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Theorem List for Metamath Proof Explorer - 14001-14100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlatmcl 14001 Closure of meet operation in a lattice. (incom 3269 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 14002* 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 14003 A lattice ordering is reflexive. (ssid 3118 analog.) (Contributed by NM, 8-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B ) 
 ->  X  .<_  X )
 
Theoremlatasymb 14004 A lattice ordering is asymetric. (eqss 3115 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 14005 A lattice ordering is asymetric. (eqss 3115 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 14006 A lattice ordering is transitive. (sstr 3108 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 14007 Deduce equality from lattice ordering. (eqssd 3117 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 14008 A lattice ordering is transitive. Deduction version of lattr 14006. (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 14009 The join of a lattice commutes. (chjcom 21915 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 14010 A join's first argument is less than or equal to the join. (chub1 21916 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 14011 A join's second argument is less than or equal to the join. (chub2 21917 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 14012 A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 21918 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 14013 Less-than-or-equal-to in terms of join. (chlejb1 21921 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 14014 Less-than-or-equal-to in terms of join. (chlejb2 21922 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 14015 Add join to both sides of a lattice ordering. (chlej1i 21882 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 14016 Add join to both sides of a lattice ordering. (chlej2i 21883 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 14017 Add join to both sides of a lattice ordering. (chlej12i 21884 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 14018 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 14019 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 14020 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 14021 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 14022 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 14023 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 14024 Lattice join is idempotent. (chjidm 21929 analog.) (Contributed by NM, 8-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X 
 .\/  X )  =  X )
 
Theoremlatmcom 14025 The join of a lattice commutes. (incom 3269 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 14026 A meet is less than or equal to its first argument. (inss1 3296 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 14027 A meet is less than or equal to its second argument. (inss2 3297 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 14028 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 3298 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 14029 Less-than-or-equal-to in terms of meet. (df-ss 3089 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 14030 Less-than-or-equal-to in terms of meet. (sseqin2 3295 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 14031 Add meet to both sides of a lattice ordering. (ssrin 3301 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 14032 Add meet to both sides of a lattice ordering. (sslin 3302 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 14033 Add join to both sides of a lattice ordering. (ss2in 3303 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 14034 Negation of less-than-or-equal-to expressed in terms of meet and less-than. (nssinpss 3308 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 14035 Equivalent expressions for "not less than" in a lattice. (chnle 21923 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 14036 Lattice join is idempotent. (inidm 3285 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X 
 ./\  X )  =  X )
 
Theoremlatabs1 14037 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 21925 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 14038 Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 21926 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 14039 An ortholattice is distributive in one ordering direction. (ledi 21949 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 14040 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 14041 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 14042 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 14043 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 14044 The least upper bound of a singleton. (chsupsn 21822 analog.) (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  Lat  /\  X  e.  B ) 
 ->  ( U `  { X } )  =  X )
 
Theoremlatjass 14045 Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 21942 analog.) (Contributed by NM, 17-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .\/  Y )  .\/  Z )  =  ( X  .\/  ( Y  .\/  Z ) ) )
 
Theoremlatj12 14046 Swap 1st and 2nd members of lattice join. (chj12 21943 analog.) (Contributed by NM, 4-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( X  .\/  ( Y 
 .\/  Z ) )  =  ( Y  .\/  ( X  .\/  Z ) ) )
 
Theoremlatj32 14047 Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 2-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .\/  Y )  .\/  Z )  =  ( ( X  .\/  Z )  .\/  Y )
 )
 
Theoremlatj13 14048 Swap 1sd and 3rd members of lattice join. (Contributed by NM, 4-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( X  .\/  ( Y 
 .\/  Z ) )  =  ( Z  .\/  ( Y  .\/  X ) ) )
 
Theoremlatj31 14049 Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 23-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .\/  Y )  .\/  Z )  =  ( ( Z  .\/  Y )  .\/  X )
 )
 
Theoremlatjrot 14050 Rotate lattice join of 3 classes. (Contributed by NM, 23-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .\/  Y )  .\/  Z )  =  ( ( Z  .\/  X )  .\/  Y )
 )
 
Theoremlatj4 14051 Rearrangement of lattice join of 4 classes. (chj4 21944 analog.) (Contributed by NM, 14-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B )
 )  ->  ( ( X  .\/  Y )  .\/  ( Z  .\/  W ) )  =  ( ( X  .\/  Z )  .\/  ( Y  .\/  W ) ) )
 
