HomeHome Metamath Proof Explorer
Theorem List (p. 303 of 328)
< 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
(1-21514)
  Hilbert Space Explorer  Hilbert Space Explorer
(21515-23037)
  Users' Mathboxes  Users' Mathboxes
(23038-32776)
 

Theorem List for Metamath Proof Explorer - 30201-30300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhl0lt1N 30201 Lattice 0 is less than lattice 1 in a Hilbert lattice. (Contributed by NM, 4-Dec-2011.) (New usage is discouraged.)
 |-  .<  =  ( lt `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  .1.  =  ( 1. `  K )   =>    |-  ( K  e.  HL  ->  .0.  .<  .1.  )
 
Theoremhlexch3 30202 A Hilbert lattice has the exchange property. (atexch 22977 analog.) (Contributed by NM, 15-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B ) 
 /\  ( P  ./\  X )  =  .0.  )  ->  ( P  .<_  ( X 
 .\/  Q )  ->  Q  .<_  ( X  .\/  P ) ) )
 
Theoremhlexch4N 30203 A Hilbert lattice has the exchange property. Part of Definition 7.8 of [MaedaMaeda] p. 32. (Contributed by NM, 15-Nov-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B ) 
 /\  ( P  ./\  X )  =  .0.  )  ->  ( P  .<_  ( X 
 .\/  Q )  <->  ( X  .\/  P )  =  ( X 
 .\/  Q ) ) )
 
Theoremhlatexchb1 30204 A version of hlexchb1 30195 for atoms. (Contributed by NM, 15-Nov-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R ) 
 ->  ( P  .<_  ( R 
 .\/  Q )  <->  ( R  .\/  P )  =  ( R 
 .\/  Q ) ) )
 
Theoremhlatexchb2 30205 A version of hlexchb2 30196 for atoms. (Contributed by NM, 7-Feb-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R ) 
 ->  ( P  .<_  ( Q 
 .\/  R )  <->  ( P  .\/  R )  =  ( Q 
 .\/  R ) ) )
 
Theoremhlatexch1 30206 Atom exchange property. (Contributed by NM, 7-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R ) 
 ->  ( P  .<_  ( R 
 .\/  Q )  ->  Q  .<_  ( R  .\/  P ) ) )
 
Theoremhlatexch2 30207 Atom exchange property. (Contributed by NM, 8-Jan-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R ) 
 ->  ( P  .<_  ( Q 
 .\/  R )  ->  Q  .<_  ( P  .\/  R ) ) )
 
TheoremhlatmstcOLDN 30208* 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 22958 analog.) (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  U  =  ( lub `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( U `  { y  e.  A  |  y  .<_  X } )  =  X )
 
Theoremhlatle 30209* The ordering of two Hilbert lattice elements is determined by the atoms under them. (chrelat3 22967 analog.) (Contributed by NM, 4-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .<_  Y  <->  A. p  e.  A  ( p  .<_  X  ->  p 
 .<_  Y ) ) )
 
Theoremhlateq 30210* The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 22969 analog.) (Contributed by NM, 24-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( A. p  e.  A  ( p  .<_  X  <-> 
 p  .<_  Y )  <->  X  =  Y ) )
 
Theoremhlrelat1 30211* An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 22959, with  /\ swapped, analog.) (Contributed by NM, 4-Dec-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X 
 .<  Y  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
 
Theoremhlrelat5N 30212* An atomistic lattice with 0 is relatively atomic, using the definition in Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  X  .<  Y ) 
 ->  E. p  e.  A  ( X  .<  ( X 
 .\/  p )  /\  p  .<_  Y ) )
 
Theoremhlrelat 30213* A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 22960 analog.) (Contributed by NM, 4-Feb-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  X  .<  Y ) 
 ->  E. p  e.  A  ( X  .<  ( X 
 .\/  p )  /\  ( X  .\/  p ) 
 .<_  Y ) )
 
