HomeHome Metamath Proof Explorer
Theorem List (p. 335 of 389)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26262)
  Hilbert Space Explorer  Hilbert Space Explorer
(26263-27786)
  Users' Mathboxes  Users' Mathboxes
(27787-38873)
 

Theorem List for Metamath Proof Explorer - 33401-33500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdleme27N 33401* Part of proof of Lemma E in [Crawley] p. 113. Eliminate the  s  =/=  t antecedent in cdleme27a 33399. (Contributed by NM, 3-Feb-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W ) ) )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W ) ) )   &    |-  D  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  N ) )   &    |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F )   &    |-  G  =  ( (
 t  .\/  U )  ./\  ( Q  .\/  (
 ( P  .\/  t
 )  ./\  W ) ) )   &    |-  O  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W ) ) )   &    |-  E  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  O ) )   &    |-  Y  =  if ( t  .<_  ( P  .\/  Q ) ,  E ,  G )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  .<_  ( t 
 .\/  V )  /\  ( V  e.  A  /\  V  .<_  W ) ) )  ->  C  .<_  ( Y  .\/  V )
 )
 
Theoremcdleme28a 33402* Lemma for cdleme25b 33386. TODO: FIX COMMENT (Contributed by NM, 4-Feb-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W ) ) )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W ) ) )   &    |-  D  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  N ) )   &    |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F )   &    |-  G  =  ( (
 t  .\/  U )  ./\  ( Q  .\/  (
 ( P  .\/  t
 )  ./\  W ) ) )   &    |-  O  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W ) ) )   &    |-  E  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  O ) )   &    |-  Y  =  if ( t  .<_  ( P  .\/  Q ) ,  E ,  G )   &    |-  V  =  ( (
 s  .\/  t )  ./\  ( X  ./\  W ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  (
 s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  (
 ( s  .\/  ( X  ./\  W ) )  =  X  /\  (
 t  .\/  ( X  ./\ 
 W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( C  .\/  ( X  ./\  W ) )  .<_  ( Y  .\/  ( X  ./\  W ) ) )
 
Theoremcdleme28b 33403* Lemma for cdleme25b 33386. TODO: FIX COMMENT (Contributed by NM, 6-Feb-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W ) ) )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W ) ) )   &    |-  D  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  N ) )   &    |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F )   &    |-  G  =  ( (
 t  .\/  U )  ./\  ( Q  .\/  (
 ( P  .\/  t
 )  ./\  W ) ) )   &    |-  O  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W ) ) )   &    |-  E  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  O ) )   &    |-  Y  =  if ( t  .<_  ( P  .\/  Q ) ,  E ,  G )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  (
 s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  (
 ( s  .\/  ( X  ./\  W ) )  =  X  /\  (
 t  .\/  ( X  ./\ 
 W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( C  .\/  ( X  ./\  W ) )  =  ( Y 
 .\/  ( X  ./\  W ) ) )
 
Theoremcdleme28c 33404* Part of proof of Lemma E in [Crawley] p. 113. Eliminate the  s  =/=  t antecedent in cdleme28b 33403. TODO: FIX COMMENT (Contributed by NM, 6-Feb-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W ) ) )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W ) ) )   &    |-  D  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  N ) )   &    |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F )   &    |-  G  =  ( (
 t  .\/  U )  ./\  ( Q  .\/  (
 ( P  .\/  t
 )  ./\  W ) ) )   &    |-  O  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W ) ) )   &    |-  E  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  O ) )   &    |-  Y  =  if ( t  .<_  ( P  .\/  Q ) ,  E ,  G )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  (
 s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( ( s  .\/  ( X  ./\ 
 W ) )  =  X  /\  ( t 
 .\/  ( X  ./\  W ) )  =  X  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( C  .\/  ( X  ./\  W ) )  =  ( Y  .\/  ( X  ./\ 
 W ) ) )
 
