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Theorem List for Metamath Proof Explorer - 29501-29600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlhp2atnle 29501 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 29502 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 29503 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 29504 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 29505 Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 29498 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 29506 Modular law for hyperplanes analogous to atmod2i2 29330 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 29507 Modular law for hyperplanes analogous to complement of atmod2i1 29329 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 29508* The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 28880. (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 29509* 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 29510 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 29511 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 29512 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 29513 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 29514 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 29515 Lemma for 4atexlem7 29543. (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 29516 Lemma for 4atexlem7 29543. (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 29517 Lemma for 4atexlem7 29543. (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 29518 Lemma for 4atexlem7 29543. (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 29519 Lemma for 4atexlem7 29543. (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 29520 Lemma for 4atexlem7 29543. (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 29521 Lemma for 4atexlem7 29543. (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 29522 Lemma for 4atexlem7 29543. (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 29523 Lemma for 4atexlem7 29543. (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 29524 Lemma for 4atexlem7 29543. (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 29525 Lemma for 4atexlem7 29543. (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 29526 Lemma for 4atexlem7 29543. (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 29527 Lemma for 4atexlem7 29543. (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 29528 Lemma for 4atexlem7 29543. (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 29529 Lemma for 4atexlem7 29543. (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 29530 Lemma for 4atexlem7 29543. (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 29531 Lemma for 4atexlem7 29543. 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 29532 Lemma for 4atexlem7 29543. (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 29533 Lemma for 4atexlem7 29543. (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 29534 Lemma for 4atexlem7 29543. (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 29535 Lemma for 4atexlem7 29543. (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 29536 Lemma for 4atexlem7 29543. (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 29537 Lemma for 4atexlem7 29543. (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 29538 Lemma for 4atexlem7 29543. (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 29539* Lemma for 4atexlem7 29543. 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 29540 Lemma for 4atexlem7 29543. (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 29541* Lemma for 4atexlem7 29543. 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 29542* Lemma for 4atexlem7 29543. (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 29543* 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 28812, 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 29544), 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 29544* 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 28812, 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 29545* More general version of 4atex 29544 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 29546* Same as 4atex2 29545 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 29547* Same as 4atex2 29545 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 29548* Same as 4atex2 29545 except that  S and 
T are zero. TODO: do we need this one or 4atex2-0aOLDN 29546 or 4atex2-0bOLDN 29547? (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 29549* More general version of 4atex 29544 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 29550* 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 29551* 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 29552 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 29553 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 29554 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 29555 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 29556 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 29557 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 29558 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 29559 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 29560 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 29561 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 29562 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 29563* 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 29564 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 29565 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 29566* 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 29567* 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 29568 Extend class notation with set of all lattice dilations.
 class  LDil
 
Syntaxcltrn 29569 Extend class notation with set of all lattice translations.
 class  LTrn
 
SyntaxcdilN 29570 Extend class notation with set of all dilations.
 class  Dil
 
SyntaxctrnN 29571 Extend class notation with set of all translations.
 class  Trn
 
Definitiondf-ldil 29572* 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 29573* 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 29574* 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 29575* 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 29576* 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 29577* 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 29578* 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 29579 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 29580 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 29581 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 29582 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 )
 
Theoremldilcnv 29583 The converse of a lattice dilation is a lattice dilation. (Contributed by NM, 10-May-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  D ) 
 ->  `' F  e.  D )
 
Theoremldilco 29584 The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  D  /\  G  e.  D )  ->  ( F  o.  G )  e.  D )
 
Theoremltrnfset 29585* The set of all lattice translations for a lattice  K. (Contributed by NM, 11-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  C  ->  (
 LTrn `  K )  =  ( w  e.  H  |->  { f  e.  ( (
 LDil `  K ) `  w )  |  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  w  /\  -.  q  .<_  w )  ->  ( ( p  .\/  ( f `  p ) )  ./\  w )  =  ( ( q 
 .\/  ( f `  q ) )  ./\  w ) ) } )
 )
 