Theoremhlrelat2 30214* A consequence of relative atomicity. (chrelat2i 22961 analog.) (Contributed by NM, 5-Feb-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( -.  X  .<_  Y  <->  E. p  e.  A  ( p  .<_  X  /\  -.  p  .<_  Y ) ) )
 
TheoremexatleN 30215 A condition for an atom to be less than or equal to a lattice element. Part of proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  .<_  X  /\  -.  Q  .<_  X  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( R  .<_  X  <->  R  =  P ) )
 
Theoremhl2at 30216* A Hilbert lattice has at least 2 atoms. (Contributed by NM, 5-Dec-2011.)
 |-  A  =  ( Atoms `  K )   =>    |-  ( K  e.  HL  ->  E. p  e.  A  E. q  e.  A  p  =/=  q )
 
Theorematex 30217 At least one atom exists. (Contributed by NM, 15-Jul-2012.)
 |-  A  =  ( Atoms `  K )   =>    |-  ( K  e.  HL  ->  A  =/=  (/) )
 
TheoremintnatN 30218 If the intersection with a non-majorizing element is an atom, the intersecting element is not an atom. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( -.  Y  .<_  X  /\  ( X  ./\  Y )  e.  A ) )  ->  -.  Y  e.  A )
 
Theorem2llnne2N 30219 Condition implying that two intersecting lines are different. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A )  /\  -.  P  .<_  ( R 
 .\/  Q ) )  ->  ( R  .\/  P )  =/=  ( R  .\/  Q ) )
 
Theorem2llnneN 30220 Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( R  .\/  P )  =/=  ( R  .\/  Q ) )
 
Theoremcvr1 30221 A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 22951 analog.) (Contributed by NM, 17-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( -.  P  .<_  X  <->  X C ( X  .\/  P ) ) )
 
Theoremcvr2N 30222 Less-than and covers equivalence in a Hilbert lattice. (chcv2 22952 analog.) (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( X  .<  ( X 
 .\/  P )  <->  X C ( X 
 .\/  P ) ) )
 
Theoremhlrelat3 30223* The Hilbert lattice is relatively atomic. Stronger version of hlrelat 30213. (Contributed by NM, 2-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  E. p  e.  A  ( X C ( X  .\/  p ) 
 /\  ( X  .\/  p )  .<_  Y ) )
 
Theoremcvrval3 30224* Binary relation expressing  Y covers  X. (Contributed by NM, 16-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  E. p  e.  A  ( -.  p  .<_  X  /\  ( X  .\/  p )  =  Y ) ) )
 
Theoremcvrval4N 30225* Binary relation expressing  Y covers  X. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <->  ( X  .<  Y  /\  E. p  e.  A  ( X  .\/  p )  =  Y ) ) )
 
Theoremcvrval5 30226* Binary relation expressing  X covers  X  ./\  Y. (Contributed by NM, 7-Dec-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( ( X  ./\  Y ) C X  <->  E. p  e.  A  ( -.  p  .<_  Y  /\  ( p  .\/  ( X 
 ./\  Y ) )  =  X ) ) )
 
Theoremcvrp 30227 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22971 analog.) (Contributed by NM, 18-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( ( X  ./\  P )  =  .0.  <->  X C ( X 
 .\/  P ) ) )
 
Theorematcvr1 30228 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) 
 ->  ( P  =/=  Q  <->  P C ( P  .\/  Q ) ) )
 
Theorematcvr2 30229 An atom is covered by its join with a different atom. (Contributed by NM, 7-Feb-2012.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) 
 ->  ( P  =/=  Q  <->  P C ( Q  .\/  P ) ) )
 
Theoremcvrexchlem 30230 Lemma for cvrexch 30231. (cvexchlem 22964 analog.) (Contributed by NM, 18-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( ( X  ./\  Y ) C Y  ->  X C ( X  .\/  Y ) ) )
 
