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Theorem List for Metamath Proof Explorer - 30801-30900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-watsN 30801* Define W-atoms corresponding to an arbitrary "fiducial (i.e. reference) atom"  d. These are all atoms not in the polarity of  { d } ), which is the hyperplane determined by  d. Definition of set W in [Crawley] p. 111. (Contributed by NM, 26-Jan-2012.)
 |-  WAtoms  =  ( k  e.  _V  |->  ( d  e.  ( Atoms `  k )  |->  ( (
 Atoms `  k )  \  ( ( _|_ P `  k ) `  { d } ) ) ) )
 
Definitiondf-pautN 30802* Define set of all projective automorphisms. This is the intended definition of automorphism in [Crawley] p. 112. (Contributed by NM, 26-Jan-2012.)
 |-  PAut  =  ( k  e.  _V  |->  { f  |  ( f : ( PSubSp `  k
 )
 -1-1-onto-> ( PSubSp `  k )  /\  A. x  e.  ( PSubSp `
  k ) A. y  e.  ( PSubSp `  k ) ( x 
 C_  y  <->  ( f `  x )  C_  ( f `
  y ) ) ) } )
 
TheoremwatfvalN 30803* The W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  P  =  ( _|_ P `  K )   &    |-  W  =  (
 WAtoms `  K )   =>    |-  ( K  e.  B  ->  W  =  ( d  e.  A  |->  ( A  \  ( ( _|_ P `  K ) `  { d }
 ) ) ) )
 
TheoremwatvalN 30804 Value of the W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  P  =  ( _|_ P `  K )   &    |-  W  =  (
 WAtoms `  K )   =>    |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( W `  D )  =  ( A  \  (
 ( _|_ P `  K ) `  { D }
 ) ) )
 
TheoremiswatN 30805 The predicate "is a W atom" (corresponding to fiducial atom  D). (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  P  =  ( _|_ P `  K )   &    |-  W  =  (
 WAtoms `  K )   =>    |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( P  e.  ( W `  D )  <->  ( P  e.  A  /\  -.  P  e.  ( ( _|_ P `  K ) `  { D } ) ) ) )
 
Theoremlhpset 30806* The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( K  e.  A  ->  H  =  { w  e.  B  |  w C  .1.  } )
 
Theoremislhp 30807 The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 11-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( K  e.  A  ->  ( W  e.  H 
 <->  ( W  e.  B  /\  W C  .1.  )
 ) )
 
Theoremislhp2 30808 The predicate "is a co-atom (lattice hyperplane)." (Contributed by NM, 18-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  A  /\  W  e.  B )  ->  ( W  e.  H  <->  W C  .1.  )
 )
 
Theoremlhpbase 30809 A co-atom is a member of the lattice base set (i.e. a lattice element). (Contributed by NM, 18-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( W  e.  H  ->  W  e.  B )
 
Theoremlhp1cvr 30810 The lattice unit covers a co-atom (lattice hyperplane). (Contributed by NM, 18-May-2012.)
 |-  .1.  =  ( 1. `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  A  /\  W  e.  H )  ->  W C  .1.  )
 
Theoremlhplt 30811 An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  P  .<_  W ) )  ->  P  .<  W )
 
Theoremlhp2lt 30812 The join of two atoms under a co-atom is strictly less than it. (Contributed by NM, 8-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  .\/  =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  P  .<_  W ) 
 /\  ( Q  e.  A  /\  Q  .<_  W ) )  ->  ( P  .\/  Q )  .<  W )
 
Theoremlhpexlt 30813* There exists an atom less than a co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
 |-  .<  =  ( lt `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<  W )
 
Theoremlhp0lt 30814 A co-atom is greater than zero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
 |-  .<  =  ( lt `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  .<  W )
 
Theoremlhpn0 30815 A co-atom is nonzero. TODO: is this needed? (Contributed by NM, 26-Apr-2013.)
 |-  .0.  =  ( 0. `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  W  =/=  .0.  )
 
Theoremlhpexle 30816* There exists an atom under a co-atom. (Contributed by NM, 26-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  p  .<_  W )
 
Theoremlhpexnle 30817* There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  -.  p  .<_  W )
 
Theoremlhpexle1lem 30818* Lemma for lhpexle1 30819 and others that eliminates restrictions on  X. (Contributed by NM, 24-Jul-2013.)
 |-  ( ph  ->  E. p  e.  A  ( p  .<_  W  /\  ps ) )   &    |-  ( ( ph  /\  ( X  e.  A  /\  X  .<_  W ) ) 
 ->  E. p  e.  A  ( p  .<_  W  /\  ps 
 /\  p  =/=  X ) )   =>    |-  ( ph  ->  E. p  e.  A  ( p  .<_  W 
 /\  ps  /\  p  =/= 
 X ) )
 
