HomeHome Metamath Proof Explorer
Theorem List (p. 296 of 323)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21811)
  Hilbert Space Explorer  Hilbert Space Explorer
(21812-23334)
  Users' Mathboxes  Users' Mathboxes
(23335-32225)
 

Theorem List for Metamath Proof Explorer - 29501-29600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremomlol 29501 An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)
 |-  ( K  e.  OML  ->  K  e.  OL )
 
Theoremomlop 29502 An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.)
 |-  ( K  e.  OML  ->  K  e.  OP )
 
Theoremomllat 29503 An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.)
 |-  ( K  e.  OML  ->  K  e.  Lat )
 
Theoremomllaw 29504 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 29505 Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 22477 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 29506 Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 22328 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 29507 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 29508 The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 22505 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 29509 Lemma for cmtcomN 29510. (cmcmlem 22483 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 29510 Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 22484 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 29511 Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 22485 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 29512 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 22487 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 29513 Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 22487 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 29514 Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 22488 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 29515 Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 22500 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 29516 Alternate definition for the commutes relation. (cmbr4i 22493 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 29517 Ordered elements commute. (lecmi 22494 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 29518 Any element commutes with itself. (cmidi 22502 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 29519 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 22510 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 29520 Foulis-Holland Theorem, part 3. Dual of omlfh1N 29519. (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 29521 Analog of modular law atmod1i2 30119 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 29522 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 )
 
19.26.8  Atomic lattices with covering property
 
Syntaxccvr 29523 Extend class notation with covers relation.
 class  <o
 
Syntaxcatm 29524 Extend class notation with atoms.
 class  Atoms
 
Syntaxcal 29525 Extend class notation with atomic lattices.
 class  AtLat
 
Syntaxclc 29526 Extend class notation with lattices with the covering property.
 class  CvLat
 
Definitiondf-covers 29527* 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 29530 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 29528* 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 29529* 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 29530* 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 23175 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 29531 The covers relation implies the less-than relation. (cvpss 23178 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 29532 There is no element between the two arguments of the covers relation. (cvnbtwn 23179 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 29533 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 29534 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 29535* Binary relation expressing  Y covers  X. Definition of covers in [Kalmbach] p. 15. (cvbr2 23176 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 29536 The covers relation implies no in-betweenness. (cvnbtwn2 23180 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 29537 The covers relation implies no in-betweenness. (cvnbtwn3 23181 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 29538 Contraposition law for the covers relation. (cvcon3 23177 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 29539 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 29540 The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 23182 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 29541 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 29542 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 29543 The covers relation is not reflexive. (cvnref 23184 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 29544 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 29545 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 29546* 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 29547 The predicate "is an atom". (ela 23232 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 29548 The predicate "is an atom". (elatcv0 23234 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 29549 An atom covers zero. (atcv0 23235 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 29550 An atom is a member of the lattice base set (i.e. a lattice element). (atelch 23237 analog.) (Contributed by NM, 10-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( P  e.  A  ->  P  e.  B )
 
Theorematssbase 29551 The set of atoms is a subset of the base set. (atssch 23236 analog.) (Contributed by NM, 21-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  A  C_  B
 
Theorem0ltat 29552 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 29553 A poset element less than or equal to an atom equals either zero or the atom. (atss 23239 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 29554 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 29555 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 29556 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 29557 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 29558 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 29559* 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.)
 |-  AtLat  =  {
 k  e.  Lat  |  ( ( 0. `  k
 )  e.  ( Base `  k )  /\  A. x  e.  ( Base `  k ) ( x  =/=  ( 0. `  k
 )  ->  E. p  e.  ( Atoms `  k ) p ( le `  k
 ) x ) ) }
 
Theoremisatl 29560* The predicate "is an atomic lattice." Every nonzero element is less than or equal to an atom. (Contributed by NM, 18-Sep-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( K  e.  AtLat  <->  ( K  e.  Lat  /\  .0.  e.  B  /\  A. x  e.  B  ( x  =/=  .0.  ->  E. y  e.  A  y  .<_  x ) ) )
 
Theorematllat 29561 An atomic lattice is a lattice. (Contributed by NM, 21-Oct-2011.)
 |-  ( K  e.  AtLat  ->  K  e.  Lat )
 
Theorematlpos 29562 An atomic lattice is a poset. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  AtLat  ->  K  e.  Poset )
 
TheoremisatliN 29563* Properties that determine an atomic lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.)
 |-  K  e.  Lat   &    |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  .0.  e.  B   &    |-  (
 ( x  e.  B  /\  x  =/=  .0.  )  ->  E. y  e.  A  y  .<_  x )   =>    |-  K  e.  AtLat
 