Theoremcvrexch 30231 A Hilbert lattice satisfies the exchange axiom. Proposition 1(iii) of [Kalmbach] p. 140 and its converse. Originally proved by Garrett Birkhoff in 1933. (cvexchi 22965 analog.) (Contributed by NM, 18-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( ( X  ./\  Y ) C Y  <->  X C ( X 
 .\/  Y ) ) )
 
Theoremcvratlem 30232 Lemma for cvrat 30233. (atcvatlem 22981 analog.) (Contributed by NM, 22-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A )
 )  /\  ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q )
 ) )  ->  ( -.  P ( le `  K ) X  ->  X  e.  A ) )
 
Theoremcvrat 30233 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 22982 analog.) (Contributed by NM, 22-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A ) )  ->  ( ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q ) )  ->  X  e.  A ) )
 
Theoremltltncvr 30234 A chained strong ordering is not a covers relation. (Contributed by NM, 18-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .<  Y  /\  Y  .<  Z )  ->  -.  X C Z ) )
 
Theoremltcvrntr 30235 Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .<  Y  /\  Y C Z )  ->  -.  X C Z ) )
 
Theoremcvrntr 30236 The covers relation is not transitive. (cvntr 22888 analog.) (Contributed by NM, 18-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  A  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X C Y  /\  Y C Z )  ->  -.  X C Z ) )
 
Theorematcvr0eq 30237 The covers relation is not transitive. (atcv0eq 22975 analog.) (Contributed by NM, 29-Nov-2011.)
 |-  .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  (  .0. 
 C ( P  .\/  Q )  <->  P  =  Q ) )
 
Theoremlnnat 30238 A line (the join of two distinct atoms) is not an atom. (Contributed by NM, 14-Jun-2012.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  -.  ( P  .\/  Q )  e.  A )
 )
 
Theorematcvrj0 30239 Two atoms covering the zero subspace are equal. (atcv1 22976 analog.) (Contributed by NM, 29-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A )  /\  X C ( P 
 .\/  Q ) )  ->  ( X  =  .0.  <->  P  =  Q ) )
 
Theoremcvrat2 30240 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 22983 analog.) (Contributed by NM, 30-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A ) 
 /\  ( P  =/=  Q 
 /\  X C ( P  .\/  Q )
 ) )  ->  X  e.  A )
 
TheorematcvrneN 30241 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) 
 /\  P C ( Q  .\/  R )
 )  ->  Q  =/=  R )
 
Theorematcvrj1 30242 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P C ( Q  .\/  R ) )
 
Theorematcvrj2b 30243 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )
 )  ->  ( ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) )  <->  P C ( Q 
 .\/  R ) ) )
 
Theorematcvrj2 30244 Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( Q  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  P C ( Q  .\/  R ) )
 
TheorematleneN 30245 Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  R  /\  P  .<_  ( Q  .\/  R ) ) )  ->  Q  =/=  R )
 
Theorematltcvr 30246 An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)
 |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  ( P  .<  ( Q  .\/  R ) 
 <->  P C ( Q 
 .\/  R ) ) )
 
Theorematle 30247* Any non-zero element has an atom under it. (Contributed by NM, 28-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  X  e.  B  /\  X  =/=  .0.  )  ->  E. p  e.  A  p  .<_  X )
 
Theorematlt 30248 Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012.)
 |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P 
 .<  ( P  .\/  Q ) 
 <->  P  =/=  Q ) )
 
Theorematlelt 30249 Transfer less-than relation from one atom to another. (Contributed by NM, 7-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  Q  .<  X ) ) 
 ->  P  .<  X )
 
Theorem2atlt 30250* Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  ->  E. q  e.  A  ( q  =/= 
 P  /\  q  .<  X ) )
 
TheorematexchcvrN 30251 Atom exchange property. Version of hlatexch2 30207 with covers relation. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  C  =  (  <o  `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) 
 /\  P  =/=  R )  ->  ( P C ( Q  .\/  R ) 
 ->  Q C ( P 
 .\/  R ) ) )
 