Theoremcdleme28 33405* Quantified version of cdleme28c 33404. (Compare cdleme24 33384.) (Contributed by NM, 7-Feb-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W ) ) )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W ) ) )   &    |-  D  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  N ) )   &    |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F )   &    |-  G  =  ( (
 t  .\/  U )  ./\  ( Q  .\/  (
 ( P  .\/  t
 )  ./\  W ) ) )   &    |-  O  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W ) ) )   &    |-  E  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  O ) )   &    |-  Y  =  if ( t  .<_  ( P  .\/  Q ) ,  E ,  G )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) 
 ->  A. s  e.  A  A. t  e.  A  ( ( ( -.  s  .<_  W  /\  ( s 
 .\/  ( X  ./\  W ) )  =  X )  /\  ( -.  t  .<_  W  /\  ( t 
 .\/  ( X  ./\  W ) )  =  X ) )  ->  ( C 
 .\/  ( X  ./\  W ) )  =  ( Y  .\/  ( X  ./\ 
 W ) ) ) )
 
Theoremcdleme29ex 33406* Lemma for cdleme29b 33407. (Compare cdleme25a 33385.) TODO: FIX COMMENT (Contributed by NM, 7-Feb-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W ) ) )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W ) ) )   &    |-  D  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  N ) )   &    |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) 
 ->  E. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
 .\/  ( X  ./\  W ) )  =  X )  /\  ( C  .\/  ( X  ./\  W ) )  e.  B ) )
 
Theoremcdleme29b 33407* Transform cdleme28 33405. (Compare cdleme25b 33386.) TODO: FIX COMMENT (Contributed by NM, 7-Feb-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W ) ) )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W ) ) )   &    |-  D  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  N ) )   &    |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) 
 ->  E. v  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( X  ./\  W ) )  =  X ) 
 ->  v  =  ( C  .\/  ( X  ./\  W ) ) ) )
 
Theoremcdleme29c 33408* Transform cdleme28b 33403. (Compare cdleme25c 33387.) TODO: FIX COMMENT (Contributed by NM, 8-Feb-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W ) ) )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W ) ) )   &    |-  D  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  N ) )   &    |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) 
 ->  E! v  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( X  ./\  W ) )  =  X ) 
 ->  v  =  ( C  .\/  ( X  ./\  W ) ) ) )
 
Theoremcdleme29cl 33409* Show closure of the unique element in cdleme28c 33404. (Contributed by NM, 8-Feb-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W ) ) )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W ) ) )   &    |-  D  =  (
 iota_ u  e.  B  A. z  e.  A  ( ( -.  z  .<_  W 
 /\  -.  z  .<_  ( P  .\/  Q )
 )  ->  u  =  N ) )   &    |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F )   &    |-  I  =  ( iota_ v  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( X  ./\ 
 W ) )  =  X )  ->  v  =  ( C  .\/  ( X  ./\  W ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) 
 ->  I  e.  B )
 
Theoremcdleme30a 33410 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Feb-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  A  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  Y  e.  B )  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  X  .<_  Y ) ) 
 ->  ( s  .\/  ( Y  ./\  W ) )  =  Y )
 
Theoremcdleme31so 33411* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Feb-2013.)
 |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( N  .\/  ( x  ./\  W ) ) ) )   &    |-  C  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( X  ./\ 
 W ) )  =  X )  ->  z  =  ( N  .\/  ( X  ./\  W ) ) ) )   =>    |-  ( X  e.  B  -> 
 [_ X  /  x ]_ O  =  C )
 
Theoremcdleme31sn 33412* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
 |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D )   &    |-  C  =  if ( R  .<_  ( P  .\/  Q ) ,  [_ R  /  s ]_ I ,  [_ R  /  s ]_ D )   =>    |-  ( R  e.  A  -> 
 [_ R  /  s ]_ N  =  C )
 
Theoremcdleme31sn1 33413* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
 |-  I  =  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  G ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D )   &    |-  C  =  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  [_ R  /  s ]_ G ) )   =>    |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ N  =  C )
 
Theoremcdleme31se 33414* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
 |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  T )  ./\  W )
 ) )   &    |-  Y  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  T )  ./\  W )
 ) )   =>    |-  ( R  e.  A  -> 
 [_ R  /  s ]_ E  =  Y )
 
