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Theorem List for Metamath Proof Explorer - 31401-31500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdlemn11 31401 Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 /\  ( J `  R )  C_  ( ( J `  Q ) 
 .(+)  ( I `  X ) ) )  ->  R  .<_  ( Q  .\/  X ) )
 
Theoremcdlemn 31402 Lemma N of [Crawley] p. 121 line 27. (Contributed by NM, 27-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  ( X  e.  B  /\  X  .<_  W ) ) )  ->  ( R  .<_  ( Q  .\/  X ) 
 <->  ( J `  R )  C_  ( ( J `
  Q )  .(+)  ( I `  X ) ) ) )
 
Theoremdihordlem6 31403* Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T ) )  ->  ( <. ( s `  G ) ,  s >.  .+  <. g ,  O >. )  =  <. ( ( s `  G )  o.  g ) ,  s >. )
 
Theoremdihordlem7 31404* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G ) ,  s >.  .+ 
 <. g ,  O >. ) ) )  ->  (
 f  =  ( ( s `  G )  o.  g )  /\  O  =  s )
 )
 
Theoremdihordlem7b 31405* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( s  e.  E  /\  g  e.  T  /\  <. f ,  O >.  =  ( <. ( s `  G ) ,  s >.  .+ 
 <. g ,  O >. ) ) )  ->  (
 f  =  g  /\  O  =  s )
 )
 
Theoremdihjustlem 31406 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B )  /\  ( Q  .\/  ( X  ./\  W ) )  =  ( R  .\/  ( X  ./\  W ) ) )  ->  (
 ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  (
 ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) ) )
 
Theoremdihjust 31407 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B )  /\  ( Q  .\/  ( X  ./\  W ) )  =  ( R  .\/  ( X  ./\  W ) ) )  ->  (
 ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  =  ( ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) ) )
 
Theoremdihord1 31408 Part of proof after Lemma N of [Crawley] p. 122. Forward ordering property. TODO: change  ( Q  .\/  ( X  ./\  W ) )  =  X to  Q  .<_  X using lhpmcvr3 30214, here and all theorems below. (Contributed by NM, 2-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( Q 
 .\/  ( X  ./\  W ) )  =  X  /\  ( R  .\/  ( Y  ./\  W ) )  =  Y  /\  X  .<_  Y ) )  ->  ( ( J `  Q )  .(+)  ( I `
  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `
  ( Y  ./\  W ) ) ) )
 
Theoremdihord2a 31409 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( Q 
 .\/  ( X  ./\  W ) )  =  X  /\  ( R  .\/  ( Y  ./\  W ) )  =  Y  /\  (
 ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  (
 ( J `  R )  .(+)  ( I `  ( Y  ./\  W ) ) ) ) ) 
 ->  Q  .<_  ( R  .\/  ( Y  ./\  W ) ) )
 
Theoremdihord2b 31410 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( J `
  Q )  .(+)  ( I `  ( X 
 ./\  W ) ) ) 
 C_  ( ( J `
  R )  .(+)  ( I `  ( Y 
 ./\  W ) ) ) )  ->  ( I `  ( X  ./\  W ) )  C_  ( ( J `  R )  .(+)  ( I `  ( Y 
 ./\  W ) ) ) )
 
Theoremdihord2cN 31411* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: needed? shorten other proof with it? (Contributed by NM, 3-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( f  e.  T  /\  ( R `  f )  .<_  ( X  ./\  W )
 ) )  ->  <. f ,  O >.  e.  ( I `  ( X  ./\  W ) ) )
 
Theoremdihord11b 31412* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( ( oc `  K ) `
  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( J `
  Q )  .(+)  ( I `  ( X 
 ./\  W ) ) ) 
 C_  ( ( J `
  N )  .(+)  ( I `  ( Y 
 ./\  W ) ) ) )  /\  ( f  e.  T  /\  ( R `  f )  .<_  ( X  ./\  W )
 ) )  ->  <. f ,  O >.  e.  (
 ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) )
 
Theoremdihord10 31413* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( ( oc `  K ) `
  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) ) 
 /\  ( f  e.  T  /\  ( R `
  f )  .<_  ( X  ./\  W )
 )  /\  ( (
 s  e.  E  /\  g  e.  T )  /\  ( R `  g
 )  .<_  ( Y  ./\  W )  /\  <. f ,  O >.  =  ( <. ( s `  G ) ,  s >.  .+ 
 <. g ,  O >. ) ) )  ->  ( R `  f )  .<_  ( Y  ./\  W )
 )
 