TheorematexchltN 30252 Atom exchange property. Version of hlatexch2 30207 with less-than ordering. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.)
 |-  .<  =  ( lt `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  P  =/=  R ) 
 ->  ( P  .<  ( Q 
 .\/  R )  ->  Q  .<  ( P  .\/  R ) ) )
 
Theoremcvrat3 30253 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22992 analog.) (Contributed by NM, 30-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A )
 )  ->  ( ( P  =/=  Q  /\  -.  Q  .<_  X  /\  P  .<_  ( X  .\/  Q ) )  ->  ( X 
 ./\  ( P  .\/  Q ) )  e.  A ) )
 
Theoremcvrat4 30254* A condition implying existence of an atom with the properties shown. Lemma 3.2.20 in [PtakPulmannova] p. 68. Also Lemma 9.2(delta) in [MaedaMaeda] p. 41. (atcvat4i 22993 analog.) (Contributed by NM, 30-Nov-2011.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A ) )  ->  ( ( X  =/=  .0.  /\  P  .<_  ( X  .\/  Q ) )  ->  E. r  e.  A  ( r  .<_  X 
 /\  P  .<_  ( Q 
 .\/  r ) ) ) )
 
Theoremcvrat42 30255* Commuted version of cvrat4 30254. (Contributed by NM, 28-Jan-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A ) )  ->  ( ( X  =/=  .0.  /\  P  .<_  ( X  .\/  Q ) )  ->  E. r  e.  A  ( r  .<_  X 
 /\  P  .<_  ( r 
 .\/  Q ) ) ) )
 
Theorem2atjm 30256 The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  .<_  X  /\  -.  Q  .<_  X ) ) 
 ->  ( ( P  .\/  Q )  ./\  X )  =  P )
 
Theorematbtwn 30257 Property of a 3rd atom  R on a line  P  .\/  Q intersecting element  X at  P. (Contributed by NM, 30-Jul-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  X  e.  B ) 
 /\  ( P  .<_  X 
 /\  -.  Q  .<_  X 
 /\  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( R  =/=  P  <->  -.  R  .<_  X ) )
 
TheorematbtwnexOLDN 30258* There exists a 3rd atom  r on a line  P  .\/  Q intersecting element  X at  P, such that  r is different from  Q and not in  X. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) ) 
 ->  E. r  e.  A  ( r  =/=  Q  /\  -.  r  .<_  X  /\  r  .<_  ( P  .\/  Q ) ) )
 
Theorematbtwnex 30259* Given atoms  P in  X and  Q not in  X, there exists an atom  r not in  X such that the line  Q  .\/  r intersects  X at  P. (Contributed by NM, 1-Aug-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  .<_  X  /\  -.  Q  .<_  X ) ) 
 ->  E. r  e.  A  ( r  =/=  Q  /\  -.  r  .<_  X  /\  P  .<_  ( Q  .\/  r
 ) ) )
 
Theorem3noncolr2 30260 Two ways to express 3 non-colinear atoms (rotated right 2 places). (Contributed by NM, 12-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( Q  =/=  R  /\  -.  P  .<_  ( Q 
 .\/  R ) ) )
 
Theorem3noncolr1N 30261 Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( R  =/=  P  /\  -.  Q  .<_  ( R 
 .\/  P ) ) )
 
Theoremhlatcon3 30262 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  -.  P  .<_  ( Q 
 .\/  R ) )
 
Theoremhlatcon2 30263 Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  -.  P  .<_  ( R 
 .\/  Q ) )
 
Theorem4noncolr3 30264 A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( P  =/=  Q 
 /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) ) 
 ->  ( Q  =/=  R  /\  -.  S  .<_  ( Q 
 .\/  R )  /\  -.  P  .<_  ( ( Q 
 .\/  R )  .\/  S ) ) )
 