Theoremcdleme31se2 33415* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 3-Apr-2013.)
 |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  t )  ./\  W ) ) )   &    |-  Y  =  ( ( P  .\/  Q )  ./\  ( [_ S  /  t ]_ D  .\/  ( ( R  .\/  S )  ./\  W )
 ) )   =>    |-  ( S  e.  A  -> 
 [_ S  /  t ]_ E  =  Y )
 
Theoremcdleme31sc 33416* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.)
 |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  X  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
 ) )   =>    |-  ( R  e.  A  -> 
 [_ R  /  s ]_ C  =  X )
 
Theoremcdleme31sde 33417* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.)
 |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  Y  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  Z  =  ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( R  .\/  S )  ./\  W )
 ) )   =>    |-  ( ( R  e.  A  /\  S  e.  A )  ->  [_ R  /  s ]_ [_ S  /  t ]_ E  =  Z )
 
Theoremcdleme31snd 33418* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Apr-2013.)
 |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W ) ) )   &    |-  E  =  ( ( O  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  O )  ./\  W )
 ) )   &    |-  O  =  ( ( S  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  S )  ./\  W )
 ) )   =>    |-  ( S  e.  A  -> 
 [_ S  /  v ]_ [_ N  /  t ]_ D  =  E )
 
Theoremcdleme31sdnN 33419* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.) (New usage is discouraged.)
 |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  N  =  if ( s  .<_  ( P 
 .\/  Q ) ,  I ,  C )   =>    |-  N  =  if (
 s  .<_  ( P  .\/  Q ) ,  I ,  [_ s  /  t ]_ D )
 
Theoremcdleme31sn1c 33420* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 1-Mar-2013.)
 |-  G  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  G ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  D )   &    |-  Y  =  ( ( P  .\/  Q )  ./\  ( E  .\/  ( ( R  .\/  t )  ./\ 
 W ) ) )   &    |-  C  =  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  Y ) )   =>    |-  ( ( R  e.  A  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ N  =  C )
 
Theoremcdleme31sn2 33421* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)
 |-  D  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  N  =  if ( s  .<_  ( P 
 .\/  Q ) ,  I ,  D )   &    |-  C  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
 ) )   =>    |-  ( ( R  e.  A  /\  -.  R  .<_  ( P  .\/  Q )
 )  ->  [_ R  /  s ]_ N  =  C )
 
Theoremcdleme31fv 33422* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 10-Feb-2013.)
 |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( N  .\/  ( x  ./\  W ) ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )   &    |-  C  =  (
 iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( X  ./\  W ) )  =  X ) 
 ->  z  =  ( N  .\/  ( X  ./\  W ) ) ) )   =>    |-  ( X  e.  B  ->  ( F `  X )  =  if (
 ( P  =/=  Q  /\  -.  X  .<_  W ) ,  C ,  X ) )
 
Theoremcdleme31fv1 33423* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 10-Feb-2013.)
 |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( N  .\/  ( x  ./\  W ) ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )   &    |-  C  =  (
 iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( X  ./\  W ) )  =  X ) 
 ->  z  =  ( N  .\/  ( X  ./\  W ) ) ) )   =>    |-  ( ( X  e.  B  /\  ( P  =/=  Q 
 /\  -.  X  .<_  W ) )  ->  ( F `  X )  =  C )
 
Theoremcdleme31fv1s 33424* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Feb-2013.)
 |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( N  .\/  ( x  ./\  W ) ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )   =>    |-  ( ( X  e.  B  /\  ( P  =/=  Q 
 /\  -.  X  .<_  W ) )  ->  ( F `  X )  = 
 [_ X  /  x ]_ O )
 
Theoremcdleme31fv2 33425* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 23-Feb-2013.)
 |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  O ,  x ) )   =>    |-  ( ( X  e.  B  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W ) ) 
 ->  ( F `  X )  =  X )
 
Theoremcdleme31id 33426* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 18-Apr-2013.)
 |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  O ,  x ) )   =>    |-  ( ( X  e.  B  /\  P  =  Q )  ->  ( F `  X )  =  X )
 