Theoremdihord11c 31414* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( ( oc `  K ) `
  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( ( J `  Q ) 
 .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  N )  .(+)  ( I `  ( Y 
 ./\  W ) ) ) 
 /\  f  e.  T  /\  ( R `  f
 )  .<_  ( X  ./\  W ) ) )  ->  E. y  e.  ( J `  N ) E. z  e.  ( I `  ( Y  ./\  W ) ) <. f ,  O >.  =  ( y  .+  z ) )
 
Theoremdihord2pre 31415* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( ( oc `  K ) `
  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( J `
  Q )  .(+)  ( I `  ( X 
 ./\  W ) ) ) 
 C_  ( ( J `
  N )  .(+)  ( I `  ( Y 
 ./\  W ) ) ) )  ->  ( X  ./\ 
 W )  .<_  ( Y 
 ./\  W ) )
 
Theoremdihord2pre2 31416* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 4-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  P  =  ( ( oc `  K ) `
  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  N )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( Q 
 .\/  ( X  ./\  W ) )  =  X  /\  ( N  .\/  ( Y  ./\  W ) )  =  Y  /\  (
 ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  (
 ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) ) ) 
 ->  ( Q  .\/  ( X  ./\  W ) ) 
 .<_  ( N  .\/  ( Y  ./\  W ) ) )
 
Theoremdihord2 31417 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. Todo: do we need 
-.  X  .<_  W and  -.  Y  .<_  W? (Contributed by NM, 4-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) ) 
 /\  ( X  e.  B  /\  Y  e.  B )  /\  ( ( Q 
 .\/  ( X  ./\  W ) )  =  X  /\  ( N  .\/  ( Y  ./\  W ) )  =  Y  /\  (
 ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  (
 ( J `  N )  .(+)  ( I `  ( Y  ./\  W ) ) ) ) ) 
 ->  X  .<_  Y )
 
Syntaxcdih 31418 Extend class notation with isomorphism H.
 class  DIsoH
 
Definitiondf-dih 31419* Define isomorphism H. (Contributed by NM, 28-Jan-2014.)
 |-  DIsoH  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  ( Base `  k
 )  |->  if ( x ( le `  k ) w ,  ( ( ( DIsoB `  k ) `  w ) `  x ) ,  ( iota_ u  e.  ( LSubSp `  ( ( DVecH `  k ) `  w ) ) A. q  e.  ( Atoms `  k ) ( ( -.  q ( le `  k ) w  /\  ( q ( join `  k ) ( x ( meet `  k ) w ) )  =  x )  ->  u  =  ( ( ( (
 DIsoC `  k ) `  w ) `  q
 ) ( LSSum `  (
 ( DVecH `  k ) `  w ) ) ( ( ( DIsoB `  k
 ) `  w ) `  ( x ( meet `  k ) w ) ) ) ) ) ) ) ) )
 
Theoremdihffval 31420* The isomorphism H for a lattice  K. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  ( K  e.  V  ->  (
 DIsoH `  K )  =  ( w  e.  H  |->  ( x  e.  B  |->  if ( x  .<_  w ,  ( ( ( DIsoB `  K ) `  w ) `  x ) ,  ( iota_ u  e.  ( LSubSp `
  ( ( DVecH `  K ) `  w ) ) A. q  e.  A  ( ( -.  q  .<_  w  /\  (
 q  .\/  ( x  ./\ 
 w ) )  =  x )  ->  u  =  ( ( ( (
 DIsoC `  K ) `  w ) `  q
 ) ( LSSum `  (
 ( DVecH `  K ) `  w ) ) ( ( ( DIsoB `  K ) `  w ) `  ( x  ./\  w ) ) ) ) ) ) ) ) )
 
Theoremdihfval 31421* Isomorphism H for a lattice  K. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( x  e.  B  |->  if ( x  .<_  W ,  ( D `  x ) ,  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  (
 q  .\/  ( x  ./\ 
 W ) )  =  x )  ->  u  =  ( ( C `  q )  .(+)  ( D `
  ( x  ./\  W ) ) ) ) ) ) ) )
 
