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Theorem List for Metamath Proof Explorer - 28501-28600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2llnne2N 28501 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 28502 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 28503 A Hilbert lattice has the covering property. Proposition 1(ii) in [Kalmbach] p. 140 (and its converse). (chcv1 22765 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 28504 Less-than and covers equivalence in a Hilbert lattice. (chcv2 22766 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 28505* The Hilbert lattice is relatively atomic. Stronger version of hlrelat 28495. (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 28506* 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 28507* 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 28508* 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 28509 A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22785 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 28510 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 28511 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 28512 Lemma for cvrexch 28513. (cvexchlem 22778 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 28513 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 22779 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 28514 Lemma for cvrat 28515. (atcvatlem 22795 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 28515 A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 22796 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 28516 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 28517 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 28518 The covers relation is not transitive. (cvntr 22702 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 28519 The covers relation is not transitive. (atcv0eq 22789 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 28520 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 28521 Two atoms covering the zero subspace are equal. (atcv1 22790 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 28522 A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 22797 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 28523 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 28524 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 28525 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 28526 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 28527 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 28528 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 28529* 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 28530 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 28531 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 28532* 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 28533 Atom exchange property. Version of hlatexch2 28489 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 28534 Atom exchange property. Version of hlatexch2 28489 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 28535 A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22806 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 28536* 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 22807 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 28537* Commuted version of cvrat4 28536. (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 28538 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 28539 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 28540* 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 28541* 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 28542 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 28543 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 28544 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 28545 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 28546 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 28547 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 28548 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 28549* 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 28550* 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 28551 Lemma for 3dim1 28560. (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 28552 Lemma for 3dim1 28560. (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 28553 Lemma for 3dim3 28562. (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 28554 Lemma for 3dim1 28560. (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 28555 Lemma for 3dim1 28560. (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 28556 Lemma for 3dim3 28562. (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 28557 Lemma for 3dim1 28560. (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 28558 Lemma for 3dim1 28560. (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 28559* Lemma for 3dim1 28560. (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 28560* 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 28561* 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 28562* 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 28563* 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 28564* 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 28565 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 28566* 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 28567 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 28568 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 28569 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 28570 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 28571* 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 28572 Two atoms are be 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 28573 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 28574 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 28575 Variation of projective geometry axiom ps-2 28571. (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 28576 Lemma for 3at 28583. (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 28577 Lemma for 3at 28583. (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 28578 Lemma for 3at 28583. (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 28579 Lemma for 3at 28583. (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 28580 Lemma for 3at 28583. (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 28581 Lemma for 3at 28583. (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 28582 Lemma for 3at 28583. (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 )
 )
 
Theorem3at 28583 Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analog of ps-1 28570 for lines and 4at 28706 for volumes. I could not find this proof in the literature on projective geometry (where it is either given as an axiom or stated as an unproved fact), but it is similar to Theorem 15 of Veblen, "The Foundations of Geometry" (1911), p. 18, which uses different axioms. This proof was written before I became aware of Veblen's, and it is possible that a shorter proof could be obtained by using Veblen's proof for hints. (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 )
 ) )
 
16.23.13  Projective geometries based on Hilbert lattices
 
Syntaxclln 28584 Extend class notation with set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice.
 class  LLines
 
Syntaxclpl 28585 Extend class notation with set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice.
 class  LPlanes
 
Syntaxclvol 28586 Extend class notation with set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice.
 class  LVols
 
Syntaxclines 28587 Extend class notation with set of all projective lines for a Hilbert lattice.
 class  Lines
 
SyntaxcpointsN 28588 Extend class notation with set of all projective points.
 class  Points
 
Syntaxcpsubsp 28589 Extend class notation with set of all projective subspaces.
 class  PSubSp
 
