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Theorem List for Metamath Proof Explorer - 32701-32800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremldual1dim 32701* Equivalent expressions for a 1-dim subspace (ray) of functionals. (Contributed by NM, 24-Oct-2014.)
 |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  N  =  ( LSpan `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( N `  { G }
 )  =  { g  e.  F  |  ( L `
  G )  C_  ( L `  g ) } )
 
Theoremldualkrsc 32702 The kernel of a non-zero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 28-Dec-2014.)
 |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  X  =/=  .0.  )   =>    |-  ( ph  ->  ( L `  ( X  .x.  G ) )  =  ( L `  G ) )
 
Theoremlkrss 32703 The kernel of a scalar product of a functional includes the kernel of the functional. (Contributed by NM, 27-Jan-2015.)
 |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  K )   =>    |-  ( ph  ->  ( L `  G ) 
 C_  ( L `  ( X  .x.  G ) ) )
 
Theoremlkrss2N 32704* Two functionals with kernels in a subset relationship. (Contributed by NM, 17-Feb-2015.) (New usage is discouraged.)
 |-  S  =  (Scalar `  W )   &    |-  R  =  ( Base `  S )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( ( K `  G )  C_  ( K `  H )  <->  E. r  e.  R  H  =  ( r  .x.  G ) ) )
 
TheoremlkreqN 32705 Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
 |-  S  =  (Scalar `  W )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .x.  =  ( .s `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  A  e.  ( R  \  {  .0.  } ) )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  G  =  ( A  .x.  H )
 )   =>    |-  ( ph  ->  ( K `  G )  =  ( K `  H ) )
 
TheoremlkrlspeqN 32706 Condition for colinear functionals to have equal kernels. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
 |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  D  =  (LDual `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  N  =  ( LSpan `  D )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  G  e.  (
 ( N `  { H } )  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( L `  G )  =  ( L `  H ) )
 
21.21.9  Ortholattices and orthomodular lattices
 
Syntaxcops 32707 Extend class notation with orthoposets.
 class  OP
 
SyntaxccmtN 32708 Extend class notation with the commutes relation.
 class  cm
 
Syntaxcol 32709 Extend class notation with orthlattices.
 class  OL
 
Syntaxcoml 32710 Extend class notation with orthomodular lattices.
 class  OML
 
Definitiondf-oposet 32711* Define the class of orthoposets, which are bounded posets with an orthocomplementation operation. Note that  (
Base p ) e. dom ( lub  p ) means there is an upper bound  1., and similarly for the  0. element. (Contributed by NM, 20-Oct-2011.) (Revised by NM, 13-Sep-2018.)
 |-  OP  =  { p  e.  Poset  |  ( ( ( Base `  p )  e.  dom  ( lub `  p )  /\  ( Base `  p )  e.  dom  ( glb `  p ) )  /\  E. o
 ( o  =  ( oc `  p ) 
 /\  A. a  e.  ( Base `  p ) A. b  e.  ( Base `  p ) ( ( ( o `  a
 )  e.  ( Base `  p )  /\  (
 o `  ( o `  a ) )  =  a  /\  ( a ( le `  p ) b  ->  ( o `
  b ) ( le `  p ) ( o `  a
 ) ) )  /\  ( a ( join `  p ) ( o `
  a ) )  =  ( 1. `  p )  /\  ( a (
 meet `  p ) ( o `  a ) )  =  ( 0. `  p ) ) ) ) }
 
Definitiondf-cmtN 32712* Define the commutes relation for orthoposets. Definition of commutes in [Kalmbach] p. 20. (Contributed by NM, 6-Nov-2011.)
 |-  cm  =  ( p  e.  _V  |->  {
 <. x ,  y >.  |  ( x  e.  ( Base `  p )  /\  y  e.  ( Base `  p )  /\  x  =  ( ( x (
 meet `  p ) y ) ( join `  p ) ( x (
 meet `  p ) ( ( oc `  p ) `  y ) ) ) ) } )
 
Definitiondf-ol 32713 Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
 |-  OL  =  ( Lat  i^i  OP )
 
Definitiondf-oml 32714* Define the class of orthomodular lattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
 |-  OML  =  { l  e.  OL  |  A. a  e.  ( Base `  l ) A. b  e.  ( Base `  l ) ( a ( le `  l
 ) b  ->  b  =  ( a ( join `  l ) ( b ( meet `  l )
 ( ( oc `  l ) `  a
 ) ) ) ) }
 