Theorem4noncolr2 30265 A way to express 4 non-colinear atoms (rotated right 2 places). (Contributed by NM, 11-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( P  =/=  Q 
 /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) ) 
 ->  ( R  =/=  S  /\  -.  P  .<_  ( R 
 .\/  S )  /\  -.  Q  .<_  ( ( R 
 .\/  S )  .\/  P ) ) )
 
Theorem4noncolr1 30266 A way to express 4 non-colinear atoms (rotated right 1 places). (Contributed by NM, 11-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( P  =/=  Q 
 /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) ) 
 ->  ( S  =/=  P  /\  -.  Q  .<_  ( S 
 .\/  P )  /\  -.  R  .<_  ( ( S 
 .\/  P )  .\/  Q ) ) )
 
Theoremathgt 30267* A Hilbert lattice, whose height is at least 4, has a chain of 4 successively covering atom joins. (Contributed by NM, 3-May-2012.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( K  e.  HL  ->  E. p  e.  A  E. q  e.  A  ( p C ( p 
 .\/  q )  /\  E. r  e.  A  ( ( p  .\/  q
 ) C ( ( p  .\/  q )  .\/  r )  /\  E. s  e.  A  (
 ( p  .\/  q
 )  .\/  r ) C ( ( ( p  .\/  q )  .\/  r )  .\/  s
 ) ) ) )
 
Theorem3dim0 30268* There exists a 3-dimensional (height-4) element i.e. a volume. (Contributed by NM, 25-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( K  e.  HL  ->  E. p  e.  A  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( p  =/=  q  /\  -.  r  .<_  ( p 
 .\/  q )  /\  -.  s  .<_  ( ( p 
 .\/  q )  .\/  r ) ) )
 
Theorem3dimlem1 30269 Lemma for 3dim1 30278. (Contributed by NM, 25-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R )  .\/  S )
 )  /\  P  =  Q )  ->  ( P  =/=  R  /\  -.  S  .<_  ( P  .\/  R )  /\  -.  T  .<_  ( ( P  .\/  R )  .\/  S )
 ) )
 
Theorem3dimlem2 30270 Lemma for 3dim1 30278. (Contributed by NM, 25-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  -.  S  .<_  ( Q  .\/  R )  /\  -.  T  .<_  ( ( Q  .\/  R )  .\/  S )
 )  /\  ( P  =/=  Q  /\  P  .<_  ( Q  .\/  R )
 ) )  ->  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( ( P  .\/  Q )  .\/  S )
 ) )
 
Theorem3dimlem3a 30271 Lemma for 3dim3 30280. (Contributed by NM, 27-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( -.  T  .<_  ( ( Q 
 .\/  R )  .\/  S )  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q 
 .\/  R )  .\/  S ) ) )  ->  -.  T  .<_  ( ( P 
 .\/  Q )  .\/  R ) )
 
Theorem3dimlem3 30272 Lemma for 3dim1 30278. (Contributed by NM, 25-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( Q  =/=  R 
 /\  -.  T  .<_  ( ( Q  .\/  R )  .\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R )  .\/  S ) ) )  ->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( ( P  .\/  Q )  .\/  R ) ) )
 
Theorem3dimlem3OLDN 30273 Lemma for 3dim1 30278. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( Q  =/=  R 
 /\  -.  T  .<_  ( ( Q  .\/  R )  .\/  S ) ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R )  /\  P  .<_  ( ( Q  .\/  R )  .\/  S ) ) )  ->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( ( P  .\/  Q )  .\/  R ) ) )
 
Theorem3dimlem4a 30274 Lemma for 3dim3 30280. (Contributed by NM, 27-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( -.  S  .<_  ( Q  .\/  R )  /\  -.  P  .<_  ( Q  .\/  R )  /\  -.  P  .<_  ( ( Q  .\/  R )  .\/  S ) ) )  ->  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) )
 
