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Theorem List for Metamath Proof Explorer - 33301-33400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlhpelim 33301 Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 33294 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 33302 Modular law for hyperplanes analogous to atmod2i2 33126 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 33303 Modular law for hyperplanes analogous to complement of atmod2i1 33125 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 33304* The Hilbert lattice is relatively atomic with respect to co-atoms (lattice hyperplanes). Dual version of hlrelat3 32676. (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 33305* 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 33306 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 33307 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 33308 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 33309 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 33310 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 33311 Lemma for 4atexlem7 33339. (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 33312 Lemma for 4atexlem7 33339. (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 33313 Lemma for 4atexlem7 33339. (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 33314 Lemma for 4atexlem7 33339. (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 33315 Lemma for 4atexlem7 33339. (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 33316 Lemma for 4atexlem7 33339. (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 33317 Lemma for 4atexlem7 33339. (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 33318 Lemma for 4atexlem7 33339. (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 33319 Lemma for 4atexlem7 33339. (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 33320 Lemma for 4atexlem7 33339. (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 33321 Lemma for 4atexlem7 33339. (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 33322 Lemma for 4atexlem7 33339. (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 33323 Lemma for 4atexlem7 33339. (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 33324 Lemma for 4atexlem7 33339. (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 33325 Lemma for 4atexlem7 33339. (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 33326 Lemma for 4atexlem7 33339. (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 33327 Lemma for 4atexlem7 33339. 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 33328 Lemma for 4atexlem7 33339. (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 33329 Lemma for 4atexlem7 33339. (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 33330 Lemma for 4atexlem7 33339. (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 33331 Lemma for 4atexlem7 33339. (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 33332 Lemma for 4atexlem7 33339. (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 33333 Lemma for 4atexlem7 33339. (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 33334 Lemma for 4atexlem7 33339. (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 33335* Lemma for 4atexlem7 33339. 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 33336 Lemma for 4atexlem7 33339. (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 33337* Lemma for 4atexlem7 33339. 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 33338* Lemma for 4atexlem7 33339. (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 33339* 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 32608, 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 33340), 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 33340* 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 32608, 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 33341* More general version of 4atex 33340 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 33342* Same as 4atex2 33341 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 33343* Same as 4atex2 33341 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 33344* Same as 4atex2 33341 except that  S and 
T are zero. TODO: do we need this one or 4atex2-0aOLDN 33342 or 4atex2-0bOLDN 33343? (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 33345* More general version of 4atex 33340 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 33346* 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 33347* 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 33348 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 33349 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 33350 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 33351 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 33352 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 33353 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 33354 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 33355 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 33356 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 33357 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 33358 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 33359* 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 33360 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 33361 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 33362* 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 33363* 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 33364 Extend class notation with set of all lattice dilations.
 class  LDil
 
Syntaxcltrn 33365 Extend class notation with set of all lattice translations.
 class  LTrn
 
SyntaxcdilN 33366 Extend class notation with set of all dilations.
 class  Dil
 
SyntaxctrnN 33367 Extend class notation with set of all translations.
 class  Trn
 
Definitiondf-ldil 33368* 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 33369* 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 33370* 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 33371* 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 33372* 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 33373* 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 33374* 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 33375 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 33376 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 33377 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 33378 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 33379 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 33380 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 33381* 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 33382* 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 33383* 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 33384* The predicate "is a lattice translation". Version of isltrn 33383 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 33385 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 33386 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 33387 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 33388 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 33389 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 33390 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 33391 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 33392 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 33393 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 33394 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 33395 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 33396 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 ) ) )
 
Theoremltrncvr 33397 Covering property of a lattice translation. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  C  =  (  <o  `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  F  e.  T  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( X C Y  <->  ( F `  X ) C ( F `  Y ) ) )
 
Theoremltrnval1 33398 Value of a lattice translation under its co-atom. (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  /\  X  .<_  W ) )  ->  ( F `  X )  =  X )
 
Theoremltrnid 33399* A lattice translation is the identity function iff all atoms not under the fiducial co-atom  W are equal to their values. (Contributed by NM, 24-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( A. p  e.  A  ( -.  p  .<_  W  ->  ( F `  p )  =  p )  <->  F  =  (  _I  |`  B ) ) )
 
Theoremltrnnid 33400* If a lattice translation is not the identity, then there is an atom not under the fiducial co-atom 
W and not equal to its translation. (Contributed by NM, 24-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  ( F `
  p )  =/= 
 p ) )
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