Theoremcdlemefrs29pre00 33427 ***START OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW*** FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 33060. (Contributed by NM, 29-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  (
 s  =  R  ->  (
 ph 
 <->  ps ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ps )  /\  s  e.  A )  ->  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
 .\/  ( R  ./\  W ) )  =  R ) 
 <->  ( -.  s  .<_  W 
 /\  ( s  .\/  ( R  ./\  W ) )  =  R ) ) )
 
Theoremcdlemefrs29bpre0 33428* TODO fix comment. (Contributed by NM, 29-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  (
 s  =  R  ->  (
 ph 
 <->  ps ) )   &    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q 
 /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ph )
 ) )  ->  N  e.  B )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ps )  ->  ( A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R  ./\  W ) ) )  <->  z  =  [_ R  /  s ]_ N ) )
 
Theoremcdlemefrs29bpre1 33429* TODO FIX COMMENT (Contributed by NM, 29-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  (
 s  =  R  ->  (
 ph 
 <->  ps ) )   &    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q 
 /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ph )
 ) )  ->  N  e.  B )   &    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  s ]_ N  e.  B )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ps )  ->  E. z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
 .\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R  ./\ 
 W ) ) ) )
 
Theoremcdlemefrs29cpre1 33430* TODO FIX COMMENT (Contributed by NM, 29-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  (
 s  =  R  ->  (
 ph 
 <->  ps ) )   &    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q 
 /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ph )
 ) )  ->  N  e.  B )   &    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  s ]_ N  e.  B )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ps )  ->  E! z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  ph )  /\  ( s 
 .\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R  ./\ 
 W ) ) ) )
 
Theoremcdlemefrs29clN 33431* TODO: NOT USED? Show closure of the unique element in cdlemefrs29cpre1 33430. (Contributed by NM, 29-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  (
 s  =  R  ->  (
 ph 
 <->  ps ) )   &    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q 
 /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ph )
 ) )  ->  N  e.  B )   &    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  s ]_ N  e.  B )   &    |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( R  ./\ 
 W ) )  =  R )  ->  z  =  ( N  .\/  ( R  ./\  W ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ps )  ->  O  e.  B )
 
Theoremcdlemefrs32fva 33432* Part of proof of Lemma E in [Crawley] p. 113. Value of  F at an atom not under  W. TODO: FIX COMMENT TODO: consolidate uses of lhpmat 33060 here and elsewhere, and presence/absence of  s 
.<_  ( P  .\/  Q
) term. Also, why can proof be shortened with cdleme29cl 33409? What is difference from cdlemefs27cl 33445? (Contributed by NM, 29-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  (
 s  =  R  ->  (
 ph 
 <->  ps ) )   &    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q 
 /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ph )
 ) )  ->  N  e.  B )   &    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  s ]_ N  e.  B )   &    |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( N  .\/  ( x  ./\  W ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ps )  ->  [_ R  /  x ]_ O  =  [_ R  /  s ]_ N )
 
Theoremcdlemefrs32fva1 33433* Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT (Contributed by NM, 29-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  (
 s  =  R  ->  (
 ph 
 <->  ps ) )   &    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  P  =/=  Q 
 /\  ( s  e.  A  /\  ( -.  s  .<_  W  /\  ph )
 ) )  ->  N  e.  B )   &    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  [_ R  /  s ]_ N  e.  B )   &    |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( N  .\/  ( x  ./\  W ) ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  O ,  x ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ps )  ->  ( F `  R )  =  [_ R  /  s ]_ N )
 
Theoremcdlemefr29exN 33434* Lemma for cdlemefs29bpre1N 33449. (Compare cdleme25a 33385.) TODO: FIX COMMENT TODO: IS THIS NEEDED? (Contributed by NM, 28-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) 
 /\  A. s  e.  A  C  e.  B )  ->  E. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
 .\/  ( X  ./\  W ) )  =  X )  /\  ( C  .\/  ( X  ./\  W ) )  e.  B ) )
 