Theoremlhpexle1 30819* There exists an atom under a co-atom different from any given element. (Contributed by NM, 24-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W 
 /\  p  =/=  X ) )
 
Theoremlhpexle2lem 30820* Lemma for lhpexle2 30821. (Contributed by NM, 19-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) ) 
 ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) )
 
Theoremlhpexle2 30821* There exists atom under a co-atom different from any two other elements. (Contributed by NM, 24-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W 
 /\  p  =/=  X  /\  p  =/=  Y ) )
 
Theoremlhpexle3lem 30822* There exists atom under a co-atom different from any 3 other atoms. TODO: study if adant*,simp* usage can be improved. (Contributed by NM, 9-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  Y  e.  A  /\  Z  e.  A )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  Z  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  =/=  Z ) ) )
 
Theoremlhpexle3 30823* There exists atom under a co-atom different from any three other elements. (Contributed by NM, 24-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W 
 /\  ( p  =/= 
 X  /\  p  =/=  Y 
 /\  p  =/=  Z ) ) )
 
Theoremlhpex2leN 30824* There exist at least two different atoms under a co-atom. This allows us to create a line under the co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  E. q  e.  A  ( p  .<_  W 
 /\  q  .<_  W  /\  p  =/=  q ) )
 
Theoremlhpoc 30825 The orthocomplement of a co-atom (lattice hyperplane) is an atom. (Contributed by NM, 18-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  B )  ->  ( W  e.  H  <->  (  ._|_  `  W )  e.  A )
 )
 
Theoremlhpoc2N 30826 The orthocomplement of an atom is a co-atom (lattice hyperplane). (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ._|_  =  ( oc `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  B )  ->  ( W  e.  A  <->  (  ._|_  `  W )  e.  H )
 )
 
Theoremlhpocnle 30827 The orthocomplement of a co-atom is not under it. (Contributed by NM, 22-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  -.  (  ._|_  `  W )  .<_  W )
 
Theoremlhpocat 30828 The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013.)
 |-  ._|_  =  ( oc `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  ._|_  `  W )  e.  A )
 
Theoremlhpocnel 30829 The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 25-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 ._|_  =  ( oc `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  ( (  ._|_  `  W )  e.  A  /\  -.  (  ._|_  `  W ) 
 .<_  W ) )
 
Theoremlhpocnel2 30830 The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 20-Feb-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
 
Theoremlhpjat1 30831 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 18-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( W  .\/  P )  =  .1.  )
 
Theoremlhpjat2 30832 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 4-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .\/  W )  =  .1.  )
 
Theoremlhpj1 30833 The join of a co-atom (hyperplane) and an element not under it is the lattice unit. (Contributed by NM, 7-Dec-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) 
 ->  ( W  .\/  X )  =  .1.  )
 
Theoremlhpmcvr 30834 The meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 7-Dec-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  ( X  ./\ 
 W ) C X )
 
Theoremlhpmcvr2 30835* Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  ( p 
 .\/  ( X  ./\  W ) )  =  X ) )
 
Theoremlhpmcvr3 30836 Specialization of lhpmcvr2 30835. TODO: Use this to simplify many uses of  ( P  .\/  ( X  ./\  W ) )  =  X to become  P  .<_  X. (Contributed by NM, 6-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) 
 ->  ( P  .<_  X  <->  ( P  .\/  ( X  ./\  W ) )  =  X ) )
 
Theoremlhpmcvr4N 30837 Specialization of lhpmcvr2 30835. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( X  e.  B  /\  -.  X  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y )  .<_  W 
 /\  P  .<_  X ) )  ->  -.  P  .<_  Y )
 
Theoremlhpmcvr5N 30838* Specialization of lhpmcvr2 30835. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X  ./\  Y )  .<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p 
 .\/  ( X  ./\  W ) )  =  X ) )
 
Theoremlhpmcvr6N 30839* Specialization of lhpmcvr2 30835. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X  ./\  Y )  .<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  p  .<_  X ) )
 
Theoremlhpm0atN 30840 If the meet of a lattice hyperplane with a nonzero element is zero, the element is an atom. (Contributed by NM, 28-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  =/=  .0.  /\  ( X  ./\  W )  =  .0.  ) ) 
 ->  X  e.  A )
 
Theoremlhpmat 30841 An element covered by the lattice unit, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  ./\  W )  =  .0.  )
 