Theoremisopos 32715* The predicate "is an orthoposet." (Contributed by NM, 20-Oct-2011.) (Revised by NM, 14-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  G  =  ( glb `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( K  e.  OP  <->  (
 ( K  e.  Poset  /\  B  e.  dom  U  /\  B  e.  dom  G )  /\  A. x  e.  B  A. y  e.  B  ( ( ( 
 ._|_  `  x )  e.  B  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x  /\  ( x  .<_  y  ->  (  ._|_  `  y )  .<_  (  ._|_  `  x ) ) )  /\  ( x  .\/  (  ._|_  `  x ) )  =  .1.  /\  ( x  ./\  (  ._|_  `  x ) )  =  .0.  ) ) )
 
Theoremopposet 32716 Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
 |-  ( K  e.  OP  ->  K  e.  Poset )
 
Theoremoposlem 32717 Lemma for orthoposet properties. (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( ( (  ._|_  `  X )  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X  .<_  Y  ->  (  ._|_  `  Y )  .<_  ( 
 ._|_  `  X ) ) )  /\  ( X 
 .\/  (  ._|_  `  X ) )  =  .1.  /\  ( X  ./\  (  ._|_  `  X ) )  =  .0.  ) )
 
Theoremop01dm 32718 Conditions necessary for zero and unit elements to exist. (Contributed by NM, 14-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  G  =  ( glb `  K )   =>    |-  ( K  e.  OP  ->  ( B  e.  dom  U 
 /\  B  e.  dom  G ) )
 
Theoremop0cl 32719 An orthoposet has a zero element. (h0elch 26906 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( K  e.  OP  ->  .0.  e.  B )
 
Theoremop1cl 32720 An orthoposet has a unit element. (helch 26894 analog.) (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( K  e.  OP  ->  .1.  e.  B )
 
Theoremop0le 32721 Orthoposet zero is less than or equal to any element. (ch0le 27092 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
 
Theoremople0 32722 An element less than or equal to zero equals zero. (chle0 27094 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .<_  .0.  <->  X  =  .0.  ) )
 
Theoremopnlen0 32723 An element not less than another is nonzero. TODO: Look for uses of necon3bd 2632 and op0le 32721 to see if this is useful elsewhere. (Contributed by NM, 5-May-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  /\  -.  X  .<_  Y )  ->  X  =/=  .0.  )
 
Theoremlub0N 32724 The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)
 |-  .1.  =  ( lub `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( K  e.  OP  ->  (  .1.  `  (/) )  =  .0.  )
 
Theoremopltn0 32725 A lattice element greater than zero is nonzero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .0.  .<  X  <->  X  =/=  .0.  ) )
 
Theoremople1 32726 Any element is less than the orthoposet unit. (chss 26880 analog.) (Contributed by NM, 23-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  .<_  .1.  )
 
Theoremop1le 32727 If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 27094 analog.) (Contributed by NM, 5-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  .1.  .<_  X  <->  X  =  .1.  )
 )
 
Theoremglb0N 32728 The greatest lower bound of the empty set is the unit element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.)
 |-  G  =  ( glb `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( K  e.  OP  ->  ( G `  (/) )  =  .1.  )
 
Theoremopoccl 32729 Closure of orthocomplement operation. (choccl 26957 analog.) (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
 
Theoremopococ 32730 Double negative law for orthoposets. (ococ 27057 analog.) (Contributed by NM, 13-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
 
Theoremopcon3b 32731 Contraposition law for orthoposets. (chcon3i 27117 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  =  Y  <->  ( 
 ._|_  `  Y )  =  (  ._|_  `  X ) ) )
 
Theoremopcon2b 32732 Orthocomplement contraposition law. (negcon2 9934 analog.) (Contributed by NM, 16-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  =  ( 
 ._|_  `  Y )  <->  Y  =  (  ._|_  `  X ) ) )
 
Theoremopcon1b 32733 Orthocomplement contraposition law. (negcon1 9933 analog.) (Contributed by NM, 24-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( (  ._|_  `  X )  =  Y  <->  (  ._|_  `  Y )  =  X )
 )
 
Theoremoplecon3 32734 Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .<_  Y  ->  ( 
 ._|_  `  Y )  .<_  ( 
 ._|_  `  X ) ) )
 