Theorem3dimlem4 30275 Lemma for 3dim1 30278. (Contributed by NM, 25-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( Q  =/=  R 
 /\  -.  S  .<_  ( Q  .\/  R )
 ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R ) )  /\  -.  P  .<_  ( ( Q 
 .\/  R )  .\/  S ) )  ->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R )
 ) )
 
Theorem3dimlem4OLDN 30276 Lemma for 3dim1 30278. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( Q  =/=  R 
 /\  -.  S  .<_  ( Q  .\/  R )
 ) )  /\  ( P  =/=  Q  /\  -.  P  .<_  ( Q  .\/  R ) )  /\  -.  P  .<_  ( ( Q 
 .\/  R )  .\/  S ) )  ->  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R )
 ) )
 
Theorem3dim1lem5 30277* Lemma for 3dim1 30278. (Contributed by NM, 26-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( u  e.  A  /\  v  e.  A  /\  w  e.  A )  /\  ( P  =/=  u  /\  -.  v  .<_  ( P  .\/  u )  /\  -.  w  .<_  ( ( P  .\/  u )  .\/  v )
 ) )  ->  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( P  =/=  q  /\  -.  r  .<_  ( P  .\/  q )  /\  -.  s  .<_  ( ( P  .\/  q )  .\/  r ) ) )
 
Theorem3dim1 30278* Construct a 3-dimensional volume (height-4 element) on top of a given atom  P. (Contributed by NM, 25-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A )  ->  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( P  =/=  q  /\  -.  r  .<_  ( P 
 .\/  q )  /\  -.  s  .<_  ( ( P 
 .\/  q )  .\/  r ) ) )
 
Theorem3dim2 30279* Construct 2 new layers on top of 2 given atoms. (Contributed by NM, 27-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) 
 ->  E. r  e.  A  E. s  e.  A  ( -.  r  .<_  ( P 
 .\/  Q )  /\  -.  s  .<_  ( ( P 
 .\/  Q )  .\/  r
 ) ) )
 
Theorem3dim3 30280* Construct a new layer on top of 3 given atoms. (Contributed by NM, 27-Jul-2012.)
 |-  .\/  =  ( join `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A ) )  ->  E. s  e.  A  -.  s  .<_  ( ( P  .\/  Q )  .\/  R ) )
 
Theorem2dim 30281* Generate a height-3 element (2-dimensional plane) from an atom. (Contributed by NM, 3-May-2012.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A )  ->  E. q  e.  A  E. r  e.  A  ( P C ( P 
 .\/  q )  /\  ( P  .\/  q ) C ( ( P 
 .\/  q )  .\/  r ) ) )
 
Theorem1dimN 30282* An atom is covered by a height-2 element (1-dimensional line). (Contributed by NM, 3-May-2012.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  P  e.  A )  ->  E. q  e.  A  P C ( P  .\/  q ) )
 
Theorem1cvrco 30283 The orthocomplement of an element covered by 1 is an atom. (Contributed by NM, 7-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B ) 
 ->  ( X C  .1.  <->  (  ._|_  `  X )  e.  A ) )
 
Theorem1cvratex 30284* There exists an atom less than an element covered by 1. (Contributed by NM, 7-May-2012.) (Revised by Mario Carneiro, 13-Jun-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |- 
 .1.  =  ( 1. `  K )   &    |-  C  =  ( 
 <o  `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  X  e.  B  /\  X C  .1.  )  ->  E. p  e.  A  p  .<  X )
 
Theorem1cvratlt 30285 An atom less than or equal to an element covered by 1 is less than the element. (Contributed by NM, 7-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  P  .<_  X ) ) 
 ->  P  .<  X )
 
Theorem1cvrjat 30286 An element covered by the lattice unit, when joined with an atom not under it, equals the lattice unit. (Contributed by NM, 30-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  /\  ( X C  .1.  /\ 
 -.  P  .<_  X ) )  ->  ( X  .\/  P )  =  .1.  )
 
