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Theorem List for Metamath Proof Explorer - 32501-32600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremolj02 32501 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 32502 The meet of an ortholattice element with one equals itself. (chm1i 26944 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 32503 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 32504 Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3678 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 32505 A rearrangement of lattice meet. (in12 3679 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 32506 A rearrangement of lattice meet. (in12 3679 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 32507 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 32508 Rearrangement of lattice meet of 4 classes. (in4 3684 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 32509 Lattice meet distributes over itself. (inindi 3685 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 32510 Lattice meet distributes over itself. (inindir 3686 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 32511 Meet with lattice zero is zero. (chm0 26979 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 32512 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 32513* 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 32514* 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 32515 An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OML  ->  K  e.  OL )
 
Theoremomlop 32516 An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)
 |-  ( K  e.  OML  ->  K  e.  OP )
 
Theoremomllat 32517 An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)
 |-  ( K  e.  OML  ->  K  e.  Lat )
 
Theoremomllaw 32518 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 32519 Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 27073 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 32520 Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 26924 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 32521 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 32522 The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 27101 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 32523 Lemma for cmtcomN 32524. (cmcmlem 27079 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 32524 Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 27080 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 32525 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 27081 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 32526 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 27083 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 32527 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 27083 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 32528 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 27084 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 32529 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 27096 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 32530 Alternate definition for the commutes relation. (cmbr4i 27089 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 32531 Ordered elements commute. (lecmi 27090 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 32532 Any element commutes with itself. (cmidi 27098 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 32533 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 27106 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 32534 Foulis-Holland Theorem, part 3. Dual of omlfh1N 32533. (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 32535 Analog of modular law atmod1i2 33133 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 32536 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 32537 Extend class notation with covers relation.
 class  <o
 
Syntaxcatm 32538 Extend class notation with atoms.
 class  Atoms
 
Syntaxcal 32539 Extend class notation with atomic lattices.
 class  AtLat
 
Syntaxclc 32540 Extend class notation with lattices with the covering property.
 class  CvLat
 
Definitiondf-covers 32541* 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 32544 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 32542* 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 32543* 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 32544* 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 27770 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 32545 The covers relation implies the less-than relation. (cvpss 27773 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 32546 There is no element between the two arguments of the covers relation. (cvnbtwn 27774 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 32547 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 32548 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 32549* Binary relation expressing  Y covers  X. Definition of covers in [Kalmbach] p. 15. (cvbr2 27771 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 32550 The covers relation implies no in-betweenness. (cvnbtwn2 27775 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 32551 The covers relation implies no in-betweenness. (cvnbtwn3 27776 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 32552 Contraposition law for the covers relation. (cvcon3 27772 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 32553 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 32554 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 27777 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 32555 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 )
 
Theoremcvrne 32556 The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  B )  /\  X C Y )  ->  X  =/=  Y )
 
TheoremcvrnrefN 32557 The covers relation is not reflexive. (cvnref 27779 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  A  /\  X  e.  B )  ->  -.  X C X )
 
Theoremcvrcmp 32558 If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  Poset  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  ( Z C X  /\  Z C Y ) )  ->  ( X  .<_  Y  <->  X  =  Y ) )
 
Theoremcvrcmp2 32559 If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  OP  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) 
 /\  ( X C Z  /\  Y C Z ) )  ->  ( X 
 .<_  Y  <->  X  =  Y ) )
 
Theorempats 32560* The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  D  ->  A  =  { x  e.  B  |  .0.  C x } )
 
Theoremisat 32561 The predicate "is an atom". (ela 27827 analog.) (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  D  ->  ( P  e.  A 
 <->  ( P  e.  B  /\  .0.  C P ) ) )
 
Theoremisat2 32562 The predicate "is an atom". (elatcv0 27829 analog.) (Contributed by NM, 18-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  D  /\  P  e.  B )  ->  ( P  e.  A  <->  .0.  C P ) )
 
Theorematcvr0 32563 An atom covers zero. (atcv0 27830 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  D  /\  P  e.  A )  ->  .0.  C P )
 
