e.  _V  |->  ( ( lub `  p ) `  ( Base `  p )
 ) )
 
Theoremp0val 14095 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 14096 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 14097 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 14098 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 14099 Extend class notation with the class of all lattices.
 class  Lat
 
Definitiondf-lat 14100* 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 ) ) }
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