Theoremdihval 31422* Value of isomorphism H for a lattice  K. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 3-Feb-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  X  e.  B ) 
 ->  ( I `  X )  =  if ( X  .<_  W ,  ( D `  X ) ,  ( iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q 
 .\/  ( X  ./\  W ) )  =  X )  ->  u  =  ( ( C `  q
 )  .(+)  ( D `  ( X  ./\  W ) ) ) ) ) ) )
 
Theoremdihvalc 31423* Value of isomorphism H for a lattice  K when  -.  X  .<_  W. (Contributed by NM, 4-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  ( I `  X )  =  (
 iota_ u  e.  S A. q  e.  A  ( ( -.  q  .<_  W  /\  ( q 
 .\/  ( X  ./\  W ) )  =  X )  ->  u  =  ( ( C `  q
 )  .(+)  ( D `  ( X  ./\  W ) ) ) ) ) )
 
Theoremdihlsscpre 31424 Closure of isomorphism H for a lattice  K when  -.  X  .<_  W. (Contributed by NM, 6-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  ( I `  X )  e.  S )
 
Theoremdihvalcqpre 31425 Value of isomorphism H for a lattice  K when  -.  X  .<_  W, given auxiliary atom  Q. (Contributed by NM, 6-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .\/  ( X 
 ./\  W ) )  =  X ) )  ->  ( I `  X )  =  ( ( C `
  Q )  .(+)  ( D `  ( X 
 ./\  W ) ) ) )
 
Theoremdihvalcq 31426 Value of isomorphism H for a lattice  K when  -.  X  .<_  W, given auxiliary atom  Q. TODO: Use dihvalcq2 31437 (with lhpmcvr3 30214 for  ( Q  .\/  ( X  ./\  W ) )  =  X simplification) that changes  C and  D to  I and make this obsolete. Do to other theorems as well. (Contributed by NM, 6-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .\/  ( X 
 ./\  W ) )  =  X ) )  ->  ( I `  X )  =  ( ( C `
  Q )  .(+)  ( D `  ( X 
 ./\  W ) ) ) )
 
Theoremdihvalb 31427 Value of isomorphism H for a lattice  K when  X  .<_  W. (Contributed by NM, 4-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  D  =  ( ( DIsoB `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( I `  X )  =  ( D `  X ) )
 
TheoremdihopelvalbN 31428* Ordered pair member of the partial isomorphism H for argument under  W. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( g  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  V  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) ) 
 ->  ( <. F ,  S >.  e.  ( I `  X )  <->  ( ( F  e.  T  /\  ( R `  F )  .<_  X )  /\  S  =  O ) ) )
 
Theoremdihvalcqat 31429 Value of isomorphism H for a lattice  K at an atom not under  W. (Contributed by NM, 27-Mar-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  J  =  ( ( DIsoC `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( I `  Q )  =  ( J `  Q ) )
 
Theoremdih1dimb 31430* Two expressions for a 1-dimensional subspace of vector space H (when  F is a nonzero vector i.e. non-identity translation). (Contributed by NM, 27-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T ) 
 ->  ( I `  ( R `  F ) )  =  ( N `  { <. F ,  O >. } ) )
 
Theoremdih1dimb2 31431* Isomorphism H at an atom under  W. (Contributed by NM, 27-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  U  =  ( (
 DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  Q  .<_  W ) ) 
 ->  E. f  e.  T  ( f  =/=  (  _I  |`  B )  /\  ( I `  Q )  =  ( N `  { <. f ,  O >. } ) ) )
 
Theoremdih1dimc 31432* Isomorphism H at an atom not under 
W. (Contributed by NM, 27-Apr-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  ( iota_ f  e.  T ( f `  P )  =  Q )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( I `  Q )  =  ( N `  { <. F ,  (  _I  |`  T )
 >. } ) )
 
Theoremdib2dim 31433 Extend dia2dim 31267 to partial isomorphism B. (Contributed by NM, 22-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoB `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( P  e.  A  /\  P  .<_  W ) )   &    |-  ( ph  ->  ( Q  e.  A  /\  Q  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  C_  (
 ( I `  P )  .(+)  ( I `  Q ) ) )
 
Theoremdih2dimb 31434 Extend dib2dim 31433 to isomorphism H. (Contributed by NM, 22-Sep-2014.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( P  e.  A  /\  P  .<_  W ) )   &    |-  ( ph  ->  ( Q  e.  A  /\  Q  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  C_  (
 ( I `  P )  .(+)  ( I `  Q ) ) )
 