Theorematbase 32564 An atom is a member of the lattice base set (i.e. a lattice element). (atelch 27832 analog.) (Contributed by NM, 10-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( P  e.  A  ->  P  e.  B )
 
Theorematssbase 32565 The set of atoms is a subset of the base set. (atssch 27831 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  A  C_  B
 
Theorem0ltat 32566 An atom is greater than zero. (Contributed by NM, 4-Jul-2012.)
 |-  .0.  =  ( 0. `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  OP  /\  P  e.  A )  ->  .0.  .<  P )
 
Theoremleatb 32567 A poset element less than or equal to an atom equals either zero or the atom. (atss 27834 analog.) (Contributed by NM, 17-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  ->  ( X 
 .<_  P  <->  ( X  =  P  \/  X  =  .0.  ) ) )
 
Theoremleat 32568 A poset element less than or equal to an atom equals either zero or the atom. (Contributed by NM, 15-Oct-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  X  .<_  P )  ->  ( X  =  P  \/  X  =  .0.  )
 )
 
Theoremleat2 32569 A nonzero poset element less than or equal to an atom equals the atom. (Contributed by NM, 6-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  ( X  =/=  .0.  /\  X  .<_  P ) ) 
 ->  X  =  P )
 
Theoremleat3 32570 A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  OP  /\  X  e.  B  /\  P  e.  A )  /\  X  .<_  P )  ->  ( X  e.  A  \/  X  =  .0.  )
 )
 
Theoremmeetat 32571 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A ) 
 ->  ( ( X  ./\  P )  =  P  \/  ( X  ./\  P )  =  .0.  ) )
 
Theoremmeetat2 32572 The meet of any element with an atom is either the atom or zero. (Contributed by NM, 30-Aug-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  OL  /\  X  e.  B  /\  P  e.  A ) 
 ->  ( ( X  ./\  P )  e.  A  \/  ( X  ./\  P )  =  .0.  ) )
 
Definitiondf-atl 32573* Define the class of atomic lattices, in which every nonzero element is greater than or equal to an atom. We also ensure the existence of a lattice zero, since a lattice by itself may not have a zero. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)
 |-  AtLat  =  {
 k  e.  Lat  |  ( ( Base `  k
 )  e.  dom  ( glb `  k )  /\  A. x  e.  ( Base `  k ) ( x  =/=  ( 0. `  k
 )  ->  E. p  e.  ( Atoms `  k ) p ( le `  k
 ) x ) ) }
 
Theoremisatl 32574* The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.) (Revised by NM, 14-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |- 
 .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  AtLat  <->  ( K  e.  Lat  /\  B  e.  dom  G  /\  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y 
 .<_  x ) ) )
 
Theorematllat 32575 An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)
 |-  ( K  e.  AtLat  ->  K  e.  Lat )
 
Theorematlpos 32576 An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  AtLat  ->  K  e.  Poset )
 
Theorematl0dm 32577 Condition necessary for zero element to exist. (Contributed by NM, 14-Sep-2018.)
 |-  B  =  ( Base `  K )   &    |-  U  =  ( lub `  K )   &    |-  G  =  ( glb `  K )   =>    |-  ( K  e.  AtLat  ->  B  e.  dom  G )
 
Theorematl0cl 32578 An atomic lattice has a zero element. We can use this in place of op0cl 32459 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( K  e.  AtLat  ->  .0.  e.  B )
 
Theorematl0le 32579 Orthoposet zero is less than or equal to any element. (ch0le 26929 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  e.  B ) 
 ->  .0.  .<_  X )
 
Theorematlle0 32580 An element less than or equal to zero equals zero. (chle0 26931 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  e.  B ) 
 ->  ( X  .<_  .0.  <->  X  =  .0.  ) )
 
Theorematlltn0 32581 A lattice element greater than zero is nonzero. (Contributed by NM, 1-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  e.  B ) 
 ->  (  .0.  .<  X  <->  X  =/=  .0.  )
 )
 