Theoremlhpmatb 30842 An element covered by the lattice unit, when conjoined with an atom, equals zero iff the atom is not under it. (Contributed by NM, 15-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  ->  ( -.  P  .<_  W  <->  ( P  ./\  W )  =  .0.  )
 )
 
Theoremlhp2at0 30843 Join and meet with different atoms under co-atom  W. (Contributed by NM, 15-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) ) 
 ->  ( ( P  .\/  U )  ./\  V )  =  .0.  )
 
Theoremlhp2atnle 30844 Inequality for 2 different atoms under co-atom  W. (Contributed by NM, 17-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) ) 
 ->  -.  V  .<_  ( P 
 .\/  U ) )
 
Theoremlhp2atne 30845 Inequality for joins with 2 different atoms under co-atom  W. (Contributed by NM, 22-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( ( U  e.  A  /\  U  .<_  W ) 
 /\  ( V  e.  A  /\  V  .<_  W ) )  /\  U  =/=  V )  ->  ( P  .\/  U )  =/=  ( Q  .\/  V ) )
 
Theoremlhp2at0nle 30846 Inequality for 2 different atoms (or an atom and zero) under co-atom  W. (Contributed by NM, 28-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  U  =/=  V )  /\  ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) ) 
 ->  -.  V  .<_  ( P 
 .\/  U ) )
 
Theoremlhp2at0ne 30847 Inequality for joins with 2 different atoms (or an atom and zero) under co-atom  W. (Contributed by NM, 28-Jul-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( ( ( U  e.  A  \/  U  =  .0.  )  /\  U  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) ) 
 /\  U  =/=  V )  ->  ( P  .\/  U )  =/=  ( Q 
 .\/  V ) )
 
Theoremlhpelim 30848 Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 30841 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  X  e.  B )  ->  ( ( P  .\/  ( X  ./\ 
 W ) )  ./\  W )  =  ( X 
 ./\  W ) )
 
Theoremlhpmod2i2 30849 Modular law for hyperplanes analogous to atmod2i2 30673 for atoms. (Contributed by NM, 9-Feb-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  Y  .<_  X ) 
 ->  ( ( X  ./\  W )  .\/  Y )  =  ( X  ./\  ( W  .\/  Y ) ) )
 
Theoremlhpmod6i1 30850 Modular law for hyperplanes analogous to complement of atmod2i1 30672 for atoms. (Contributed by NM, 1-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  X  .<_  W ) 
 ->  ( X  .\/  ( Y  ./\  W ) )  =  ( ( X 
 .\/  Y )  ./\  W ) )
 
Theoremlhprelat3N 30851* The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 30223. (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .<  =  ( lt `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  E. w  e.  H  ( X  .<_  ( Y  ./\  w )  /\  ( Y  ./\  w ) C Y ) )
 
Theoremcdlemb2 30852* Given two atoms not under the fiducial (reference) co-atom  W, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 30-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  -.  r  .<_  ( P  .\/  Q ) ) )
 
Theoremlhple 30853 Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( ( P  .\/  X )  ./\  W )  =  X )
 
Theoremlhpat 30854 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  ( ( P  .\/  Q )  ./\  W )  e.  A )
 
Theoremlhpat4N 30855 Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( U  e.  A  /\  U  .<_  W ) )  ->  ( ( P  .\/  U )  ./\  W )  =  U )
 
Theoremlhpat2 30856 Create an atom under a co-atom. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 21-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  R  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  R  e.  A )
 
Theoremlhpat3 30857 There is only one atom under both 
P  .\/  Q and co-atom  W. (Contributed by NM, 21-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  R  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) 
 /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q 
 /\  S  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( -.  S  .<_  W  <->  S  =/=  R ) )
 
Theorem4atexlemk 30858 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   =>    |-  ( ph  ->  K  e.  HL )
 
Theorem4atexlemw 30859 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   =>    |-  ( ph  ->  W  e.  H )
 
Theorem4atexlempw 30860 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   =>    |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
 
Theorem4atexlemp 30861 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   =>    |-  ( ph  ->  P  e.  A )
 
Theorem4atexlemq 30862 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   =>    |-  ( ph  ->  Q  e.  A )
 
Theorem4atexlems 30863 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   =>    |-  ( ph  ->  S  e.  A )
 
Theorem4atexlemt 30864 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   =>    |-  ( ph  ->  T  e.  A )
 
Theorem4atexlemutvt 30865 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   =>    |-  ( ph  ->  ( U  .\/  T )  =  ( V  .\/  T )
 )
 
Theorem4atexlempnq 30866 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   =>    |-  ( ph  ->  P  =/=  Q )
 
Theorem4atexlemnslpq 30867 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   =>    |-  ( ph  ->  -.  S  .<_  ( P  .\/  Q )
 )
 