Theoremdih2dimbALTN 31435 Extend dia2dim 31267 to isomorphism H. (This version combines dib2dim 31433 and dih2dimb 31434 for shorter overall proof, but may be less easy to understand. TODO: decide which to use.) (Contributed by NM, 22-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( P  e.  A  /\  P  .<_  W ) )   &    |-  ( ph  ->  ( Q  e.  A  /\  Q  .<_  W ) )   =>    |-  ( ph  ->  ( I `  ( P  .\/  Q ) )  C_  (
 ( I `  P )  .(+)  ( I `  Q ) ) )
 
Theoremdihopelvalcqat 31436* Ordered pair member of the partial isomorphism H for atom argument not under  W. TODO: remove .t hypothesis. (Contributed by NM, 30-Mar-2014.)
 |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  G  =  (
 iota_ g  e.  T ( g `  P )  =  Q )   &    |-  F  e.  _V   &    |-  S  e.  _V   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( I `  Q ) 
 <->  ( F  =  ( S `  G ) 
 /\  S  e.  E ) ) )
 
Theoremdihvalcq2 31437 Value of isomorphism H for a lattice  K when  -.  X  .<_  W, given auxiliary atom  Q. (Contributed by NM, 26-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) ) 
 ->  ( I `  X )  =  ( ( I `  Q )  .(+)  ( I `  ( X 
 ./\  W ) ) ) )
 
Theoremdihopelvalcpre 31438* Member of value of isomorphism H for a lattice  K when  -.  X  .<_  W, given auxiliary atom  Q. TODO: refactor to be shorter and more understandable; add lemmas? (Contributed by NM, 13-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  G  =  ( iota_ g  e.  T ( g `  P )  =  Q )   &    |-  F  e.  _V   &    |-  S  e.  _V   &    |-  Z  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  N  =  ( (
 DIsoB `  K ) `  W )   &    |-  C  =  ( ( DIsoC `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  V  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  O  =  ( a  e.  E ,  b  e.  E  |->  ( h  e.  T  |->  ( ( a `  h )  o.  (
 b `  h )
 ) ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .\/  ( X 
 ./\  W ) )  =  X ) )  ->  ( <. F ,  S >.  e.  ( I `  X )  <->  ( ( F  e.  T  /\  S  e.  E )  /\  ( R `  ( F  o.  `' ( S `  G ) ) )  .<_  X ) ) )
 
Theoremdihopelvalc 31439* Member of value of isomorphism H for a lattice  K when  -.  X  .<_  W, given auxiliary atom  Q. (Contributed by NM, 13-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  G  =  ( iota_ g  e.  T ( g `  P )  =  Q )   &    |-  F  e.  _V   &    |-  S  e.  _V   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q  .\/  ( X 
 ./\  W ) )  =  X ) )  ->  ( <. F ,  S >.  e.  ( I `  X )  <->  ( ( F  e.  T  /\  S  e.  E )  /\  ( R `  ( F  o.  `' ( S `  G ) ) )  .<_  X ) ) )
 
Theoremdihlss 31440 The value of isomorphism H is a subspace. (Contributed by NM, 6-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  ->  ( I `  X )  e.  S )
 
Theoremdihss 31441 The value of isomorphism H is a set of vectors. (Contributed by NM, 14-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B )  ->  ( I `  X )  C_  V )
 
Theoremdihssxp 31442 An isomorphism H value is included in the vector space (expressed as  T  X.  E). (Contributed by NM, 26-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( I `  X ) 
 C_  ( T  X.  E ) )
 
Theoremdihopcl 31443 Closure of an ordered pair (vector) member of a value of isomorphism H. (Contributed by NM, 26-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  <. F ,  S >.  e.  ( I `  X ) )   =>    |-  ( ph  ->  ( F  e.  T  /\  S  e.  E )
 )
 
TheoremxihopellsmN 31444* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  A  =  ( s  e.  E ,  t  e.  E  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
 t `  f )
 ) ) )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  L  =  ( LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( <. F ,  S >.  e.  ( ( I `  X )  .(+)  ( I `
  Y ) )  <->  E. g E. t E. h E. u ( (
 <. g ,  t >.  e.  ( I `  X )  /\  <. h ,  u >.  e.  ( I `  Y ) )  /\  ( F  =  (
 g  o.  h ) 
 /\  S  =  ( t A u ) ) ) ) )
 
