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Theorem List for Metamath Proof Explorer - 35301-35400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlhpoc2N 35301 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 35302 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 35303 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 35304 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 35305 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 35306 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 35307 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 35308 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 35309 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 35310* 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 35311 Specialization of lhpmcvr2 35310. 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 35312 Specialization of lhpmcvr2 35310. (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 35313* Specialization of lhpmcvr2 35310. (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 35314* Specialization of lhpmcvr2 35310. (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 35315 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 35316 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 35317 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 35318 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 35319 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 35320 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 35321 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 35322 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 35323 Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 35316 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 35324 Modular law for hyperplanes analogous to atmod2i2 35148 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 35325 Modular law for hyperplanes analogous to complement of atmod2i1 35147 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 35326* The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 34698. (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 35327* 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 35328 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 35329 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 35330 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 35331 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 35332 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 35333 Lemma for 4atexlem7 35361. (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 35334 Lemma for 4atexlem7 35361. (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 35335 Lemma for 4atexlem7 35361. (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 35336 Lemma for 4atexlem7 35361. (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 35337 Lemma for 4atexlem7 35361. (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 35338 Lemma for 4atexlem7 35361. (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 35339 Lemma for 4atexlem7 35361. (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 35340 Lemma for 4atexlem7 35361. (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 35341 Lemma for 4atexlem7 35361. (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 35342 Lemma for 4atexlem7 35361. (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 35343 Lemma for 4atexlem7 35361. (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 35344 Lemma for 4atexlem7 35361. (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 35345 Lemma for 4atexlem7 35361. (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 35346 Lemma for 4atexlem7 35361. (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 35347 Lemma for 4atexlem7 35361. (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 35348 Lemma for 4atexlem7 35361. (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 35349 Lemma for 4atexlem7 35361. 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 35350 Lemma for 4atexlem7 35361. (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 35351 Lemma for 4atexlem7 35361. (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 35352 Lemma for 4atexlem7 35361. (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 35353 Lemma for 4atexlem7 35361. (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 35354 Lemma for 4atexlem7 35361. (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 35355 Lemma for 4atexlem7 35361. (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 35356 Lemma for 4atexlem7 35361. (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 35357* Lemma for 4atexlem7 35361. 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 35358 Lemma for 4atexlem7 35361. (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 35359* Lemma for 4atexlem7 35361. 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 35360* Lemma for 4atexlem7 35361. (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 35361* 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 34630, 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 35362), 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 35362* 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 34630, 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 35363* More general version of 4atex 35362 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 35364* Same as 4atex2 35363 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 35365* Same as 4atex2 35363 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 35366* Same as 4atex2 35363 except that  S and 
T are zero. TODO: do we need this one or 4atex2-0aOLDN 35364 or 4atex2-0bOLDN 35365? (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 35367* More general version of 4atex 35362 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 35368* 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 35369* 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 35370 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 35371 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 35372 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 35373 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 35374 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 35375 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 ) ) )
 
Theoremlautcnv 35376 The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)
 |-  I  =  ( LAut `  K )   =>    |-  (
 ( K  e.  V  /\  F  e.  I ) 
 ->  `' F  e.  I
 )
 
Theoremlautlt 35377 Less-than property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<  =  ( lt `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B ) )  ->  ( X  .<  Y  <->  ( F `  X )  .<  ( F `
  Y ) ) )
 
Theoremlautcvr 35378 Covering property of a lattice automorphism. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  ( ( K  e.  A  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B ) )  ->  ( X C Y  <->  ( F `  X ) C ( F `  Y ) ) )
 
Theoremlautj 35379 Meet property of a lattice automorphism. (Contributed by NM, 25-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  (
 ( K  e.  Lat  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B )
 )  ->  ( F `  ( X  .\/  Y ) )  =  (
 ( F `  X )  .\/  ( F `  Y ) ) )
 
Theoremlautm 35380 Meet property of a lattice automorphism. (Contributed by NM, 19-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  (
 ( K  e.  Lat  /\  ( F  e.  I  /\  X  e.  B  /\  Y  e.  B )
 )  ->  ( F `  ( X  ./\  Y ) )  =  ( ( F `  X ) 
 ./\  ( F `  Y ) ) )
 
Theoremlauteq 35381* A lattice automorphism argument is equal to its value if all atoms are equal to their values. (Contributed by NM, 24-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  (
 ( ( K  e.  HL  /\  F  e.  I  /\  X  e.  B ) 
 /\  A. p  e.  A  ( F `  p )  =  p )  ->  ( F `  X )  =  X )
 
Theoremidlaut 35382 The identity function is a lattice automorphism. (Contributed by NM, 18-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  I  =  ( LAut `  K )   =>    |-  ( K  e.  A  ->  (  _I  |`  B )  e.  I )
 
Theoremlautco 35383 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)
 |-  I  =  ( LAut `  K )   =>    |-  (
 ( K  e.  V  /\  F  e.  I  /\  G  e.  I )  ->  ( F  o.  G )  e.  I )
 