Theoremisat3 32582* The predicate "is an atom". (elat2 27828 analog.) (Contributed by NM, 27-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  AtLat  ->  ( P  e.  A  <->  ( P  e.  B  /\  P  =/=  .0.  /\  A. x  e.  B  ( x  .<_  P  ->  ( x  =  P  \/  x  =  .0.  )
 ) ) ) )
 
Theorematn0 32583 An atom is not zero. (atne0 27833 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A ) 
 ->  P  =/=  .0.  )
 
Theorematnle0 32584 An atom is not less than or equal to zero. (Contributed by NM, 17-Oct-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  -.  P  .<_  .0.  )
 
Theorematlen0 32585 A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  /\  P  .<_  X )  ->  X  =/=  .0.  )
 
Theorematcmp 32586 If two atoms are comparable, they are equal. (atsseq 27835 analog.) (Contributed by NM, 13-Oct-2011.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .<_  Q  <->  P  =  Q ) )
 
Theorematncmp 32587 Frequently-used variation of atcmp 32586. (Contributed by NM, 29-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( -.  P  .<_  Q  <->  P  =/=  Q ) )
 
Theorematnlt 32588 Two atoms cannot satisfy the less than relation. (Contributed by NM, 7-Feb-2012.)
 |-  .<  =  ( lt `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  -.  P  .<  Q )
 
Theorematcvreq0 32589 An element covered by an atom must be zero. (atcveq0 27836 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  AtLat  /\  X  e.  B  /\  P  e.  A )  ->  ( X C P  <->  X  =  .0.  ) )
 
TheorematncvrN 32590 Two atoms cannot satisfy the covering relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
 |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  -.  P C Q )
 
Theorematlex 32591* Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 27848 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  e.  B  /\  X  =/=  .0.  )  ->  E. y  e.  A  y  .<_  X )
 
Theorematnle 32592 Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 27864 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( -.  P  .<_  X  <->  ( P  ./\  X )  =  .0.  )
 )
 
Theorematnem0 32593 The meet of distinct atoms is zero. (atnemeq0 27865 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  ( P  ./\  Q )  =  .0.  ) )
 
Theorematlatmstc 32594* An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in [Kalmbach] p. 140; also remark in [BeltramettiCassinelli] p. 98. (hatomistici 27850 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .1.  =  ( lub `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B )  ->  (  .1.  `  { y  e.  A  |  y  .<_  X } )  =  X )
 
Theorematlatle 32595* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 27859 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  A. p  e.  A  ( p  .<_  X  ->  p 
 .<_  Y ) ) )
 
Theorematlrelat1 32596* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 27851, with  /\ swapped, analog.) (Contributed by NM, 4-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  X  e.  B  /\  Y  e.  B )  ->  ( X 
 .<  Y  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
 
Definitiondf-cvlat 32597* Define the class of atomic lattices with the covering property. (This is actually the exchange property, but they are equivalent. The literature usually uses the covering property terminology.) (Contributed by NM, 5-Nov-2012.)
 |-  CvLat  =  {
 k  e.  AtLat  |  A. a  e.  ( Atoms `  k ) A. b  e.  ( Atoms `  k ) A. c  e.  ( Base `  k ) ( ( -.  a ( le `  k ) c  /\  a ( le `  k ) ( c ( join `  k ) b ) )  ->  b ( le `  k ) ( c ( join `  k
 ) a ) ) }
 
Theoremiscvlat 32598* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  CvLat  <->  ( K  e.  AtLat  /\  A. p  e.  A  A. q  e.  A  A. x  e.  B  ( ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q
 ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
 
Theoremiscvlat2N 32599* The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( K  e.  CvLat  <->  ( K  e.  AtLat  /\  A. p  e.  A  A. q  e.  A  A. x  e.  B  ( ( ( p  ./\  x )  =  .0.  /\  p  .<_  ( x  .\/  q )
 )  ->  q  .<_  ( x  .\/  p )
 ) ) )
 
Theoremcvlatl 32600 An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  CvLat  ->  K  e.  AtLat )
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