Theoremcdlemefr27cl 33435 Part of proof of Lemma E in [Crawley] p. 113. Closure of  N. (Contributed by NM, 23-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  N  =  if ( s  .<_  ( P 
 .\/  Q ) ,  I ,  C )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( s  e.  A  /\  -.  s  .<_  ( P 
 .\/  Q )  /\  P  =/=  Q ) )  ->  N  e.  B )
 
Theoremcdlemefr32sn2aw 33436* Show that  [_ R  / 
s ]_ N is an atom not under  W when  -.  R  .<_  ( P  .\/  Q ). (Contributed by NM, 28-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  N  =  if ( s  .<_  ( P 
 .\/  Q ) ,  I ,  C )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  -.  R  .<_  ( P  .\/  Q )
 )  ->  ( [_ R  /  s ]_ N  e.  A  /\  -.  [_ R  /  s ]_ N  .<_  W ) )
 
Theoremcdlemefr32snb 33437* Show closure of  [_ R  /  s ]_ N. (Contributed by NM, 28-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  N  =  if ( s  .<_  ( P 
 .\/  Q ) ,  I ,  C )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  -.  R  .<_  ( P  .\/  Q )
 )  ->  [_ R  /  s ]_ N  e.  B )
 
Theoremcdlemefr29bpre0N 33438* TODO fix comment. (Contributed by NM, 28-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  N  =  if ( s  .<_  ( P 
 .\/  Q ) ,  I ,  C )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  -.  R  .<_  ( P  .\/  Q )
 )  ->  ( A. s  e.  A  (
 ( ( -.  s  .<_  W  /\  -.  s  .<_  ( P  .\/  Q ) )  /\  ( s 
 .\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R  ./\ 
 W ) ) )  <-> 
 z  =  [_ R  /  s ]_ N ) )
 
Theoremcdlemefr29clN 33439* Show closure of the unique element in cdleme29c 33408. TODO fix comment. TODO Not needed? (Contributed by NM, 29-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  N  =  if ( s  .<_  ( P 
 .\/  Q ) ,  I ,  C )   &    |-  O  =  (
 iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( R  ./\  W ) )  =  R ) 
 ->  z  =  ( N  .\/  ( R  ./\  W ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  -.  R  .<_  ( P  .\/  Q )
 )  ->  O  e.  B )
 
Theoremcdleme43frv1snN 33440* Value of  [_ R  / 
s ]_ N when  -.  R  .<_  ( P  .\/  Q
). (Contributed by NM, 30-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  N  =  if ( s  .<_  ( P 
 .\/  Q ) ,  I ,  C )   &    |-  X  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
 ) )   =>    |-  ( ( R  e.  A  /\  -.  R  .<_  ( P  .\/  Q )
 )  ->  [_ R  /  s ]_ N  =  X )
 
Theoremcdlemefr32fvaN 33441* Part of proof of Lemma E in [Crawley] p. 113. Value of  F at an atom not under  W. TODO: FIX COMMENT (Contributed by NM, 29-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  N  =  if ( s  .<_  ( P 
 .\/  Q ) ,  I ,  C )   &    |-  O  =  (
 iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( N  .\/  ( x  ./\  W ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  -.  R  .<_  ( P  .\/  Q )
 )  ->  [_ R  /  x ]_ O  =  [_ R  /  s ]_ N )
 
Theoremcdlemefr32fva1 33442* Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT (Contributed by NM, 29-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  N  =  if ( s  .<_  ( P 
 .\/  Q ) ,  I ,  C )   &    |-  O  =  (
 iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( N  .\/  ( x  ./\  W ) ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  -.  R  .<_  ( P  .\/  Q )
 )  ->  ( F `  R )  =  [_ R  /  s ]_ N )
 
Theoremcdlemefr31fv1 33443* Value of  ( F `  R ) when  -.  R  .<_  ( P  .\/  Q
). TODO This may be useful for shortening others that now use riotasv 31996 3d . TODO: FIX COMMENT (Contributed by NM, 30-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  N  =  if ( s  .<_  ( P 
 .\/  Q ) ,  I ,  C )   &    |-  O  =  (
 iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( N  .\/  ( x  ./\  W ) ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  O ,  x ) )   &    |-  X  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  -.  R  .<_  ( P  .\/  Q )
 )  ->  ( F `  R )  =  X )
 