Theorem4atexlemkl 30868 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   =>    |-  ( ph  ->  K  e.  Lat )
 
Theorem4atexlemkc 30869 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   =>    |-  ( ph  ->  K  e.  CvLat
 )
 
Theorem4atexlemwb 30870 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( ph  ->  W  e.  ( Base `  K )
 )
 
Theorem4atexlempsb 30871 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .\/ 
 =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ph  ->  ( P  .\/  S )  e.  ( Base `  K )
 )
 
Theorem4atexlemqtb 30872 Lemma for 4atexlem7 30886. (Contributed by NM, 24-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .\/ 
 =  ( join `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  ( ph  ->  ( Q  .\/  T )  e.  ( Base `  K )
 )
 
Theorem4atexlempns 30873 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ph  ->  P  =/=  S )
 
Theorem4atexlemswapqr 30874 Lemma for 4atexlem7 30886. Swap  Q and  R, so that theorems involving  C can be reused for  D. Note that  U must be expanded because it involves  Q. (Contributed by NM, 25-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  ( ph  ->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( S  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W  /\  ( P  .\/  Q )  =  ( R  .\/  Q ) )  /\  ( T  e.  A  /\  ( ( ( P 
 .\/  R )  ./\  W ) 
 .\/  T )  =  ( V  .\/  T )
 ) )  /\  ( P  =/=  R  /\  -.  S  .<_  ( P  .\/  R ) ) ) )
 
Theorem4atexlemu 30875 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  ( ph  ->  U  e.  A )
 
Theorem4atexlemv 30876 Lemma for 4atexlem7 30886. (Contributed by NM, 23-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  V  =  ( ( P  .\/  S )  ./\  W )   =>    |-  ( ph  ->  V  e.  A )
 
Theorem4atexlemunv 30877 Lemma for 4atexlem7 30886. (Contributed by NM, 21-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  V  =  ( ( P  .\/  S )  ./\  W )   =>    |-  ( ph  ->  U  =/=  V )
 
Theorem4atexlemtlw 30878 Lemma for 4atexlem7 30886. (Contributed by NM, 24-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  V  =  ( ( P  .\/  S )  ./\  W )   =>    |-  ( ph  ->  T  .<_  W )
 
Theorem4atexlemntlpq 30879 Lemma for 4atexlem7 30886. (Contributed by NM, 24-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  V  =  ( ( P  .\/  S )  ./\  W )   =>    |-  ( ph  ->  -.  T  .<_  ( P  .\/  Q )
 )
 
Theorem4atexlemc 30880 Lemma for 4atexlem7 30886. (Contributed by NM, 24-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  V  =  ( ( P  .\/  S )  ./\  W )   &    |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )   =>    |-  ( ph  ->  C  e.  A )
 
Theorem4atexlemnclw 30881 Lemma for 4atexlem7 30886. (Contributed by NM, 24-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  V  =  ( ( P  .\/  S )  ./\  W )   &    |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )   =>    |-  ( ph  ->  -.  C  .<_  W )
 
Theorem4atexlemex2 30882* Lemma for 4atexlem7 30886. Show that when  C  =/=  S,  C satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  V  =  ( ( P  .\/  S )  ./\  W )   &    |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )   =>    |-  ( ( ph  /\  C  =/=  S ) 
 ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
 
Theorem4atexlemcnd 30883 Lemma for 4atexlem7 30886. (Contributed by NM, 24-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  V  =  ( ( P  .\/  S )  ./\  W )   &    |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )   &    |-  D  =  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )   =>    |-  ( ph  ->  C  =/=  D )
 
Theorem4atexlemex4 30884* Lemma for 4atexlem7 30886. Show that when  C  =  S,  D satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012.)
 |-  ( ph 
 <->  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  ( T  e.  A  /\  ( U  .\/  T )  =  ( V  .\/  T ) ) )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) ) )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  V  =  ( ( P  .\/  S )  ./\  W )   &    |-  C  =  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )   &    |-  D  =  ( ( R  .\/  T )  ./\  ( P  .\/  S ) )   =>    |-  ( ( ph  /\  C  =  S ) 
 ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
 
Theorem4atexlemex6 30885* Lemma for 4atexlem7 30886. (Contributed by NM, 25-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  ( ( P  .\/  R )  =  ( Q 
 .\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
 
