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Theorem List for Metamath Proof Explorer - 28501-28600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremisopiN 28501* Properties that determine an orthoposet (constructed structure version). (Contributed by NM, 13-Sep-2011.) (New usage is discouraged.)
 |-  K  e.  Poset   &    |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  .0.  e.  B   &    |-  .1.  e.  B   &    |-  ( x  e.  B  ->  (  ._|_  `  x )  e.  B )   &    |-  ( x  e.  B  ->  (  ._|_  `  (  ._|_  `  x ) )  =  x )   &    |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .<_  y  ->  (  ._|_  `  y )  .<_  ( 
 ._|_  `  x ) ) )   &    |-  ( x  e.  B  ->  ( x  .\/  (  ._|_  `  x ) )  =  .1.  )   &    |-  ( x  e.  B  ->  ( x  ./\  (  ._|_  `  x ) )  =  .0.  )   =>    |-  K  e.  OP
 
Theoremopposet 28502 Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
 |-  ( K  e.  OP  ->  K  e.  Poset )
 
Theoremoposlem 28503 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.  ) )
 
Theoremop0cl 28504 An orthoposet has a zero element. (h0elch 21759 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( K  e.  OP  ->  .0.  e.  B )
 
Theoremop1cl 28505 An orthoposet has a unit element. (helch 21748 analog.) (Contributed by NM, 22-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( K  e.  OP  ->  .1.  e.  B )
 
Theoremop0le 28506 Orthoposet zero is less than or equal to any element. (ch0le 21945 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  .0.  .<_  X )
 
Theoremople0 28507 An element less than or equal to zero equals zero. (chle0 21947 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 28508 An element not less than another is nonzero. TODO: Look for uses of necon3bd 2456 and op0le 28506 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 28509 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 28510 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 28511 Any element is less than the orthoposet unit. (chss 21734 analog.) (Contributed by NM, 23-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .1.  =  ( 1. `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  X  .<_  .1.  )
 
Theoremop1le 28512 If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 21947 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 28513 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 28514 Closure of orthocomplement operation. (choccl 21810 analog.) (Contributed by NM, 20-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
 
Theoremopococ 28515 Double negative law for orthoposets. (ococ 21910 analog.) (Contributed by NM, 13-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
 
Theoremopcon3b 28516 Contraposition law for orthoposets. (chcon3i 21970 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 28517 Orthocomplement contraposition law. (negcon2 9033 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 28518 Orthocomplement contraposition law. (negcon1 9032 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 28519 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 28520 Contraposition law for orthoposets. (chsscon3 22004 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 28521 Contraposition law for strict ordering in orthoposets. (chsscon1 22005 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 28522 Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.)
 |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( K  e.  OP  ->  (  ._|_  `  .1.  )  =  .0.  )
 
Theoremopoc0 28523 Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.)
 |-  .0.  =  ( 0. `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  ._|_  =  ( oc `  K )   =>    |-  ( K  e.  OP  ->  (  ._|_  `  .0.  )  =  .1.  )
 
Theoremopltcon3b 28524 Contraposition law for strict ordering in orthoposets. (chpsscon3 22007 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 28525 Contraposition law for strict ordering in orthoposets. (chpsscon1 22008 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 28526 Contraposition law for strict ordering in orthoposets. (chsscon2 22006 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 28527 Law of excluded middle for orthoposets. (chjo 22019 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 28528 Law of contradiction for orthoposets. (chocin 21999 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 28529* The orthocomplement of the unique poset element such that  ps. (riotaneg 9662 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 28530* 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 28531 Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 22106 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 28532 The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  <->  ( K  e.  Lat  /\  K  e.  OP )
 )
 
Theoremollat 28533 An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  ->  K  e.  Lat )
 
Theoremolop 28534 An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OL  ->  K  e.  OP )
 
TheoremolposN 28535 An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
 |-  ( K  e.  OL  ->  K  e.  Poset )
 
TheoremisolatiN 28536 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 28537 DeMorgan's law for meet in an ortholattice. (chdmm1 22029 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 28538 DeMorgan's law for meet in an ortholattice. (chdmm2 22030 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 28539 DeMorgan's law for meet in an ortholattice. (chdmm3 22031 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 28540 DeMorgan's law for meet in an ortholattice. (chdmm4 22032 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 28541 DeMorgan's law for join in an ortholattice. (chdmj1 22033 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 28542 DeMorgan's law for join in an ortholattice. (chdmj2 22034 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 28543 DeMorgan's law for join in an ortholattice. (chdmj3 22035 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 28544 DeMorgan's law for join in an ortholattice. (chdmj4 22036 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 28545 An ortholattice element joined with zero equals itself. (chj0 22001 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 28546 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 28547 The meet of an ortholattice element with one equals itself. (chm1i 21960 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 28548 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 28549 Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3321 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 28550 A rearrangement of lattice meet. (in12 3322 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 28551 A rearrangement of lattice meet. (in12 3322 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 28552 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 28553 Rearrangement of lattice meet of 4 classes. (in4 3327 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 28554 Lattice meet distributes over itself. (inindi 3328 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 28555 Lattice meet distributes over itself. (inindir 3329 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 28556 Meet with lattice zero is zero. (chm0 21995 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 28557 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 28558* 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 28559* 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 28560 An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OML  ->  K  e.  OL )
 