Syntaxcpmap 28590 Extend class notation with projective map.
 class  pmap
 
Definitiondf-llines 28591* Define the set of all "lattice lines" (lattice elements which cover an atom) in a Hilbert lattice  k, in other words all elements of height 2 (or lattice dimension 2 or projective dimension 1). (Contributed by NM, 16-Jun-2012.)
 |-  LLines  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. p  e.  ( Atoms `  k ) p ( 
 <o  `  k ) x } )
 
Definitiondf-lplanes 28592* Define the set of all "lattice planes" (lattice elements which cover a line) in a Hilbert lattice  k, in other words all elements of height 3 (or lattice dimension 3 or projective dimension 2). (Contributed by NM, 16-Jun-2012.)
 |-  LPlanes  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. p  e.  ( LLines `  k ) p ( 
 <o  `  k ) x } )
 
Definitiondf-lvols 28593* Define the set of all 3-dimensional "lattice volumes" (lattice elements which cover a plane) in a Hilbert lattice  k, in other words all elements of height 4 (or lattice dimension 4 or projective dimension 3). (Contributed by NM, 1-Jul-2012.)
 |-  LVols  =  ( k  e.  _V  |->  { x  e.  ( Base `  k )  |  E. p  e.  ( LPlanes `  k ) p ( 
 <o  `  k ) x } )
 
Definitiondf-lines 28594* Define set of all projective lines for a Hilbert lattice (actually in any set at all, for simplicity). The join of two distinct atoms equals a line. Definition of lines in item 1 of [Holland95] p. 222. (Contributed by NM, 19-Sep-2011.)
 |-  Lines  =  ( k  e.  _V  |->  { s  |  E. q  e.  ( Atoms `  k ) E. r  e.  ( Atoms `  k ) ( q  =/=  r  /\  s  =  { p  e.  ( Atoms `  k )  |  p ( le `  k
 ) ( q (
 join `  k ) r ) } ) }
 )
 
Definitiondf-pointsN 28595* Define set of all projective points in a Hilbert lattice (actually in any set at all, for simplicity). A projective point is the singleton of a lattice atom. Definition 15.1 of [MaedaMaeda] p. 61. Note that item 1 in [Holland95] p. 222 defines a point as the atom itself, but this leads to a complicated subspace ordering that may be either membership or inclusion based on its arguments. (Contributed by NM, 2-Oct-2011.)
 |-  Points  =  ( k  e.  _V  |->  { q  |  E. p  e.  ( Atoms `  k )
 q  =  { p } } )
 
Definitiondf-psubsp 28596* Define set of all projective subspaces. Based on definition of subspace in [Holland95] p. 212. (Contributed by NM, 2-Oct-2011.)
 |-  PSubSp  =  ( k  e.  _V  |->  { s  |  ( s 
 C_  ( Atoms `  k
 )  /\  A. p  e.  s  A. q  e.  s  A. r  e.  ( Atoms `  k )
 ( r ( le `  k ) ( p ( join `  k )
 q )  ->  r  e.  s ) ) }
 )
 
Definitiondf-pmap 28597* Define projective map for  k at  a. Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)
 |-  pmap  =  ( k  e.  _V  |->  ( a  e.  ( Base `  k )  |->  { p  e.  ( Atoms `  k )  |  p ( le `  k ) a } ) )
 
Theoremllnset 28598* The set of lattice lines in a Hilbert lattice. (Contributed by NM, 16-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 LLines `  K )   =>    |-  ( K  e.  D  ->  N  =  { x  e.  B  |  E. p  e.  A  p C x } )
 
Theoremislln 28599* The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 LLines `  K )   =>    |-  ( K  e.  D  ->  ( X  e.  N 
 <->  ( X  e.  B  /\  E. p  e.  A  p C X ) ) )
 
Theoremislln4 28600* The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  N  =  (
 LLines `  K )   =>    |-  ( ( K  e.  D  /\  X  e.  B )  ->  ( X  e.  N  <->  E. p  e.  A  p C X ) )
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