TheorempautsetN 35384* The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   &    |-  M  =  ( PAut `  K )   =>    |-  ( K  e.  B  ->  M  =  { f  |  ( f : S -1-1-onto-> S  /\  A. x  e.  S  A. y  e.  S  ( x  C_  y  <->  ( f `  x )  C_  ( f `
  y ) ) ) } )
 
TheoremispautN 35385* The predictate "is a projective automorphism." (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
 |-  S  =  ( PSubSp `  K )   &    |-  M  =  ( PAut `  K )   =>    |-  ( K  e.  B  ->  ( F  e.  M  <->  ( F : S
 -1-1-onto-> S  /\  A. x  e.  S  A. y  e.  S  ( x  C_  y 
 <->  ( F `  x )  C_  ( F `  y ) ) ) ) )
 
Syntaxcldil 35386 Extend class notation with set of all lattice dilations.
 class  LDil
 
Syntaxcltrn 35387 Extend class notation with set of all lattice translations.
 class  LTrn
 
SyntaxcdilN 35388 Extend class notation with set of all dilations.
 class  Dil
 
SyntaxctrnN 35389 Extend class notation with set of all translations.
 class  Trn
 
Definitiondf-ldil 35390* Define set of all lattice dilations. Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
 |-  LDil  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { f  e.  ( LAut `  k )  |  A. x  e.  ( Base `  k ) ( x ( le `  k
 ) w  ->  (
 f `  x )  =  x ) } )
 )
 
Definitiondf-ltrn 35391* Define set of all lattice translations. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
 |-  LTrn  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  { f  e.  ( (
 LDil `  k ) `  w )  |  A. p  e.  ( Atoms `  k ) A. q  e.  ( Atoms `  k ) ( ( -.  p ( le `  k ) w  /\  -.  q
 ( le `  k
 ) w )  ->  ( ( p (
 join `  k ) ( f `  p ) ) ( meet `  k
 ) w )  =  ( ( q (
 join `  k ) ( f `  q ) ) ( meet `  k
 ) w ) ) } ) )
 
Definitiondf-dilN 35392* Define set of all dilations. Definition of dilation in [Crawley] p. 111. (Contributed by NM, 30-Jan-2012.)
 |-  Dil  =  ( k  e.  _V  |->  ( d  e.  ( Atoms `  k )  |->  { f  e.  ( PAut `  k )  |  A. x  e.  ( PSubSp `  k ) ( x 
 C_  ( ( WAtoms `  k ) `  d
 )  ->  ( f `  x )  =  x ) } ) )
 
Definitiondf-trnN 35393* Define set of all translations. Definition of translation in [Crawley] p. 111. (Contributed by NM, 4-Feb-2012.)
 |-  Trn  =  ( k  e.  _V  |->  ( d  e.  ( Atoms `  k )  |->  { f  e.  ( ( Dil `  k ) `  d )  |  A. q  e.  ( ( WAtoms `
  k ) `  d ) A. r  e.  ( ( WAtoms `  k
 ) `  d )
 ( ( q ( +P `  k
 ) ( f `  q ) )  i^i  ( ( _|_P `  k ) `  { d } ) )  =  ( ( r ( +P `  k
 ) ( f `  r ) )  i^i  ( ( _|_P `  k ) `  { d } ) ) }
 ) )
 
Theoremldilfset 35394* The mapping from fiducial co-atom  w to its set of lattice dilations. (Contributed by NM, 11-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  (
 LAut `  K )   =>    |-  ( K  e.  C  ->  ( LDil `  K )  =  ( w  e.  H  |->  { f  e.  I  |  A. x  e.  B  ( x  .<_  w  ->  ( f `  x )  =  x ) } ) )
 
Theoremldilset 35395* The set of lattice dilations for a fiducial co-atom  W. (Contributed by NM, 11-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  (
 LAut `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( K  e.  C  /\  W  e.  H )  ->  D  =  { f  e.  I  |  A. x  e.  B  ( x  .<_  W  ->  ( f `  x )  =  x ) }
 )
 
Theoremisldil 35396* The predicate "is a lattice dilation". Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  (
 LAut `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( K  e.  C  /\  W  e.  H )  ->  ( F  e.  D  <->  ( F  e.  I  /\  A. x  e.  B  ( x  .<_  W 
 ->  ( F `  x )  =  x )
 ) ) )
 
Theoremldillaut 35397 A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( LAut `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  D ) 
 ->  F  e.  I )
 
Theoremldil1o 35398 A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  D ) 
 ->  F : B -1-1-onto-> B )
 
Theoremldilval 35399 Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  D  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F `  X )  =  X )
 
Theoremidldil 35400 The identity function is a lattice dilation. (Contributed by NM, 18-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( K  e.  A  /\  W  e.  H )  ->  (  _I  |`  B )  e.  D )
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