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Theorem List for Metamath Proof Explorer - 29701-29800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdleme31se 29701* 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 29702* 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 29703* 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 29704* 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 29705* 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 29706* 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 29707* 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 29708* 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 29709* 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 29710* 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 29711* 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 29712* 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 29713* 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 29714 ***START OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW*** FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 29349. (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 29715* 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 29716* 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 29717* 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 29718* TODO: NOT USED? Show closure of the unique element in cdlemefrs29cpre1 29717. (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 29719* 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 29349 here and elsewhere, and presence/absence of  s 
.<_  ( P  .\/  Q
) term. Also, why can proof be shortened with cdleme29cl 29696? What is difference from cdlemefs27cl 29732? (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 29720* 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 29721* Lemma for cdlemefs29bpre1N 29736. (Compare cdleme25a 29672.) 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 29722 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 29723* 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 29724* 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 29725* 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 29726* Show closure of the unique element in cdleme29c 29695. 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 29727* 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 29728* 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 29729* 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 29730* Value of  ( F `  R ) when  -.  R  .<_  ( P  .\/  Q
). TODO This may be useful for shortening others that now use riotasv 6285 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 29731 FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 29349. (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 29732* 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 29685 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 29733* 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 29734* 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 29735* 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 29736* 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 29737* 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 29738* Show closure of the unique element in cdleme29c 29695. (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 29739* 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 29740* 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 29741* 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 29349 here and elsewhere, and presence/absence of  s 
.<_  ( P  .\/  Q
) term. Also, why can proof be shortened with cdleme27cl 29685? What is difference from cdlemefs27cl 29732? (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 29742* 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 29743* Value of  ( F `  R ) when  R  .<_  ( P  .\/  Q ). TODO This may be useful for shortening others that now use riotasv 6285 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 29744* 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 29745* 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 29748 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 29746* 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 29747* Explicit expansion of cdlemefr45 29746. TODO: use to shorten cdlemefr45 29746 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 29748* 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 29749* Explicit expansion of cdlemefs45 29748. TODO: use to shorten cdlemefs45 29748 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 29750* Explicit expansion of cdlemefs45 29748. TODO: use to shorten cdlemefs45 29748 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 29751* 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 29752* 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 29753* 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 29754* 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 29755* 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  =  ( ( 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.  B )
 
Theoremcdleme32fva 29756* Part of proof of Lemma D in [Crawley] p. 113. Value of  F at an atom not under  W. (Contributed by NM, 2-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 )   &    |-  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 ) ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q )  ->  [_ R  /  x ]_ O  =  [_ R  /  s ]_ N )
 
Theoremcdleme32fva1 29757* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 2-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 )   &    |-  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 ) ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  P  =/=  Q )  ->  ( F `  R )  =  [_ R  /  s ]_ N )
 
Theoremcdleme32fvaw 29758* Show that  ( F `  R ) is an atom not under  W when  R is an atom not under  W. (Contributed by NM, 18-Apr-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 )   &    |-  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 ) ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  (
 ( F `  R )  e.  A  /\  -.  ( F `  R )  .<_  W ) )
 
Theoremcdleme32fvcl 29759* Part of proof of Lemma D in [Crawley] p. 113. Closure of the function  F. (Contributed by NM, 10-Feb-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 )   &    |-  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 ) ) 
 /\  X  e.  B )  ->  ( F `  X )  e.  B )
 
Theoremcdleme32a 29760* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 19-Feb-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 )   &    |-  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 ) ) 
 /\  ( X  e.  B  /\  ( P  =/=  Q 
 /\  -.  X  .<_  W ) )  /\  (
 ( s  e.  A  /\  -.  s  .<_  W ) 
 /\  ( s  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  =  ( N  .\/  ( X  ./\ 
 W ) ) )
 
Theoremcdleme32b 29761* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 19-Feb-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 )   &    |-  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 ) ) 
 /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  ( s  .\/  ( X 
 ./\  W ) )  =  X  /\  X  .<_  Y ) )  ->  ( F `  Y )  =  ( N  .\/  ( Y  ./\  W ) ) )
 
Theoremcdleme32c 29762* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 19-Feb-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 )   &    |-  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 ) ) 
 /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  ( s  .\/  ( X 
 ./\  W ) )  =  X  /\  X  .<_  Y ) )  ->  ( F `  X )  .<_  ( F `  Y ) )
 
Theoremcdleme32d 29763* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 20-Feb-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 )   &    |-  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 ) ) 
 /\  ( X  e.  B  /\  Y  e.  B  /\  ( P  =/=  Q  /\  -.  X  .<_  W ) )  /\  X  .<_  Y )  ->  ( F `  X )  .<_  ( F `
  Y ) )
 
Theoremcdleme32e 29764* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 20-Feb-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 )   &    |-  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 ) ) 
 /\  ( ( X  e.  B  /\  Y  e.  B )  /\  -.  ( P  =/=  Q  /\  -.  X  .<_  W )  /\  ( P  =/=  Q  /\  -.  Y  .<_  W ) ) 
 /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  ( s  .\/  ( Y 
 ./\  W ) )  =  Y  /\  X  .<_  Y ) )  ->  ( F `  X )  .<_  ( F `  Y ) )
 
Theoremcdleme32f 29765* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 20-Feb-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 )   &    |-  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