Theoremcdlemefs29pre00N 33444 FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 33060. (Contributed by NM, 27-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  R  .<_  ( P  .\/  Q ) )  /\  s  e.  A )  ->  (
 ( ( -.  s  .<_  W  /\  s  .<_  ( P  .\/  Q )
 )  /\  ( s  .\/  ( R  ./\  W ) )  =  R )  <-> 
 ( -.  s  .<_  W 
 /\  ( s  .\/  ( R  ./\  W ) )  =  R ) ) )
 
Theoremcdlemefs27cl 33445* Part of proof of Lemma E in [Crawley] p. 113. Closure of  N. TODO FIX COMMENT This is the start of a re-proof of cdleme27cl 33398 etc. with the  s  .<_  ( P 
.\/  Q ) condition (so as to not have the  C hypothesis). (Contributed by NM, 24-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ u  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  u  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( s  e.  A  /\  -.  s  .<_  W ) 
 /\  s  .<_  ( P 
 .\/  Q )  /\  P  =/=  Q ) )  ->  N  e.  B )
 
Theoremcdlemefs32sn1aw 33446* Show that  [_ R  / 
s ]_ N is an atom not under  W when  R  .<_  ( P 
.\/  Q ). (Contributed by NM, 24-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   &    |-  Y  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  t )  ./\ 
 W ) ) )   &    |-  Z  =  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  Y ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P  .\/  Q ) )  ->  ( [_ R  /  s ]_ N  e.  A  /\  -.  [_ R  /  s ]_ N  .<_  W ) )
 
Theoremcdlemefs32snb 33447* Show closure of  [_ R  /  s ]_ N. (Contributed by NM, 24-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  R  .<_  ( P 
 .\/  Q ) )  ->  [_ R  /  s ]_ N  e.  B )
 
Theoremcdlemefs29bpre0N 33448* TODO FIX COMMENT (Contributed by NM, 26-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  R  .<_  ( P 
 .\/  Q ) )  ->  ( A. s  e.  A  ( ( ( -.  s  .<_  W  /\  s  .<_  ( P  .\/  Q ) )  /\  ( s 
 .\/  ( R  ./\  W ) )  =  R )  ->  z  =  ( N  .\/  ( R  ./\ 
 W ) ) )  <-> 
 z  =  [_ R  /  s ]_ N ) )
 
Theoremcdlemefs29bpre1N 33449* TODO FIX COMMENT (Contributed by NM, 27-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  R  .<_  ( P 
 .\/  Q ) )  ->  E. z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  s  .<_  ( P  .\/  Q )
 )  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) 
 ->  z  =  ( N  .\/  ( R  ./\  W ) ) ) )
 
Theoremcdlemefs29cpre1N 33450* TODO FIX COMMENT (Contributed by NM, 26-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  R  .<_  ( P 
 .\/  Q ) )  ->  E! z  e.  B  A. s  e.  A  ( ( ( -.  s  .<_  W  /\  s  .<_  ( P  .\/  Q )
 )  /\  ( s  .\/  ( R  ./\  W ) )  =  R ) 
 ->  z  =  ( N  .\/  ( R  ./\  W ) ) ) )
 
Theoremcdlemefs29clN 33451* Show closure of the unique element in cdleme29c 33408. (Contributed by NM, 27-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   &    |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( R  ./\ 
 W ) )  =  R )  ->  z  =  ( N  .\/  ( R  ./\  W ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  R  .<_  ( P 
 .\/  Q ) )  ->  O  e.  B )
 
Theoremcdleme43fsv1snlem 33452* Value of  [_ R  / 
s ]_ N when  R  .<_  ( P  .\/  Q ). (Contributed by NM, 30-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   &    |-  Y  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\ 
 W ) ) )   &    |-  Z  =  ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( R  .\/  S )  ./\ 
 W ) ) )   &    |-  V  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  t )  ./\ 
 W ) ) )   &    |-  X  =  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  V ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) 
 /\  ( R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P 
 .\/  Q ) ) ) 
 ->  [_ R  /  s ]_ N  =  Z )
 