Theoremoplecon3b 32735 Contraposition law for orthoposets. (chsscon3 27151 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .<_  Y  <->  (  ._|_  `  Y )  .<_  (  ._|_  `  X ) ) )
 
Theoremoplecon1b 32736 Contraposition law for strict ordering in orthoposets. (chsscon1 27152 analog.) (Contributed by NM, 6-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( (  ._|_  `  X )  .<_  Y  <->  (  ._|_  `  Y )  .<_  X ) )
 
Theoremopoc1 32737 Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)
 |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( K  e.  OP  ->  (  ._|_  `  .1.  )  =  .0.  )
 
Theoremopoc0 32738 Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)
 |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( K  e.  OP  ->  (  ._|_  `  .0.  )  =  .1.  )
 
Theoremopltcon3b 32739 Contraposition law for strict ordering in orthoposets. (chpsscon3 27154 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .<  Y  <->  (  ._|_  `  Y )  .<  (  ._|_  `  X ) ) )
 
Theoremopltcon1b 32740 Contraposition law for strict ordering in orthoposets. (chpsscon1 27155 analog.) (Contributed by NM, 5-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( (  ._|_  `  X )  .<  Y  <->  (  ._|_  `  Y )  .<  X ) )
 
Theoremopltcon2b 32741 Contraposition law for strict ordering in orthoposets. (chsscon2 27153 analog.) (Contributed by NM, 5-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .<  (  ._|_  `  Y )  <->  Y  .<  (  ._|_  `  X ) ) )
 
Theoremopexmid 32742 Law of excluded middle for orthoposets. (chjo 27166 analog.) (Contributed by NM, 13-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  .\/ 
 =  ( join `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  .\/  (  ._|_  `  X )
 )  =  .1.  )
 
Theoremopnoncon 32743 Law of contradiction for orthoposets. (chocin 27146 analog.) (Contributed by NM, 13-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  ./\ 
 =  ( meet `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  ./\  (  ._|_  `  X )
 )  =  .0.  )
 
TheoremriotaocN 32744* The orthocomplement of the unique poset element such that  ps. (riotaneg 10593 analog.) (Contributed by NM, 16-Jan-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  ( x  =  (  ._|_  `  y )  ->  ( ph  <->  ps ) )   =>    |-  ( ( K  e.  OP  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  B  ph )  =  (  ._|_  `  ( iota_ y  e.  B  ps ) ) )
 
TheoremcmtfvalN 32745* Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( K  e.  A  ->  C  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  B  /\  x  =  ( ( x  ./\  y )  .\/  ( x  ./\  (  ._|_  `  y ) ) ) ) } )
 
TheoremcmtvalN 32746 Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 27235 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X C Y  <->  X  =  ( ( X 
 ./\  Y )  .\/  ( X  ./\  (  ._|_  `  Y ) ) ) ) )
 
Theoremisolat 32747 The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  <->  ( K  e.  Lat  /\  K  e.  OP )
 )
 
Theoremollat 32748 An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  ->  K  e.  Lat )
 
Theoremolop 32749 An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  ->  K  e.  OP )
 
TheoremolposN 32750 An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
 |-  ( K  e.  OL  ->  K  e.  Poset )
 
TheoremisolatiN 32751 Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
 |-  K  e.  Lat   &    |-  K  e.  OP   =>    |-  K  e.  OL
 
Theoremoldmm1 32752 De Morgan's law for meet in an ortholattice. (chdmm1 27176 analog.) (Contributed by NM, 6-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  ( X  ./\  Y ) )  =  ( (  ._|_  `  X )  .\/  (  ._|_  `  Y ) ) )
 
Theoremoldmm2 32753 De Morgan's law for meet in an ortholattice. (chdmm2 27177 analog.) (Contributed by NM, 6-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  (
 (  ._|_  `  X )  ./\ 
 Y ) )  =  ( X  .\/  (  ._|_  `  Y ) ) )
 
Theoremoldmm3N 32754 De Morgan's law for meet in an ortholattice. (chdmm3 27178 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  ( X  ./\  (  ._|_  `  Y ) ) )  =  ( (  ._|_  `  X )  .\/  Y ) )
 
Theoremoldmm4 32755 De Morgan's law for meet in an ortholattice. (chdmm4 27179 analog.) (Contributed by NM, 6-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  (
 (  ._|_  `  X )  ./\  (  ._|_  `  Y ) ) )  =  ( X  .\/  Y )
 )
 