Theoremdihopellsm 31445* Ordered pair membership in a subspace sum of isomorphism H values. (Contributed by NM, 26-Sep-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  A  =  ( v  e.  E ,  w  e.  E  |->  ( i  e.  T  |->  ( ( v `  i )  o.  ( w `  i ) ) ) )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  L  =  (
 LSubSp `  U )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( <. F ,  S >.  e.  ( ( I `  X )  .(+)  ( I `
  Y ) )  <->  E. g E. t E. h E. u ( (
 <. g ,  t >.  e.  ( I `  X )  /\  <. h ,  u >.  e.  ( I `  Y ) )  /\  ( F  =  (
 g  o.  h ) 
 /\  S  =  ( t A u ) ) ) ) )
 
Theoremdihord6apre 31446* Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B ) )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  q )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) )  /\  ( I `  X ) 
 C_  ( I `  Y ) )  ->  X  .<_  Y )
 
Theoremdihord3 31447 The isomorphism H for a lattice  K is order-preserving in the region under co-atom  W. (Contributed by NM, 6-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) ) 
 ->  ( ( I `  X )  C_  ( I `
  Y )  <->  X  .<_  Y ) )
 
Theoremdihord4 31448 The isomorphism H for a lattice  K is order-preserving in the region not under co-atom  W. TODO: reformat q e. A /\ -. q .<_ W to eliminate adant*. (Contributed by NM, 6-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  -.  Y  .<_  W ) )  ->  ( ( I `  X )  C_  ( I `  Y )  <->  X  .<_  Y ) )
 
Theoremdihord5b 31449 Part of proof that isomorphism H is order-preserving. TODO: eliminate 3ad2ant1; combine w/ other way to have one lhpmcvr2 (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  -.  Y  .<_  W ) )  /\  X  .<_  Y )  ->  ( I `  X )  C_  ( I `  Y ) )
 
Theoremdihord6b 31450 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( Y  e.  B  /\  Y  .<_  W ) )  /\  X  .<_  Y )  ->  ( I `  X )  C_  ( I `  Y ) )
 
Theoremdihord6a 31451 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( Y  e.  B  /\  Y  .<_  W ) )  /\  ( I `
  X )  C_  ( I `  Y ) )  ->  X  .<_  Y )
 
Theoremdihord5apre 31452 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( X  e.  B  /\  X  .<_  W ) 
 /\  ( Y  e.  B  /\  -.  Y  .<_  W ) )  /\  ( I `  X )  C_  ( I `  Y ) )  ->  X  .<_  Y )
 
Theoremdihord5a 31453 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  -.  Y  .<_  W ) )  /\  ( I `
  X )  C_  ( I `  Y ) )  ->  X  .<_  Y )
 
Theoremdihord 31454 The isomorphism H is order-preserving. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( I `  X ) 
 C_  ( I `  Y )  <->  X  .<_  Y ) )
 
Theoremdih11 31455 The isomorphism H is one-to-one. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( I `  X )  =  ( I `  Y )  <->  X  =  Y ) )
 
Theoremdihf11lem 31456 Functionality of the isomorphism H. (Contributed by NM, 6-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  I : B --> S )
 
Theoremdihf11 31457 The isomorphism H for a lattice  K is a one-to-one function. . Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  I : B -1-1-> S )
 
Theoremdihfn 31458 Functionality and domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  B )
 
Theoremdihdm 31459 Domain of isomorphism H. (Contributed by NM, 9-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  dom  I  =  B )
 
Theoremdihcl 31460 Closure of isomorphism H. (Contributed by NM, 8-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B ) 
 ->  ( I `  X )  e.  ran  I )
 
Theoremdihcnvcl 31461 Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I
 )  ->  ( `' I `  X )  e.  B )
 
Theoremdihcnvid1 31462 The converse isomorphism of an isomorphism. (Contributed by NM, 5-Aug-2014.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B ) 
 ->  ( `' I `  ( I `  X ) )  =  X )
 
Theoremdihcnvid2 31463 The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I
 )  ->  ( I `  ( `' I `  X ) )  =  X )
 
Theoremdihcnvord 31464 Ordering property for converse of isomorphism H. (Contributed by NM, 17-Aug-2014.)
 |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Y  e.  ran  I )   =>    |-  ( ph  ->  (
 ( `' I `  X )  .<_  ( `' I `  Y )  <->  X  C_  Y ) )
 