Theorem1cvrat 30287 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 30-Apr-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B )  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) ) 
 ->  ( ( P  .\/  Q )  ./\  X )  e.  A )
 
Theoremps-1 30288 The join of two atoms  R  .\/  S (specifying a projective geometry line) is determined uniquely by any two atoms (specifying two points) less than or equal to that join. Part of Lemma 16.4 of [MaedaMaeda] p. 69, showing projective space postulate PS1 in [MaedaMaeda] p. 67. (Contributed by NM, 15-Nov-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  ( ( P  .\/  Q )  .<_  ( R  .\/  S )  <->  ( P  .\/  Q )  =  ( R  .\/  S ) ) )
 
Theoremps-2 30289* Lattice analog for the projective geometry axiom, "if a line intersects two sides of a triangle at different points then it also intersects the third side." Projective space condition PS2 in [MaedaMaeda] p. 68 and part of Theorem 16.4 in [MaedaMaeda] p. 69. (Contributed by NM, 1-Dec-2011.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( -.  P  .<_  ( Q 
 .\/  R )  /\  S  =/=  T )  /\  ( S  .<_  ( P  .\/  Q )  /\  T  .<_  ( Q  .\/  R )
 ) ) )  ->  E. u  e.  A  ( u  .<_  ( P 
 .\/  R )  /\  u  .<_  ( S  .\/  T ) ) )
 
Theorem2atjlej 30290 Two atoms are different if their join majorizes the join of two different atoms. (Contributed by NM, 4-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q ) 
 .<_  ( R  .\/  S ) ) )  ->  R  =/=  S )
 
Theoremhlatexch3N 30291 Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( Q  =/=  R  /\  ( P  .\/  Q )  =  ( P  .\/  R ) ) ) 
 ->  ( P  .\/  Q )  =  ( Q  .\/  R ) )
 
Theoremhlatexch4 30292 Exchange 2 atoms. (Contributed by NM, 13-May-2013.)
 |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) 
 /\  ( R  e.  A  /\  S  e.  A )  /\  ( P  =/=  R 
 /\  Q  =/=  S  /\  ( P  .\/  Q )  =  ( R  .\/  S ) ) ) 
 ->  ( P  .\/  R )  =  ( Q  .\/  S ) )
 
Theoremps-2b 30293 Variation of projective geometry axiom ps-2 30289. (Contributed by NM, 3-Jul-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T 
 /\  ( S  .<_  ( P  .\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) ) 
 ->  ( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/= 
 .0.  )
 
Theorem3atlem1 30294 Lemma for 3at 30301. (Contributed by NM, 22-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
 )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  -.  P  .<_  ( T  .\/  U )  /\  -.  Q  .<_  ( P  .\/  U )
 )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S 
 .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  U ) )
 
Theorem3atlem2 30295 Lemma for 3at 30301. (Contributed by NM, 22-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
 )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/=  U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U ) )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  U )
 )
 
Theorem3atlem3 30296 Lemma for 3at 30301. (Contributed by NM, 23-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
 )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  U 
 /\  -.  Q  .<_  ( P  .\/  U )
 )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S 
 .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  U ) )
 
Theorem3atlem4 30297 Lemma for 3at 30301. (Contributed by NM, 23-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  R ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  R )
 )
 
Theorem3atlem5 30298 Lemma for 3at 30301. (Contributed by NM, 23-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
 )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q 
 /\  -.  Q  .<_  ( P  .\/  U )
 )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S 
 .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  U ) )
 
Theorem3atlem6 30299 Lemma for 3at 30301. (Contributed by NM, 23-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
 )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q 
 /\  Q  .<_  ( P 
 .\/  U ) )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U )
 )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  U ) )
 
Theorem3atlem7 30300 Lemma for 3at 30301. (Contributed by NM, 23-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
 )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q )  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T )  .\/  U )
 )
    < 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 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32776
  Copyright terms: Public domain < Previous  Next >