Theoremoldmj1 32756 De Morgan's law for join in an ortholattice. (chdmj1 27180 analog.) (Contributed by NM, 6-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  ( X  .\/  Y ) )  =  ( (  ._|_  `  X )  ./\  (  ._|_  `  Y ) ) )
 
Theoremoldmj2 32757 De Morgan's law for join in an ortholattice. (chdmj2 27181 analog.) (Contributed by NM, 7-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  (
 (  ._|_  `  X )  .\/  Y ) )  =  ( X  ./\  (  ._|_  `  Y ) ) )
 
Theoremoldmj3 32758 De Morgan's law for join in an ortholattice. (chdmj3 27182 analog.) (Contributed by NM, 7-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  ( X  .\/  (  ._|_  `  Y ) ) )  =  ( (  ._|_  `  X )  ./\  Y ) )
 
Theoremoldmj4 32759 De Morgan's law for join in an ortholattice. (chdmj4 27183 analog.) (Contributed by NM, 7-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  (  ._|_  `  (
 (  ._|_  `  X )  .\/  (  ._|_  `  Y ) ) )  =  ( X  ./\  Y )
 )
 
Theoremolj01 32760 An ortholattice element joined with zero equals itself. (chj0 27148 analog.) (Contributed by NM, 19-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  .\/  .0.  )  =  X )
 
Theoremolj02 32761 An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B )  ->  (  .0.  .\/  X )  =  X )
 
Theoremolm11 32762 The meet of an ortholattice element with one equals itself. (chm1i 27107 analog.) (Contributed by NM, 22-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .1.  )  =  X )
 
Theoremolm12 32763 The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B )  ->  (  .1.  ./\  X )  =  X )
 
TheoremlatmassOLD 32764 Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3672 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  ./\  Y )  ./\  Z )  =  ( X 
 ./\  ( Y  ./\  Z ) ) )
 
Theoremlatm12 32765 A rearrangement of lattice meet. (in12 3673 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  ./\  ( Y  ./\  Z ) )  =  ( Y 
 ./\  ( X  ./\  Z ) ) )
 
Theoremlatm32 32766 A rearrangement of lattice meet. (in12 3673 analog.) (Contributed by NM, 13-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  ./\  Y )  ./\  Z )  =  ( ( X  ./\  Z )  ./\ 
 Y ) )
 
Theoremlatmrot 32767 Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  ./\  Y )  ./\  Z )  =  ( ( Z  ./\  X )  ./\ 
 Y ) )
 
Theoremlatm4 32768 Rearrangement of lattice meet of 4 classes. (in4 3678 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  ( Z  e.  B  /\  W  e.  B ) )  ->  ( ( X  ./\  Y )  ./\  ( Z  ./\  W ) )  =  ( ( X  ./\  Z )  ./\  ( Y  ./\  W ) ) )
 
TheoremlatmmdiN 32769 Lattice meet distributes over itself. (inindi 3679 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  ./\  ( Y  ./\  Z ) )  =  ( ( X  ./\  Y )  ./\  ( X  ./\  Z ) ) )
 
Theoremlatmmdir 32770 Lattice meet distributes over itself. (inindir 3680 analog.) (Contributed by NM, 6-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   =>    |-  (
 ( K  e.  OL  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  ./\  Y )  ./\  Z )  =  ( ( X  ./\  Z )  ./\  ( Y  ./\  Z ) ) )
 
Theoremolm01 32771 Meet with lattice zero is zero. (chm0 27142 analog.) (Contributed by NM, 8-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B )  ->  ( X  ./\  .0.  )  =  .0.  )
 
Theoremolm02 32772 Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B )  ->  (  .0.  ./\  X )  =  .0.  )
 
Theoremisoml 32773* The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( K  e.  OML  <->  ( K  e.  OL  /\  A. x  e.  B  A. y  e.  B  ( x  .<_  y 
 ->  y  =  ( x  .\/  ( y  ./\  (  ._|_  `  x )
 ) ) ) ) )
 
TheoremisomliN 32774* Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
 |-  K  e.  OL   &    |-  B  =  (
 Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .<_  y  ->  y  =  ( x  .\/  (
 y  ./\  (  ._|_  `  x ) ) ) ) )   =>    |-  K  e.  OML
 