Theoremomlop 28561 An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)
 |-  ( K  e.  OML  ->  K  e.  OP )
 
Theoremomllat 28562 An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)
 |-  ( K  e.  OML  ->  K  e.  Lat )
 
Theoremomllaw 28563 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 28564 Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 22107 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 28565 Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 21940 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 28566 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 28567 The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 22135 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 28568 Lemma for cmtcomN 28569. (cmcmlem 22113 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 28569 Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 22114 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 28570 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 22115 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 28571 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 22117 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 28572 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 22117 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 28573 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 22118 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 28574 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 22130 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 28575 Alternate definition for the commutes relation. (cmbr4i 22123 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 28576 Ordered elements commute. (lecmi 22124 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 28577 Any element commutes with itself. (cmidi 22132 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 28578 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 22140 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 28579 Foulis-Holland Theorem, part 3. Dual of omlfh1N 28578. (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 28580 Analog of modular law atmod1i2 29178 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 28581 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 )
 
16.24.11  Atomic lattices with covering property
 
Syntaxccvr 28582 Extend class notation with covers relation.
 class  <o
 
Syntaxcatm 28583 Extend class notation with atoms.
 class  Atoms
 
Syntaxcal 28584 Extend class notation with atomic lattices.
 class  AtLat
 
Syntaxclc 28585 Extend class notation with lattices with the covering property.
 class  CvLat
 
Definitiondf-covers 28586* 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 28589 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 28587* 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 28588* 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 28589* 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 22787 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 ) ) ) )
 
Theoremcvrlt 28590 The covers relation implies the less-than relation. (cvpss 22790 analog.) (Contributed by NM, 8-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<  Y )
 
Theoremcvrnbtwn 28591 There is no element between the two arguments of the covers relation. (cvnbtwn 22791 analog.) (Contributed by NM, 18-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) 
 /\  X C Y )  ->  -.  ( X  .<  Z  /\  Z  .<  Y ) )
 
Theoremncvr1 28592 No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B )  ->  -.  .1.  C X )
 
TheoremcvrletrN 28593 Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  C  =  ( 
 <o  `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X C Y  /\  Y  .<_  Z )  ->  X  .<  Z ) )
 
Theoremcvrval2 28594* Binary relation expressing  Y covers  X. Definition of covers in [Kalmbach] p. 15. (cvbr2 22788 analog.) (Contributed by NM, 16-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  C  =  ( 
 <o  `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  ( X  .<  Y 
 /\  A. z  e.  B  ( ( X  .<  z 
 /\  z  .<_  Y ) 
 ->  z  =  Y ) ) ) )
 
Theoremcvrnbtwn2 28595 The covers relation implies no in-betweenness. (cvnbtwn2 22792 analog.) (Contributed by NM, 17-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  C  =  ( 
 <o  `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X 
 .<  Z  /\  Z  .<_  Y )  <->  Z  =  Y ) )
 
Theoremcvrnbtwn3 28596 The covers relation implies no in-betweenness. (cvnbtwn3 22793 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  C  =  ( 
 <o  `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y )  ->  ( ( X 
 .<_  Z  /\  Z  .<  Y )  <->  X  =  Z ) )
 
Theoremcvrcon3b 28597 Contraposition law for the covers relation. (cvcon3 22789 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X C Y  <->  ( 
 ._|_  `  Y ) C (  ._|_  `  X ) ) )
 
Theoremcvrle 28598 The covers relation implies the less-than-or-equal relation. (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  .<_  Y )
 
Theoremcvrnbtwn4 28599 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 22794 analog.) (Contributed by NM, 18-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X C Y ) 
 ->  ( ( X  .<_  Z 
 /\  Z  .<_  Y )  <-> 
 ( X  =  Z  \/  Z  =  Y ) ) )
 
Theoremcvrnle 28600 The covers relation implies the negation of the reverse less-than-or-equal relation. (Contributed by NM, 18-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  -.  Y  .<_  X )
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