Theoremlatj4rot 14052 Rotate lattice join of 4 classes. (Contributed by NM, 11-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B )
 )  ->  ( ( X  .\/  Y )  .\/  ( Z  .\/  W ) )  =  ( ( W  .\/  X )  .\/  ( Y  .\/  Z ) ) )
 
Theoremlatjjdi 14053 Lattice join distributes over itself. (Contributed by NM, 30-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( X  .\/  ( Y 
 .\/  Z ) )  =  ( ( X  .\/  Y )  .\/  ( X  .\/  Z ) ) )
 
Theoremlatjjdir 14054 Lattice join distributes over itself. (Contributed by NM, 2-Aug-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   =>    |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .\/  Y )  .\/  Z )  =  ( ( X  .\/  Z )  .\/  ( Y  .\/  Z ) ) )
 
Theoremmod1ile 14055 The weak direction of the modular law (e.g. pmod1i 28941, atmod1i1 28950) that holds in any lattice. (Contributed by NM, 11-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  .<_  Z  ->  ( X  .\/  ( Y  ./\  Z ) )  .<_  ( ( X 
 .\/  Y )  ./\  Z ) ) )
 
Theoremmod2ile 14056 The weak direction of the modular law (e.g. pmod2iN 28942) that holds in any lattice. (Contributed by NM, 11-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( Z  .<_  X  ->  ( ( X  ./\  Y )  .\/  Z )  .<_  ( X  ./\  ( Y  .\/  Z ) ) ) )
 
Syntaxccla 14057 Extend class notation with complete lattices.
 class  CLat
 
Definitiondf-clat 14058* Define the class of all complete lattices. (Contributed by NM, 18-Oct-2012.)
 |- 
 CLat  =  { p  e.  Poset  |  A. s
 ( s  C_  ( Base `  p )  ->  ( ( ( lub `  p ) `  s
 )  e.  ( Base `  p )  /\  (
 ( glb `  p ) `  s )  e.  ( Base `  p ) ) ) }
 
Theoremisclat 14059* The predicate "is an complete lattice." (Contributed by NM, 18-Oct-2012.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  G  =  ( glb `  K )   =>    |-  ( K  e.  CLat  <->  ( K  e.  Poset  /\  A. s ( s  C_  B  ->  ( ( U `
  s )  e.  B  /\  ( G `
  s )  e.  B ) ) ) )
 
Theoremclatlem 14060 Lemma for properties of a complete lattice. (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  G  =  ( glb `  K )   =>    |-  ( ( K  e.  CLat  /\  S  C_  B )  ->  ( ( U `  S )  e.  B  /\  ( G `  S )  e.  B )
 )
 
Theoremclatlubcl 14061 LUB always exists in a complete lattice. (chsupcl 21749 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( U `  S )  e.  B )
 
Theoremclatglbcl 14062 GLB always exists in a complete lattice. (chintcl 21741 analog.) (Contributed by NM, 14-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( G `  S )  e.  B )
 
Theoremisclati 14063* Properties that determine a complete lattice. (Contributed by NM, 12-Sep-2011.)
 |-  K  e.  Poset   &    |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  G  =  ( glb `  K )   &    |-  ( s  C_  B  ->  ( U `  s
 )  e.  B )   &    |-  ( s  C_  B  ->  ( G `  s )  e.  B )   =>    |-  K  e.  CLat
 
Theoremclatl 14064 A complete lattice is a lattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  CLat  ->  K  e.  Lat )
 
Theoremisglbd 14065* Properties that determine the greatest lower bound of a complete lattice. (Contributed by Mario Carneiro, 19-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   &    |-  (
 ( ph  /\  y  e.  S )  ->  H  .<_  y )   &    |-  ( ( ph  /\  x  e.  B  /\  A. y  e.  S  x  .<_  y )  ->  x  .<_  H )   &    |-  ( ph  ->  K  e.  CLat )   &    |-  ( ph  ->  S 
 C_  B )   &    |-  ( ph  ->  H  e.  B )   =>    |-  ( ph  ->  ( G `  S )  =  H )
 
Theoremlublem 14066* Lemma for least upper bound properties in a complete lattice. (Contributed by NM, 19-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( A. y  e.  S  y  .<_  ( U `
  S )  /\  A. z  e.  B  (
 A. y  e.  S  y  .<_  z  ->  ( U `  S )  .<_  z ) ) )
 
Theoremlubub 14067 The LUB of a complete lattice subset is an upper bound. (Contributed by NM, 19-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B  /\  X  e.  S )  ->  X  .<_  ( U `  S ) )
 