Theoremomlol 32775 An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OML  ->  K  e.  OL )
 
Theoremomlop 32776 An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)
 |-  ( K  e.  OML  ->  K  e.  OP )
 
Theoremomllat 32777 An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)
 |-  ( K  e.  OML  ->  K  e.  Lat )
 
Theoremomllaw 32778 The orthomodular law. (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  Y  =  ( X  .\/  ( Y  ./\  (  ._|_  `  X ) ) ) ) )
 
Theoremomllaw2N 32779 Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 27236 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  ( X  .\/  ( (  ._|_  `  X )  ./\  Y ) )  =  Y ) )
 
Theoremomllaw3 32780 Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 27087 analog.) (Contributed by NM, 19-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  Y 
 /\  ( Y  ./\  (  ._|_  `  X )
 )  =  .0.  )  ->  X  =  Y ) )
 
Theoremomllaw4 32781 Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |- 
 ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  ( (  ._|_  `  ( ( 
 ._|_  `  X )  ./\  Y ) )  ./\  Y )  =  X ) )
 
Theoremomllaw5N 32782 The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 27264 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  (
 (  ._|_  `  X )  ./\  ( X  .\/  Y ) ) )  =  ( X  .\/  Y ) )
 
TheoremcmtcomlemN 32783 Lemma for cmtcomN 32784. (cmcmlem 27242 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  ->  Y C X ) )
 
TheoremcmtcomN 32784 Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 27243 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  Y C X ) )
 
Theoremcmt2N 32785 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 27244 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  X C (  ._|_  `  Y ) ) )
 
Theoremcmt3N 32786 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 27246 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  ( 
 ._|_  `  X ) C Y ) )
 
Theoremcmt4N 32787 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 27246 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  ( 
 ._|_  `  X ) C (  ._|_  `  Y ) ) )
 
Theoremcmtbr2N 32788 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 27247 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  X  =  ( ( X 
 .\/  Y )  ./\  ( X  .\/  (  ._|_  `  Y ) ) ) ) )
 
Theoremcmtbr3N 32789 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 27259 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  ( X  ./\  ( (  ._|_  `  X )  .\/  Y ) )  =  ( X  ./\  Y )
 ) )
 
Theoremcmtbr4N 32790 Alternate definition for the commutes relation. (cmbr4i 27252 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  ( X  ./\  ( (  ._|_  `  X )  .\/  Y ) )  .<_  Y ) )
 
TheoremlecmtN 32791 Ordered elements commute. (lecmi 27253 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  X C Y ) )
 
TheoremcmtidN 32792 Any element commutes with itself. (cmidi 27261 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  OML  /\  X  e.  B ) 
 ->  X C X )
 
Theoremomlfh1N 32793 Foulis-Holland Theorem, part 1. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Part of Theorem 5 in [Kalmbach] p. 25. (fh1 27269 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X C Y  /\  X C Z ) )  ->  ( X  ./\  ( Y  .\/  Z ) )  =  (
 ( X  ./\  Y ) 
 .\/  ( X  ./\  Z ) ) )
 
Theoremomlfh3N 32794 Foulis-Holland Theorem, part 3. Dual of omlfh1N 32793. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X C Y  /\  X C Z ) )  ->  ( X  .\/  ( Y  ./\  Z ) )  =  ( ( X  .\/  Y )  ./\  ( X  .\/  Z ) ) )
 
Theoremomlmod1i2N 32795 Analog of modular law atmod1i2 33393 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  ( cm `  K )   =>    |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( X  .<_  Z  /\  Y C Z ) ) 
 ->  ( X  .\/  ( Y  ./\  Z ) )  =  ( ( X 
 .\/  Y )  ./\  Z ) )
 
TheoremomlspjN 32796 Contraction of a Sasaki projection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OML  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  X  .<_  Y ) 
 ->  ( ( X  .\/  (  ._|_  `  Y )
 )  ./\  Y )  =  X )
 
21.21.10  Atomic lattices with covering property
 
Syntaxccvr 32797 Extend class notation with covers relation.
 class  <o
 
Syntaxcatm 32798 Extend class notation with atoms.
 class  Atoms
 
Syntaxcal 32799 Extend class notation with atomic lattices.
 class  AtLat
 
Syntaxclc 32800 Extend class notation with lattices with the covering property.
 class  CvLat
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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