Theoremltrnset 29586* The set of lattice translations for a fiducial co-atom  W. (Contributed by NM, 11-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( K  e.  B  /\  W  e.  H )  ->  T  =  { f  e.  D  |  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W 
 /\  -.  q  .<_  W )  ->  ( ( p  .\/  ( f `  p ) )  ./\  W )  =  ( ( q  .\/  ( f `  q ) )  ./\  W ) ) } )
 
Theoremisltrn 29587* The predicate "is a lattice translation". Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( K  e.  B  /\  W  e.  H )  ->  ( F  e.  T  <->  ( F  e.  D  /\  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W )  ->  ( ( p  .\/  ( F `  p ) )  ./\  W )  =  ( ( q  .\/  ( F `  q ) )  ./\  W )
 ) ) ) )
 
Theoremisltrn2N 29588* The predicate "is a lattice translation". Version of isltrn 29587 that considers only different  p and  q. TODO: Can this eliminate some separate proofs for the 
p  =  q case? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( K  e.  B  /\  W  e.  H )  ->  ( F  e.  T  <->  ( F  e.  D  /\  A. p  e.  A  A. q  e.  A  ( ( -.  p  .<_  W  /\  -.  q  .<_  W  /\  p  =/=  q )  ->  (
 ( p  .\/  ( F `  p ) ) 
 ./\  W )  =  ( ( q  .\/  ( F `  q ) ) 
 ./\  W ) ) ) ) )
 
Theoremltrnu 29589 Uniqueness property of a lattice translation value for atoms not under the fiducial co-atom  W. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 20-May-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( ( K  e.  V  /\  W  e.  H ) 
 /\  F  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 ->  ( ( P  .\/  ( F `  P ) )  ./\  W )  =  ( ( Q  .\/  ( F `  Q ) )  ./\  W )
 )
 
Theoremltrnldil 29590 A lattice translation is a lattice dilation. (Contributed by NM, 20-May-2012.)
 |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( LDil `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T ) 
 ->  F  e.  D )
 
Theoremltrnlaut 29591 A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( LAut `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T ) 
 ->  F  e.  I )
 
Theoremltrn1o 29592 A lattice translation is a one-to-one onto function. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T ) 
 ->  F : B -1-1-onto-> B )
 
Theoremltrncl 29593 Closure of a lattice translation. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T  /\  X  e.  B )  ->  ( F `  X )  e.  B )
 
Theoremltrn11 29594 One-to-one property of a lattice translation. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( ( F `  X )  =  ( F `  Y ) 
 <->  X  =  Y ) )
 
Theoremltrncnvnid 29595 If a translation is different from the identity, so is its converse. (Contributed by NM, 17-Jun-2013.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  `' F  =/=  (  _I  |`  B ) )
 
TheoremltrncoidN 29596 Two translations are equal if the composition of one with the converse of the other is the zero translation. This is an analog of vector subtraction. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  ->  ( ( F  o.  `' G )  =  (  _I  |`  B )  <->  F  =  G ) )
 
Theoremltrnle 29597 Less-than or equal property of a lattice translation. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( X  .<_  Y  <->  ( F `  X )  .<_  ( F `
  Y ) ) )
 
TheoremltrncnvleN 29598 Less-than or equal property of lattice translation converse. (Contributed by NM, 10-May-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( X  .<_  Y  <->  ( `' F `  X )  .<_  ( `' F `  Y ) ) )
 
Theoremltrnm 29599 Lattice translation of a meet. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( F `  ( X  ./\  Y ) )  =  ( ( F `  X ) 
 ./\  ( F `  Y ) ) )
 
Theoremltrnj 29600 Lattice translation of a meet. TODO: change antecedent to  K  e.  HL (Contributed by NM, 25-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .\/  =  ( join `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( F `  ( X  .\/  Y ) )  =  (
 ( F `  X )  .\/  ( F `  Y ) ) )
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