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Theorem List for Metamath Proof Explorer - 32201-32300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremlkreqN 32201 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 32202 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 32203 Extend class notation with orthoposets.
 class  OP
 
SyntaxccmtN 32204 Extend class notation with the commutes relation.
 class  cm
 
Syntaxcol 32205 Extend class notation with orthlattices.
 class  OL
 
Syntaxcoml 32206 Extend class notation with orthomodular lattices.
 class  OML
 
Definitiondf-oposet 32207* 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 32208* 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 32209 Define the class of ortholattices. Definition from [Kalmbach] p. 16. (Contributed by NM, 18-Sep-2011.)
 |-  OL  =  ( Lat  i^i  OP )
 
Definitiondf-oml 32210* 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 32211* 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 32212 Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
 |-  ( K  e.  OP  ->  K  e.  Poset )
 
Theoremoposlem 32213 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 32214 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 32215 An orthoposet has a zero element. (h0elch 26600 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( K  e.  OP  ->  .0.  e.  B )
 
Theoremop1cl 32216 An orthoposet has a unit element. (helch 26588 analog.) (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( K  e.  OP  ->  .1.  e.  B )
 
Theoremop0le 32217 Orthoposet zero is less than or equal to any element. (ch0le 26786 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
 
Theoremople0 32218 An element less than or equal to zero equals zero. (chle0 26788 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 32219 An element not less than another is nonzero. TODO: Look for uses of necon3bd 2617 and op0le 32217 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 32220 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 32221 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 32222 Any element is less than the orthoposet unit. (chss 26574 analog.) (Contributed by NM, 23-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  .<_  .1.  )
 
Theoremop1le 32223 If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 26788 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 32224 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 32225 Closure of orthocomplement operation. (choccl 26651 analog.) (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
 
Theoremopococ 32226 Double negative law for orthoposets. (ococ 26751 analog.) (Contributed by NM, 13-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
 
Theoremopcon3b 32227 Contraposition law for orthoposets. (chcon3i 26811 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 32228 Orthocomplement contraposition law. (negcon2 9910 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 32229 Orthocomplement contraposition law. (negcon1 9909 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 32230 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 32231 Contraposition law for orthoposets. (chsscon3 26845 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 32232 Contraposition law for strict ordering in orthoposets. (chsscon1 26846 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 32233 Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)
 |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( K  e.  OP  ->  (  ._|_  `  .1.  )  =  .0.  )
 
Theoremopoc0 32234 Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)
 |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( K  e.  OP  ->  (  ._|_  `  .0.  )  =  .1.  )
 
Theoremopltcon3b 32235 Contraposition law for strict ordering in orthoposets. (chpsscon3 26848 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 32236 Contraposition law for strict ordering in orthoposets. (chpsscon1 26849 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 32237 Contraposition law for strict ordering in orthoposets. (chsscon2 26847 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 32238 Law of excluded middle for orthoposets. (chjo 26860 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 32239 Law of contradiction for orthoposets. (chocin 26840 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 32240* The orthocomplement of the unique poset element such that  ps. (riotaneg 10560 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 32241* 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 32242 Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 26929 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 32243 The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  <->  ( K  e.  Lat  /\  K  e.  OP )
 )
 
Theoremollat 32244 An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  ->  K  e.  Lat )
 
Theoremolop 32245 An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  ->  K  e.  OP )
 
TheoremolposN 32246 An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
 |-  ( K  e.  OL  ->  K  e.  Poset )
 
TheoremisolatiN 32247 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 32248 De Morgan's law for meet in an ortholattice. (chdmm1 26870 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 32249 De Morgan's law for meet in an ortholattice. (chdmm2 26871 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 32250 De Morgan's law for meet in an ortholattice. (chdmm3 26872 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 32251 De Morgan's law for meet in an ortholattice. (chdmm4 26873 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 32252 De Morgan's law for join in an ortholattice. (chdmj1 26874 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 32253 De Morgan's law for join in an ortholattice. (chdmj2 26875 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 32254 De Morgan's law for join in an ortholattice. (chdmj3 26876 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 32255 De Morgan's law for join in an ortholattice. (chdmj4 26877 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 32256 An ortholattice element joined with zero equals itself. (chj0 26842 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 32257 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 32258 The meet of an ortholattice element with one equals itself. (chm1i 26801 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 32259 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 32260 Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3651 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 32261 A rearrangement of lattice meet. (in12 3652 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 32262 A rearrangement of lattice meet. (in12 3652 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 32263 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 32264 Rearrangement of lattice meet of 4 classes. (in4 3657 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 32265 Lattice meet distributes over itself. (inindi 3658 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 32266 Lattice meet distributes over itself. (inindir 3659 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 32267 Meet with lattice zero is zero. (chm0 26836 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 32268 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 32269* 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 32270* 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 32271 An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OML  ->  K  e.  OL )
 
Theoremomlop 32272 An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)
 |-  ( K  e.  OML  ->  K  e.  OP )
 
Theoremomllat 32273 An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)
 |-  ( K  e.  OML  ->  K  e.  Lat )
 
Theoremomllaw 32274 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 32275 Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 26930 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 32276 Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 26781 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 32277 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 32278 The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 26958 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 32279 Lemma for cmtcomN 32280. (cmcmlem 26936 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 32280 Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 26937 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 32281 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 26938 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 32282 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 26940 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 32283 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 26940 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 32284 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 26941 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 32285 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 26953 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 32286 Alternate definition for the commutes relation. (cmbr4i 26946 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 32287 Ordered elements commute. (lecmi 26947 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 32288 Any element commutes with itself. (cmidi 26955 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 32289 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 26963 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 32290 Foulis-Holland Theorem, part 3. Dual of omlfh1N 32289. (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 32291 Analog of modular law atmod1i2 32889 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 32292 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 32293 Extend class notation with covers relation.
 class  <o
 
Syntaxcatm 32294 Extend class notation with atoms.
 class  Atoms
 
Syntaxcal 32295 Extend class notation with atomic lattices.
 class  AtLat
 
Syntaxclc 32296 Extend class notation with lattices with the covering property.
 class  CvLat
 
Definitiondf-covers 32297* Define the covers relation ("is covered by") for posets. " a is covered by  b " means that  a is strictly less than  b and there is nothing in between. See cvrval 32300 for the relation form. (Contributed by NM, 18-Sep-2011.)
 |-  <o  =  ( p  e.  _V  |->  {
 <. a ,  b >.  |  ( ( a  e.  ( Base `  p )  /\  b  e.  ( Base `  p ) ) 
 /\  a ( lt `  p ) b  /\  -. 
 E. z  e.  ( Base `  p ) ( a ( lt `  p ) z  /\  z ( lt `  p ) b ) ) }
 )
 
Definitiondf-ats 32298* Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.)
 |-  Atoms  =  ( p  e.  _V  |->  { a  e.  ( Base `  p )  |  ( 0. `  p ) (  <o  `  p )
 a } )
 
Theoremcvrfval 32299* Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( K  e.  A  ->  C  =  { <. x ,  y >.  |  ( ( x  e.  B  /\  y  e.  B )  /\  x  .<  y  /\  -. 
 E. z  e.  B  ( x  .<  z  /\  z  .<  y ) ) } )
 
Theoremcvrval 32300* Binary relation expressing  B covers  A, which means that  B is larger than  A and there is nothing in between. Definition 3.2.18 of [PtakPulmannova] p. 68. (cvbr 27627 analog.) (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X C Y  <->  ( X  .<  Y  /\  -. 
 E. z  e.  B  ( X  .<  z  /\  z  .<  Y ) ) ) )
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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 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38873
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