Theoremcdleme43fsv1sn 33453* Value of  [_ R  / 
s ]_ N when  R  .<_  ( P  .\/  Q ). (Contributed by NM, 30-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   &    |-  Y  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\ 
 W ) ) )   &    |-  Z  =  ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( R  .\/  S )  ./\ 
 W ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R 
 .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )
 ) )  ->  [_ R  /  s ]_ N  =  Z )
 
Theoremcdlemefs32fvaN 33454* Part of proof of Lemma E in [Crawley] p. 113. Value of  F at an atom not under  W. TODO: FIX COMMENT TODO: consolidate uses of lhpmat 33060 here and elsewhere, and presence/absence of  s 
.<_  ( P  .\/  Q
) term. Also, why can proof be shortened with cdleme27cl 33398? What is difference from cdlemefs27cl 33445? (Contributed by NM, 29-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   &    |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( N  .\/  ( x  ./\  W ) ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  R  .<_  ( P 
 .\/  Q ) )  ->  [_ R  /  x ]_ O  =  [_ R  /  s ]_ N )
 
Theoremcdlemefs32fva1 33455* Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT (Contributed by NM, 29-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   &    |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( N  .\/  ( x  ./\  W ) ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  O ,  x ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P  .\/  Q ) )  ->  ( F `
  R )  = 
 [_ R  /  s ]_ N )
 
Theoremcdlemefs31fv1 33456* Value of  ( F `  R ) when  R  .<_  ( P  .\/  Q ). TODO This may be useful for shortening others that now use riotasv 31996 3d . TODO: FIX COMMENT. ***END OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW***
       "cdleme3xsn1aw" decreased using "cdlemefs32sn1aw"
       "cdleme32sn1aw" decreased from 3302 to 36 using "cdlemefs32sn1aw".
       "cdleme32sn2aw" decreased from 1687 to 26 using "cdlemefr32sn2aw".
       "cdleme32snaw" decreased from 376 to 375 using "cdlemefs32sn1aw".
       "cdleme32snaw" decreased from 375 to 368 using "cdlemefr32sn2aw".
       "cdleme35sn3a" decreased from 547 to 523 using "cdleme43frv1sn".
       
(Contributed by NM, 27-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   &    |-  O  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( N  .\/  ( x  ./\  W ) ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  O ,  x ) )   &    |-  Y  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  Z  =  ( ( P  .\/  Q )  ./\  ( Y  .\/  ( ( R  .\/  S )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R 
 .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )
 ) )  ->  ( F `  R )  =  Z )
 
Theoremcdlemefr44 33457* Value of f(r) when r is an atom not under pq, using more compact hypotheses. TODO: eliminate and use cdlemefr45 instead? TODO FIX COMMENT (Contributed by NM, 31-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  O  =  (
 iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( if ( s  .<_  ( P 
 .\/  Q ) ,  I ,  [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  O ,  x ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  ( F `  R )  = 
 [_ R  /  t ]_ D )
 
Theoremcdlemefs44 33458* Value of fs(r) when r is an atom under pq and s is any atom not under pq, using more compact hypotheses. TODO: eliminate and use cdlemefs45 33461 instead TODO FIX COMMENT (Contributed by NM, 31-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  O  =  (
 iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W 
 /\  ( s  .\/  ( x  ./\  W ) )  =  x ) 
 ->  z  =  ( if ( s  .<_  ( P 
 .\/  Q ) ,  I ,  [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  O ,  x ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) 
 /\  ( R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( F `  R )  =  [_ R  /  s ]_ [_ S  /  t ]_ E )
 
Theoremcdlemefr45 33459* Value of f(r) when r is an atom not under pq, using very compact hypotheses. TODO FIX COMMENT (Contributed by NM, 1-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  -.  R  .<_  ( P  .\/  Q )
 )  ->  ( F `  R )  =  [_ R  /  t ]_ D )
 