Theorematl0cl 29564 An atomic lattice has a zero element. We can use this in place of op0cl 29445 for lattices without orthocomplements. (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( K  e.  AtLat  ->  .0.  e.  B )
 
Theorematl0le 29565 Orthoposet zero is less than or equal to any element. (ch0le 22333 analog.) (Contributed by NM, 12-Oct-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   =>    |-  ( ( K  e.  AtLat  /\  X  e.  B ) 
 ->  .0.  .<_  X )
 
Theorematlle0 29566 An element less than or equal to zero equals zero. (chle0 22335 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 29567 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 29568* The predicate "is an atom". (elat2 23233 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 29569 An atom is not zero. (atne0 23238 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  AtLat  /\  P  e.  A ) 
 ->  P  =/=  .0.  )
 
Theorematnle0 29570 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 29571 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 29572 If two atoms are comparable, they are equal. (atsseq 23240 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 29573 Frequently-used variation of atcmp 29572. (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 29574 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 29575 An element covered by an atom must be zero. (atcveq0 23241 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 29576 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 29577* Every nonzero element of an atomic lattice is greater than or equal to an atom. (hatomic 23253 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 29578 Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 23269 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 29579 The meet of distinct atoms is zero. (atnemeq0 23270 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 29580* 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 23255 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 29581* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 23264 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 29582* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 23256, 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 29583* 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 29584* 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 29585* 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 29586 An atomic lattice with the covering property is an atomic lattice. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  CvLat  ->  K  e.  AtLat )
 
Theoremcvllat 29587 An atomic lattice with the covering property is a lattice. (Contributed by NM, 5-Nov-2012.)
 |-  ( K  e.  CvLat  ->  K  e.  Lat )
 
TheoremcvlposN 29588 An atomic lattice with the covering property is a poset. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
 |-  ( K  e.  CvLat  ->  K  e.  Poset )
 
Theoremcvlexch1 29589 An atomic covering lattice has the exchange property. (Contributed by NM, 6-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q )  ->  Q  .<_  ( X 
 .\/  P ) ) )
 
Theoremcvlexch2 29590 An atomic covering lattice has the exchange property. (Contributed by NM, 6-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( Q  .\/  X )  ->  Q  .<_  ( P 
 .\/  X ) ) )
 
Theoremcvlexchb1 29591 An atomic covering lattice has the exchange property. (Contributed by NM, 16-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( X  .\/  Q ) 
 <->  ( X  .\/  P )  =  ( X  .\/  Q ) ) )
 
Theoremcvlexchb2 29592 An atomic covering lattice has the exchange property. (Contributed by NM, 22-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  -.  P  .<_  X )  ->  ( P  .<_  ( Q  .\/  X ) 
 <->  ( P  .\/  X )  =  ( Q  .\/  X ) ) )
 
Theoremcvlexch3 29593 An atomic covering lattice has the exchange property. (atexch 23274 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  ./\  X )  =  .0.  )  ->  ( P  .<_  ( X 
 .\/  Q )  ->  Q  .<_  ( X  .\/  P ) ) )
 
Theoremcvlexch4N 29594 An atomic covering lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (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  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  ./\  X )  =  .0.  )  ->  ( P  .<_  ( X 
 .\/  Q )  <->  ( X  .\/  P )  =  ( X 
 .\/  Q ) ) )
 
Theoremcvlatexchb1 29595 A version of cvlexchb1 29591 for atoms. (Contributed by NM, 5-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R )  ->  ( P  .<_  ( R  .\/  Q )  <->  ( R  .\/  P )  =  ( R  .\/  Q ) ) )
 
Theoremcvlatexchb2 29596 A version of cvlexchb2 29592 for atoms. (Contributed by NM, 5-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R )  ->  ( P  .<_  ( Q  .\/  R )  <->  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
 
Theoremcvlatexch1 29597 Atom exchange property. (Contributed by NM, 5-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R )  ->  ( P  .<_  ( R  .\/  Q )  ->  Q  .<_  ( R  .\/  P ) ) )
 
Theoremcvlatexch2 29598 Atom exchange property. (Contributed by NM, 5-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R )  ->  ( P  .<_  ( Q  .\/  R )  ->  Q  .<_  ( P  .\/  R ) ) )
 
Theoremcvlatexch3 29599 Atom exchange property. (Contributed by NM, 29-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q 
 /\  P  =/=  R ) )  ->  ( P 
 .<_  ( Q  .\/  R )  ->  ( P  .\/  Q )  =  ( P 
 .\/  R ) ) )
 
Theoremcvlcvr1 29600 The covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 23248 analog.) (Contributed by NM, 5-Nov-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  ( -.  P  .<_  X  <->  X C ( X 
 .\/  P ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 32201-32225
  Copyright terms: Public domain < Previous  Next >