Theorem4atexlem7 30886* Whenever there are at least 4 atoms under  P  .\/  Q (specifically,  P,  Q,  r, and  ( P  .\/  Q
)  ./\  W), there are also at least 4 atoms under  P  .\/  S. This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p  \/ q/0 and hence p  \/ s/0 contains at least four atoms..." Note that by cvlsupr2 30155, our  ( P  .\/  r )  =  ( Q  .\/  r ) is a shorter way to express  r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ). With a longer proof, the condition  -.  S  .<_  ( P  .\/  Q ) could be eliminated (see 4atex 30887), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  S  e.  A )  /\  ( P  =/=  Q 
 /\  -.  S  .<_  ( P  .\/  Q )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P 
 .\/  z )  =  ( S  .\/  z
 ) ) )
 
Theorem4atex 30887* Whenever there are at least 4 atoms under  P  .\/  Q (specifically,  P,  Q,  r, and  ( P  .\/  Q
)  ./\  W), there are also at least 4 atoms under  P  .\/  S. This proves the statement in Lemma E of [Crawley] p. 114, last line, "...p  \/ q/0 and hence p  \/ s/0 contains at least four atoms..." Note that by cvlsupr2 30155, our  ( P  .\/  r )  =  ( Q  .\/  r ) is a shorter way to express  r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ). (Contributed by NM, 27-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  S  e.  A )  /\  ( P  =/=  Q 
 /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P 
 .\/  z )  =  ( S  .\/  z
 ) ) )
 
Theorem4atex2 30888* More general version of 4atex 30887 for a line  S 
.\/  T not necessarily connected to  P  .\/  Q. (Contributed by NM, 27-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  T  e.  A  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) ) ) ) 
 ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( S  .\/  z )  =  ( T  .\/  z ) ) )
 
Theorem4atex2-0aOLDN 30889* Same as 4atex2 30888 except that  S is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( S 
 .\/  z )  =  ( T  .\/  z
 ) ) )
 
Theorem4atex2-0bOLDN 30890* Same as 4atex2 30888 except that  T is zero. (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  T  =  ( 0. `  K )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) ) ) ) 
 ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( S  .\/  z )  =  ( T  .\/  z ) ) )
 
Theorem4atex2-0cOLDN 30891* Same as 4atex2 30888 except that  S and 
T are zero. TODO: do we need this one or 4atex2-0aOLDN 30889 or 4atex2-0bOLDN 30890? (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  T  =  ( 0. `  K )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) ) ) ) 
 ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( S  .\/  z )  =  ( T  .\/  z ) ) )
 
Theorem4atex3 30892* More general version of 4atex 30887 for a line  S 
.\/  T not necessarily connected to  P  .\/  Q. (Contributed by NM, 29-May-2013.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  ( T  e.  A  /\  S  =/=  T )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/=  S  /\  z  =/=  T  /\  z  .<_  ( S  .\/  T )
 ) ) )
 
Theoremlautset 30893* The set of lattice automorphisms. (Contributed by NM, 11-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  ( K  e.  A  ->  I  =  { f  |  ( f : B -1-1-onto-> B  /\  A. x  e.  B  A. y  e.  B  ( x  .<_  y  <->  ( f `  x )  .<_  ( f `
  y ) ) ) } )
 
Theoremislaut 30894* The predictate "is a lattice automorphism." (Contributed by NM, 11-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  ( K  e.  A  ->  ( F  e.  I  <->  ( F : B -1-1-onto-> B  /\  A. x  e.  B  A. y  e.  B  ( x  .<_  y  <-> 
 ( F `  x )  .<_  ( F `  y ) ) ) ) )
 
Theoremlautle 30895 Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  ( ( ( K  e.  V  /\  F  e.  I )  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
  Y ) ) )
 
Theoremlaut1o 30896 A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  (
 ( K  e.  A  /\  F  e.  I ) 
 ->  F : B -1-1-onto-> B )
 
Theoremlaut11 30897 One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  (
 ( ( K  e.  V  /\  F  e.  I
 )  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( ( F `  X )  =  ( F `  Y )  <->  X  =  Y ) )
 
Theoremlautcl 30898 A lattice automorphism value belongs to the base set. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  (
 ( ( K  e.  V  /\  F  e.  I
 )  /\  X  e.  B )  ->  ( F `
  X )  e.  B )
 
TheoremlautcnvclN 30899 Reverse closure of a lattice automorphism. (Contributed by NM, 25-May-2012.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  (
 ( ( K  e.  V  /\  F  e.  I
 )  /\  X  e.  B )  ->  ( `' F `  X )  e.  B )
 
Theoremlautcnvle 30900 Less-than or equal property of lattice automorphism converse. (Contributed by NM, 19-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  ( ( ( K  e.  V  /\  F  e.  I )  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X  .<_  Y  <->  ( `' F `  X )  .<_  ( `' F `  Y ) ) )
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