Theoremdihcnv11 31465 The converse of isomorphism H is one-to-one. (Contributed by NM, 17-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ran  I )   &    |-  ( ph  ->  Y  e.  ran  I )   =>    |-  ( ph  ->  (
 ( `' I `  X )  =  ( `' I `  Y )  <->  X  =  Y )
 )
 
Theoremdihsslss 31466 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  ran  I  C_  S )
 
Theoremdihrnlss 31467 The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  X  e.  S )
 
Theoremdihrnss 31468 The isomorphism H maps to a set of vectors. (Contributed by NM, 14-Mar-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  V  =  ( Base `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  ran  I )  ->  X  C_  V )
 
Theoremdihvalrel 31469 The value of isomorphism H is a relation. (Contributed by NM, 9-Mar-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Rel  ( I `  X ) )
 
Theoremdih0 31470 The value of isomorphism H at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 9-Mar-2014.)
 |-  .0.  =  ( 0. `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  O  =  ( 0g `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .0.  )  =  { O } )
 
Theoremdih0bN 31471 A lattice element is zero iff its isomorphism is the zero subspace. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  Z  =  ( 0g `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( X  =  .0.  <->  ( I `  X )  =  { Z } ) )
 
Theoremdih0vbN 31472 A vector is zero iff its span is the isomorphism of lattice zero. (Contributed by NM, 16-Aug-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  U )   &    |-  Z  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( X  =  Z  <->  ( N `  { X } )  =  ( I `  .0.  ) ) )
 
Theoremdih0cnv 31473 The isomorphism H converse value of the zero subspace is the lattice zero. (Contributed by NM, 19-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  Z  =  ( 0g `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( `' I `  { Z } )  =  .0.  )
 
Theoremdih0rn 31474 The zero subspace belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  {  .0.  }  e.  ran 
 I )
 
Theoremdih0sb 31475 A subspace is zero iff the converse of its isomorphism is lattice zero. (Contributed by NM, 17-Aug-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  .0.  =  ( 0. `  K )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  U )   &    |-  Z  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ran  I )   =>    |-  ( ph  ->  ( X  =  { Z } 
 <->  ( `' I `  X )  =  .0.  ) )
 
Theoremdih1 31476 The value of isomorphism H at the lattice unit is the set of all vectors. (Contributed by NM, 13-Mar-2014.)
 |-  .1.  =  ( 1. `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( I `  .1.  )  =  V )
 
Theoremdih1rn 31477 The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   =>    |-  (
 ( K  e.  HL  /\  W  e.  H ) 
 ->  V  e.  ran  I
 )
 
Theoremdih1cnv 31478 The isomorphism H converse value of the full vector space is the lattice one. (Contributed by NM, 19-Jun-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  .1.  =  ( 1. `  K )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  (
 Base `  U )   =>    |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( `' I `  V )  =  .1.  )
 
TheoremdihwN 31479* Value of isomorphism H at the fiducial hyperplane  W. (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   =>    |-  ( ph  ->  ( I `  W )  =  ( T  X.  {  .0.  } ) )
 
Theoremdihmeetlem1N 31480* Isomorphism H of a conjunction. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  q )   &    |-  .0.  =  ( h  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) ) 
 ->  ( I `  ( X  ./\  Y ) )  =  ( ( I `
  X )  i^i  ( I `  Y ) ) )
 
Theoremdihglblem5apreN 31481* A conjunction property of isomorphism H. TODO: reduce antecedent size; general review for shorter proof. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  q )   &    |-  .0.  =  ( h  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) 
 ->  ( I `  ( X  ./\  W ) )  =  ( ( I `
  X )  i^i  ( I `  W ) ) )
 
Theoremdihglblem5aN 31482 A conjunction property of isomorphism H. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B ) 
 ->  ( I `  ( X  ./\  W ) )  =  ( ( I `
  X )  i^i  ( I `  W ) ) )
 
Theoremdihglblem2aN 31483* Lemma for isomorphism H of a GLB. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\ 
 W ) }   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) ) 
 ->  T  =/=  (/) )
 
Theoremdihglblem2N 31484* The GLB of a set of lattice elements  S is the same as that of the set  T with elements of  S cut down to be under  W. (Contributed by NM, 19-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\ 
 W ) }   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  C_  B  /\  ( G `  S ) 
 .<_  W )  ->  ( G `  S )  =  ( G `  T ) )
 
