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Theorem List for Metamath Proof Explorer - 29401-29500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem4atexlemqtb 29401 Lemma for 4atexlem7 29415. (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 29402 Lemma for 4atexlem7 29415. (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 29403 Lemma for 4atexlem7 29415. 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 29404 Lemma for 4atexlem7 29415. (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 29405 Lemma for 4atexlem7 29415. (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 29406 Lemma for 4atexlem7 29415. (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 29407 Lemma for 4atexlem7 29415. (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 29408 Lemma for 4atexlem7 29415. (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 29409 Lemma for 4atexlem7 29415. (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 29410 Lemma for 4atexlem7 29415. (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 29411* Lemma for 4atexlem7 29415. 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 29412 Lemma for 4atexlem7 29415. (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 29413* Lemma for 4atexlem7 29415. 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 29414* Lemma for 4atexlem7 29415. (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 29415* 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 28684, 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 29416), 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 29416* 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 28684, 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 29417* More general version of 4atex 29416 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 29418* Same as 4atex2 29417 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 29419* Same as 4atex2 29417 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 29420* Same as 4atex2 29417 except that  S and 
T are zero. TODO: do we need this one or 4atex2-0aOLDN 29418 or 4atex2-0bOLDN 29419? (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 29421* More general version of 4atex 29416 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 29422* 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 29423* 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 29424 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 29425 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 29426 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 29427 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 29428 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 29429 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 29430 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 29431 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 29432 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 29433 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 29434 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 29435* 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 29436 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 29437 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 29438* 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 29439* 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 29440 Extend class notation with set of all lattice dilations.
 class  LDil
 
Syntaxcltrn 29441 Extend class notation with set of all lattice translations.
 class  LTrn
 
SyntaxcdilN 29442 Extend class notation with set of all dilations.
 class  Dil
 
SyntaxctrnN 29443 Extend class notation with set of all translations.
 class  Trn
 
Definitiondf-ldil 29444* 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 29445* 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 29446* 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 29447* 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 29448* 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 29449* 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 29450* 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 29451 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 29452 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 29453 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 29454 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 29455 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 29456 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 29457* 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 29458* 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 29459* 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 29460* The predicate "is a lattice translation". Version of isltrn 29459 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 29461 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 29462 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 29463 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 29464 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 29465 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 29466 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 29467 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 29468 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 29469 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 29470 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 29471 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 29472 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 29473 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 29474 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 29475* 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 29476* 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 ) )
 
Theoremltrnatb 29477 The lattice translation of an atom is an atom. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B )  ->  ( P  e.  A  <->  ( F `  P )  e.  A ) )
 
Theoremltrncnvatb 29478 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)
 |-  B  =  ( Base `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B )  ->  ( P  e.  A  <->  ( `' F `  P )  e.  A ) )
 
Theoremltrnel 29479 The lattice translation of an atom not under the fiducial co-atom is also an atom not under the fiducial co-atom. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 22-May-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
 
Theoremltrnat 29480 The lattice translation of an atom is also an atom. TODO: See if this can shorten some ltrnel 29479 uses. (Contributed by NM, 25-May-2012.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A )  ->  ( F `  P )  e.  A )
 
Theoremltrncnvat 29481 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 9-May-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A )  ->  ( `' F `  P )  e.  A )
 
Theoremltrncnvel 29482 The converse of the lattice translation of an atom not under the fiducial co-atom. (Contributed by NM, 10-May-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( `' F `  P )  e.  A  /\  -.  ( `' F `  P ) 
 .<_  W ) )
 
TheoremltrncoelN 29483 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 29479 uses. (Contributed by NM, 1-May-2013.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  (
 ( F `  ( G `  P ) )  e.  A  /\  -.  ( F `  ( G `
  P ) ) 
 .<_  W ) )
 
Theoremltrncoat 29484 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 29479, ltrnat 29480 uses. (Contributed by NM, 1-May-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) 
 /\  P  e.  A )  ->  ( F `  ( G `  P ) )  e.  A )
 
Theoremltrncoval 29485 Two ways to express value of translation composition. (Contributed by NM, 31-May-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T ) 
 /\  P  e.  A )  ->  ( ( F  o.  G ) `  P )  =  ( F `  ( G `  P ) ) )
 
Theoremltrncnv 29486 The converse of a lattice translation is a lattice translation. (Contributed by NM, 10-May-2013.)
 |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  `' F  e.  T )
 
Theoremltrn11at 29487 Frequently used one-to-one property of lattice translation atoms. (Contributed by NM, 5-May-2013.)
 |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q ) ) 
 ->  ( F `  P )  =/=  ( F `  Q ) )
 
Theoremltrneq2 29488* The equality of two translations is determined by their equality at atoms. (Contributed by NM, 2-Mar-2014.)
 |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  ->  ( A. p  e.  A  ( F `  p )  =  ( G `  p )  <->  F  =  G ) )
 