Theoremcdlemefr45e 33460* Explicit expansion of cdlemefr45 33459. TODO: use to shorten cdlemefr45 33459 uses? TODO FIX COMMENT (Contributed by NM, 10-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  -.  R  .<_  ( P  .\/  Q )
 )  ->  ( F `  R )  =  ( ( R  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  R )  ./\  W )
 ) ) )
 
Theoremcdlemefs45 33461* Value of fs(r) when r is an atom under pq and s is any atom not under pq, using very compact hypotheses. TODO FIX COMMENT (Contributed by NM, 1-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R 
 .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )
 ) )  ->  ( F `  R )  = 
 [_ R  /  s ]_ [_ S  /  t ]_ E )
 
Theoremcdlemefs45ee 33462* Explicit expansion of cdlemefs45 33461. TODO: use to shorten cdlemefs45 33461 uses? Should  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) be assigned to a hypothesis letter? TODO FIX COMMENT (Contributed by NM, 10-Apr-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R 
 .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )
 ) )  ->  ( F `  R )  =  ( ( P  .\/  Q )  ./\  ( (
 ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )  .\/  (
 ( R  .\/  S )  ./\  W ) ) ) )
 
Theoremcdlemefs45eN 33463* Explicit expansion of cdlemefs45 33461. TODO: use to shorten cdlemefs45 33461 uses? TODO FIX COMMENT (Contributed by NM, 10-Apr-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q 
 /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  (
 s  .\/  ( x  ./\ 
 W ) )  =  x )  ->  z  =  ( if ( s 
 .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  E ) ) , 
 [_ s  /  t ]_ D )  .\/  ( x  ./\  W ) ) ) ) ,  x ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R 
 .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )
 ) )  ->  ( F `  R )  =  ( ( P  .\/  Q )  ./\  ( ( F `  S )  .\/  ( ( R  .\/  S )  ./\  W )
 ) ) )
 
Theoremcdleme32sn1awN 33464* Show that  [_ R  / 
s ]_ N is an atom not under  W when  R  .<_  ( P 
.\/  Q ). (Contributed by NM, 6-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   &    |-  Y  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  t )  ./\ 
 W ) ) )   &    |-  Z  =  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  Y ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P  .\/  Q ) )  ->  ( [_ R  /  s ]_ N  e.  A  /\  -.  [_ R  /  s ]_ N  .<_  W ) )
 
Theoremcdleme41sn3a 33465* Show that  [_ R  / 
s ]_ N is under  P  .\/  Q when  R  .<_  ( P 
.\/  Q ). (Contributed by NM, 19-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   &    |-  Y  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  t )  ./\ 
 W ) ) )   &    |-  Z  =  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q ) )  ->  y  =  Y ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  R  .<_  ( P  .\/  Q ) )  ->  [_ R  /  s ]_ N  .<_  ( P  .\/  Q )
 )
 
Theoremcdleme32sn2awN 33466* Show that  [_ R  / 
s ]_ N is an atom not under  W when  -.  R  .<_  ( P  .\/  Q ). (Contributed by NM, 6-Mar-2013.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  -.  R  .<_  ( P  .\/  Q )
 )  ->  ( [_ R  /  s ]_ N  e.  A  /\  -.  [_ R  /  s ]_ N  .<_  W ) )
 
Theoremcdleme32snaw 33467* Show that  [_ R  / 
s ]_ N is an atom not under  W. (Contributed by NM, 6-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W ) ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W ) ) )   &    |-  I  =  (
 iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W 
 /\  -.  t  .<_  ( P  .\/  Q )
 )  ->  y  =  E ) )   &    |-  N  =  if ( s  .<_  ( P  .\/  Q ) ,  I ,  C )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) )  ->  ( [_ R  /  s ]_ N  e.  A  /\  -.  [_ R  /  s ]_ N  .<_  W ) )
 
Theoremcdleme32snb 33468* Show closure of  [_ R  /  s ]_ N. (Contributed by NM, 1-Mar-2013.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W ) ) )   &    |-  D  =  (