Theoremlubl 14068* The LUB of a complete lattice subset is a least bound. (Contributed by NM, 19-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B  /\  X  e.  B )  ->  ( A. y  e.  S  y  .<_  X  ->  ( U `  S ) 
 .<_  X ) )
 
Theoremlubss 14069 Subset law for least upper bounds. (chsupss 21751 analog.) (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( U `  S ) 
 .<_  ( U `  T ) )
 
Theoremlubel 14070 An element of a set is less than or equal to the least upper bound of the set. (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   =>    |-  (
 ( K  e.  CLat  /\  X  e.  S  /\  S  C_  B )  ->  X  .<_  ( U `  S ) )
 
Theoremlubun 14071 The LUB of a union. (Contributed by NM, 5-Mar-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  U  =  ( lub `  K )   =>    |-  ( ( K  e.  CLat  /\  S  C_  B  /\  T  C_  B )  ->  ( U `  ( S  u.  T ) )  =  ( ( U `
  S )  .\/  ( U `  T ) ) )
 
Theoremclatglb 14072* Properties of greatest lower bound of a complete lattice. (Contributed by NM, 5-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B )  ->  ( A. y  e.  S  ( G `  S )  .<_  y  /\  A. z  e.  B  (
 A. y  e.  S  z  .<_  y  ->  z  .<_  ( G `  S ) ) ) )
 
Theoremclatglble 14073 A greatest lower bound is a least element. (Contributed by NM, 5-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  S  C_  B  /\  X  e.  S )  ->  ( G `  S )  .<_  X )
 
Theoremclatleglb 14074* Two ways of expressing "less than or equal to the greatest lower bound." (Contributed by NM, 5-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  X  e.  B  /\  S  C_  B )  ->  ( X  .<_  ( G `
  S )  <->  A. y  e.  S  X  .<_  y ) )
 
Theoremclatglbss 14075 Subset law for greatest lower bound. (Contributed by Mario Carneiro, 16-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  G  =  ( glb `  K )   =>    |-  (
 ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( G `  T ) 
 .<_  ( G `  S ) )
 
9.2.3  The dual of an ordered set
 
Syntaxcodu 14076 Class function defining dual orders.
 class ODual
 
Definitiondf-odu 14077 Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 14081, oduleval 14079, and oduleg 14080 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 14126. (Contributed by Stefan O'Rear, 29-Jan-2015.)

 |- ODual  =  ( w  e.  _V  |->  ( w sSet  <. ( le ` 
 ndx ) ,  `' ( le `  w )
 >. ) )
 
Theoremoduval 14078 Value of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  .<_  =  ( le `  O )   =>    |-  D  =  ( O sSet  <. ( le `  ndx ) ,  `'  .<_  >. )
 
Theoremoduleval 14079 Value of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  .<_  =  ( le `  O )   =>    |-  `'  .<_  =  ( le `  D )
 
Theoremoduleg 14080 Truth of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  .<_  =  ( le `  O )   &    |-  G  =  ( le `  D )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A G B 
 <->  B  .<_  A ) )
 
Theoremodubas 14081 Base set of an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  B  =  (
 Base `  O )   =>    |-  B  =  (
 Base `  D )
 
Theorempospropd 14082* Posethood is determined only by structure components and only by the value of the relation within the base set. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  ( ph  ->  K  e.  V )   &    |-  ( ph  ->  L  e.  W )   &    |-  ( ph  ->  B  =  (
 Base `  K ) )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( le `  K ) y  <->  x ( le `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  Poset 
 <->  L  e.  Poset ) )
 
Theoremodupos 14083 Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   =>    |-  ( O  e.  Poset  ->  D  e.  Poset )
 
Theoremoduposb 14084 Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   =>    |-  ( O  e.  V  ->  ( O  e.  Poset  <->  D  e.  Poset ) )
 
Theoremmeet0 14085 Lemma for odujoin 14090. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  ( meet `  (/) )  =  (/)
 
Theoremjoin0 14086 Lemma for odumeet 14088. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  ( join `  (/) )  =  (/)
 
Theoremoduglb 14087 Greatest lower bounds in a dual order are least upper bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  U  =  ( lub `  O )   =>    |-  ( O  e.  V  ->  U  =  ( glb `  D ) )
 
Theoremodumeet 14088 Meets in a dual order are joins in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  .\/  =  ( join `  O )   =>    |-  .\/  =  ( meet `  D )
 