Theoremltrneq 29489* The equality of two translations is determined by their equality at atoms not under co-atom  W. (Contributed by NM, 20-Jun-2013.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  ->  ( A. p  e.  A  ( -.  p  .<_  W  ->  ( F `  p )  =  ( G `  p ) )  <->  F  =  G ) )
 
Theoremidltrn 29490 The identity function is a lattice translation. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 18-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
 
Theoremltrnmw 29491 Property of lattice translation value. Remark below Lemma B in [Crawley] p. 112. TODO: Can this be used in more places? (Contributed by NM, 20-May-2012.)
 |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |- 
 .0.  =  ( 0. `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  W )  =  .0.  )
 
TheoremdilfsetN 29492* The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  W  =  ( WAtoms `  K )   &    |-  M  =  ( PAut `  K )   &    |-  L  =  ( Dil `  K )   =>    |-  ( K  e.  B  ->  L  =  ( d  e.  A  |->  { f  e.  M  |  A. x  e.  S  ( x  C_  ( W `  d ) 
 ->  ( f `  x )  =  x ) } ) )
 
TheoremdilsetN 29493* The set of dilations for a fiducial atom  D. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  W  =  ( WAtoms `  K )   &    |-  M  =  ( PAut `  K )   &    |-  L  =  ( Dil `  K )   =>    |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( L `  D )  =  {
 f  e.  M  |  A. x  e.  S  ( x  C_  ( W `
  D )  ->  ( f `  x )  =  x ) } )
 
TheoremisdilN 29494* The predicate "is a dilation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  W  =  ( WAtoms `  K )   &    |-  M  =  ( PAut `  K )   &    |-  L  =  ( Dil `  K )   =>    |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( F  e.  ( L `  D )  <-> 
 ( F  e.  M  /\  A. x  e.  S  ( x  C_  ( W `
  D )  ->  ( F `  x )  =  x ) ) ) )
 
TheoremtrnfsetN 29495* The mapping from fiducial atom to set of translations. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  W  =  ( WAtoms `  K )   &    |-  M  =  ( PAut `  K )   &    |-  L  =  ( Dil `  K )   &    |-  T  =  ( Trn `  K )   =>    |-  ( K  e.  C  ->  T  =  ( d  e.  A  |->  { f  e.  ( L `  d
 )  |  A. q  e.  ( W `  d
 ) A. r  e.  ( W `  d ) ( ( q  .+  (
 f `  q )
 )  i^i  (  ._|_  ` 
 { d } )
 )  =  ( ( r  .+  ( f `
  r ) )  i^i  (  ._|_  `  { d } ) ) }
 ) )
 
TheoremtrnsetN 29496* The set of translations for a fiducial atom  D. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  W  =  ( WAtoms `  K )   &    |-  M  =  ( PAut `  K )   &    |-  L  =  ( Dil `  K )   &    |-  T  =  ( Trn `  K )   =>    |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( T `  D )  =  {
 f  e.  ( L `
  D )  | 
 A. q  e.  ( W `  D ) A. r  e.  ( W `  D ) ( ( q  .+  ( f `
  q ) )  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( f `  r
 ) )  i^i  (  ._|_  `  { D }
 ) ) } )
 
TheoremistrnN 29497* The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
 |-  A  =  ( Atoms `  K )   &    |-  S  =  ( PSubSp `  K )   &    |-  .+  =  ( + P `  K )   &    |- 
 ._|_  =  ( _|_ P `
  K )   &    |-  W  =  ( WAtoms `  K )   &    |-  M  =  ( PAut `  K )   &    |-  L  =  ( Dil `  K )   &    |-  T  =  ( Trn `  K )   =>    |-  ( ( K  e.  B  /\  D  e.  A )  ->  ( F  e.  ( T `  D )  <-> 
 ( F  e.  ( L `  D )  /\  A. q  e.  ( W `
  D ) A. r  e.  ( W `  D ) ( ( q  .+  ( F `
  q ) )  i^i  (  ._|_  `  { D } ) )  =  ( ( r  .+  ( F `  r ) )  i^i  (  ._|_  ` 
 { D } )
 ) ) ) )
 
Syntaxctrl 29498 Extend class notation with set of all traces of lattice translations.
 class  trL
 
Definitiondf-trl 29499* Define trace of a lattice translation. (Contributed by NM, 20-May-2012.)
 |-  trL  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( f  e.  ( (
 LTrn `  k ) `  w )  |->  ( iota_ x  e.  ( Base `  k
 ) A. p  e.  ( Atoms `  k ) ( -.  p ( le `  k ) w  ->  x  =  ( ( p ( join `  k
 ) ( f `  p ) ) (
 meet `  k ) w ) ) ) ) ) )
 
Theoremtrlfset 29500* The set of all traces of lattice translations for a lattice  K. (Contributed by NM, 20-May-2012.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  C  ->  ( trL `  K )  =  ( w  e.  H  |->  ( f  e.  (
 ( LTrn `  K ) `  w )  |->  ( iota_ x  e.  B A. p  e.  A  ( -.  p  .<_  w  ->  x  =  ( ( p  .\/  ( f `  p ) )  ./\  w ) ) ) ) ) )
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