Theoremdihglblem3N 31485* Isomorphism H of a lattice glb. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\ 
 W ) }   &    |-  J  =  ( ( DIsoB `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
 .<_  W )  ->  ( I `  ( G `  T ) )  = 
 |^|_ x  e.  T  ( I `  x ) )
 
Theoremdihglblem3aN 31486* Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\ 
 W ) }   &    |-  J  =  ( ( DIsoB `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  ( G `  S ) 
 .<_  W )  ->  ( I `  ( G `  S ) )  = 
 |^|_ x  e.  T  ( I `  x ) )
 
Theoremdihglblem4 31487* Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  T  =  { u  e.  B  |  E. v  e.  S  u  =  ( v  ./\ 
 W ) }   &    |-  J  =  ( ( DIsoB `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) ) ) 
 ->  ( I `  ( G `  S ) ) 
 C_  |^|_ x  e.  S  ( I `  x ) )
 
Theoremdihglblem5 31488* Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  S  =  (
 LSubSp `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( T  C_  B  /\  T  =/=  (/) ) ) 
 ->  |^|_ x  e.  T  ( I `  x )  e.  S )
 
Theoremdihmeetlem2N 31489 Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  G  =  ( iota_ h  e.  T ( h `  P )  =  q )   &    |-  .0.  =  ( h  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W )  /\  ( Y  e.  B  /\  Y  .<_  W ) ) 
 ->  ( I `  ( X  ./\  Y ) )  =  ( ( I `
  X )  i^i  ( I `  Y ) ) )
 
TheoremdihglbcpreN 31490* Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane  W. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .<_  =  ( le `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  F  =  ( iota_ g  e.  T ( g `  P )  =  q )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/=  (/) )  /\  -.  ( G `  S )  .<_  W )  ->  ( I `  ( G `
  S ) )  =  |^|_ x  e.  S  ( I `  x ) )
 
TheoremdihglbcN 31491* Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane  W. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  G  =  ( glb `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  .<_  =  ( le `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  C_  B  /\  S  =/= 
 (/) )  /\  -.  ( G `  S ) 
 .<_  W )  ->  ( I `  ( G `  S ) )  = 
 |^|_ x  e.  S  ( I `  x ) )
 
TheoremdihmeetcN 31492 Isomorphism H of a lattice meet when the meet is not under the fiducial hyperplane  W. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  -.  ( X  ./\  Y )  .<_  W )  ->  ( I `  ( X 
 ./\  Y ) )  =  ( ( I `  X )  i^i  ( I `
  Y ) ) )
 
TheoremdihmeetbN 31493 Isomorphism H of a lattice meet when one element is under the fiducial hyperplane  W. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  ( Y  e.  B  /\  Y  .<_  W ) )  ->  ( I `  ( X 
 ./\  Y ) )  =  ( ( I `  X )  i^i  ( I `
  Y ) ) )
 
TheoremdihmeetbclemN 31494 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y ) 
 .<_  W )  ->  ( I `  ( X  ./\  Y ) )  =  ( ( ( I `  X )  i^i  ( I `
  Y ) )  i^i  ( I `  W ) ) )
 
Theoremdihmeetlem3N 31495 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (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 )  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y ) 
 .<_  W )  /\  (
 ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( Q  .\/  ( X  ./\  W ) )  =  X ) 
 /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .\/  ( Y 
 ./\  W ) )  =  Y ) )  ->  Q  =/=  R )
 
Theoremdihmeetlem4preN 31496* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   &    |-  G  =  ( iota_ g  e.  T ( g `
  P )  =  Q )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( ( I `  Q )  i^i  ( I `  ( X  ./\  W ) ) )  =  {  .0.  } )
 
Theoremdihmeetlem4N 31497 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  (
 LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  ( ( I `  Q )  i^i  ( I `  ( X  ./\  W ) ) )  =  {  .0.  } )
 
Theoremdihmeetlem5 31498 Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  ( X  ./\  ( Y  .\/  Q ) )  =  (
 ( X  ./\  Y ) 
 .\/  Q ) )
 
Theoremdihmeetlem6 31499 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) ) 
 ->  -.  ( X  ./\  ( Y  .\/  Q ) )  .<_  W )
 
Theoremdihmeetlem7N 31500 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B ) 
 /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  ( ( ( X 
 ./\  Y )  .\/  p )  ./\  Y )  =  ( X  ./\  Y ) )
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