Theoremodulub 14089 Least upper bounds in a dual order are greatest lower bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  L  =  ( glb `  O )   =>    |-  ( O  e.  V  ->  L  =  ( lub `  D ) )
 
Theoremodujoin 14090 Joins in a dual order are meets in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   &    |-  ./\  =  ( meet `  O )   =>    |-  ./\  =  ( join `  D )
 
Theoremodulatb 14091 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   =>    |-  ( O  e.  V  ->  ( O  e.  Lat  <->  D  e.  Lat ) )
 
Theoremoduclatb 14092 Being a complete lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   =>    |-  ( O  e.  CLat  <->  D  e.  CLat )
 
Theoremodulat 14093 Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
 |-  D  =  (ODual `  O )   =>    |-  ( O  e.  Lat  ->  D  e.  Lat )
 
Theoremposlubmo 14094* Least upper bounds in a poset are unique if they exist. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |- 
 .<_  =  ( le `  K )   &    |-  B  =  ( Base `  K )   =>    |-  ( ( K  e.  Poset  /\  S  C_  B )  ->  E* x ( x  e.  B  /\  ( A. y  e.  S  y  .<_  x  /\  A. z  e.  B  ( A. y  e.  S  y  .<_  z  ->  x  .<_  z ) ) ) )
 
Theoremposlubd 14095* Properties which determine a least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |- 
 .<_  =  ( le `  K )   &    |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  ( ph  ->  K  e.  Poset )   &    |-  ( ph  ->  S  C_  B )   &    |-  ( ph  ->  T  e.  B )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  x  .<_  T )   &    |-  ( ( ph  /\  y  e.  B  /\  A. x  e.  S  x  .<_  y ) 
 ->  T  .<_  y )   =>    |-  ( ph  ->  ( U `  S )  =  T )
 
Theoremposlubdg 14096* Properties which determine a least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |- 
 .<_  =  ( le `  K )   &    |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  U  =  ( lub `  K ) )   &    |-  ( ph  ->  K  e.  Poset )   &    |-  ( ph  ->  S 
 C_  B )   &    |-  ( ph  ->  T  e.  B )   &    |-  ( ( ph  /\  x  e.  S )  ->  x  .<_  T )   &    |-  ( ( ph  /\  y  e.  B  /\  A. x  e.  S  x  .<_  y )  ->  T  .<_  y )   =>    |-  ( ph  ->  ( U `  S )  =  T )
 
Theoremposglbd 14097* Properties which determine a greatest lower bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |- 
 .<_  =  ( le `  K )   &    |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  G  =  ( glb `  K ) )   &    |-  ( ph  ->  K  e.  Poset )   &    |-  ( ph  ->  S 
 C_  B )   &    |-  ( ph  ->  T  e.  B )   &    |-  ( ( ph  /\  x  e.  S )  ->  T  .<_  x )   &    |-  ( ( ph  /\  y  e.  B  /\  A. x  e.  S  y 
 .<_  x )  ->  y  .<_  T )   =>    |-  ( ph  ->  ( G `  S )  =  T )
 
9.2.4  Subset order structures
 
Syntaxcipo 14098 Class function defining inclusion posets.
 class toInc
 
Definitiondf-ipo 14099* For any family of sets, define the poset of that family ordered by inclusion. See ipobas 14102, ipolerval 14103, and ipole 14105 for its contract.

EDITORIAL: I'm not thrilled with the name. Any suggestions? (Contributed by Stefan O'Rear, 30-Jan-2015.) (New usage is discouraged.)

 |- toInc  =  ( f  e.  _V  |->  [_
 { <. x ,  y >.  |  ( { x ,  y }  C_  f  /\  x  C_  y ) }  /  o ]_ ( { <. ( Base `  ndx ) ,  f >. , 
 <. (TopSet `  ndx ) ,  (ordTop `  o ) >. }  u.  { <. ( le `  ndx ) ,  o >. ,  <. ( oc
 `  ndx ) ,  ( x  e.  f  |->  U. { y  e.  f  |  ( y  i^i  x )  =  (/) } ) >. } ) )
 
Theoremipostr 14100 The structure of df-ipo 14099 is a structure defining indexes up to 11. (Contributed by Mario Carneiro, 25-Oct-2015.)
 |-  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. (TopSet `  ndx ) ,  J >. }  u.  { <. ( le `  ndx ) ,  .<_  >. ,  <. ( oc `  ndx ) ,  ._|_  >. } ) Struct  <. 1 , ; 1 1 >.
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