HomeHome Metamath Proof Explorer
Theorem List (p. 308 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

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

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21494)
  Hilbert Space Explorer  Hilbert Space Explorer
(21495-23017)
  Users' Mathboxes  Users' Mathboxes
(23018-31433)
 

Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdihval 30701* 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 30702* 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 30703 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 30704 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 30705 Value of isomorphism H for a lattice  K when  -.  X  .<_  W, given auxiliary atom  Q. TODO: Use dihvalcq2 30716 (with lhpmcvr3 29493 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 30706 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 30707* 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 30708 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 30709* 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 30710* 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 30711* 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 30712 Extend dia2dim 30546 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 30713 Extend dib2dim 30712 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 30714 Extend dia2dim 30546 to isomorphism H. (This version combines dib2dim 30712 and dih2dimb 30713 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 30715* 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 30716 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 30717* 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 30718* 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 30719 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 30720 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 30721 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 30722 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 30723* 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 30724* 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 30725* 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 30726 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 30727 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 30728 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 30729 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 30730 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 30731 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 30732 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 30733 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 30734 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 30735 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 30736 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 30737 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 30738 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 30739 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 30740 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 30741 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 30742 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 30743 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 30744 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 30745 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 30746 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 30747 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 30748 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 30749 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 30750 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 30751 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 30752 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 30753 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 30754 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 30755 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 30756 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 30757 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 30758* 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 30759* 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 30760* 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 30761 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 30762* 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 30763* 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 30764* 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 30765* 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 30766* 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 30767* 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 30768 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 30769* 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 30770* 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 30771 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 30772 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 30773 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 30774 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 30775* 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 30776 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 30777 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 30778 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 30779 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 ) )
 
Theoremdihjatc1 30780 Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change  .\/ order of  ( X  ./\  Y )  .\/  Q here and down? (Contributed by NM, 6-Apr-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  /\  Y  e.  B ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q 
 .<_  X  /\  ( X 
 ./\  Y )  .<_  W ) )  ->  ( I `  ( ( X  ./\  Y )  .\/  Q )
 )  =  ( ( I `  Q ) 
 .(+)  ( I `  ( X  ./\  Y ) ) ) )
 
Theoremdihjatc2N 30781 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.)
 |-  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  /\  Y  e.  B ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q 
 .<_  X  /\  ( X 
 ./\  Y )  .<_  W ) )  ->  ( I `  ( Q  .\/  ( X  ./\  Y ) ) )  =  ( ( I `  Q ) 
 .(+)  ( I `  ( X  ./\  Y ) ) ) )
 
Theoremdihjatc3 30782 Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-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  /\  Y  e.  B ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( Q 
 .<_  X  /\  ( X 
 ./\  Y )  .<_  W ) )  ->  ( I `  ( ( X  ./\  Y )  .\/  Q )
 )  =  ( ( I `  ( X 
 ./\  Y ) )  .(+)  ( I `  Q ) ) )
 
Theoremdihmeetlem8N 30783 Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change  .\/ order of  ( X  ./\  Y )  .\/  p here and down? (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  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  /\  Y  e.  B ) 
 /\  ( p  e.  A  /\  -.  p  .<_  W )  /\  ( p  .<_  X  /\  ( X  ./\  Y )  .<_  W ) )  ->  ( I `  ( ( X 
 ./\  Y )  .\/  p ) )  =  (
 ( I `  p )  .(+)  ( I `  ( X  ./\  Y ) ) ) )
 
Theoremdihmeetlem9N 30784 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  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  /\  Y  e.  B ) 
 /\  p  e.  A )  ->  ( ( ( I `  p ) 
 .(+)  ( I `  ( X  ./\  Y ) ) )  i^i  ( I `
  Y ) )  =  ( ( I `
  ( X  ./\  Y ) )  .(+)  ( ( I `  p )  i^i  ( I `  Y ) ) ) )
 
Theoremdihmeetlem10N 30785 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  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  /\  Y  e.  B ) 
 /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) ) 
 ->  ( I `  (
 ( X  ./\  Y ) 
 .\/  p ) )  =  ( ( I `
  X )  i^i  ( I `  ( Y  .\/  p ) ) ) )
 
Theoremdihmeetlem11N 30786 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  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  /\  Y  e.  B ) 
 /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X ) ) 
 ->  ( ( I `  ( ( X  ./\  Y )  .\/  p )
 )  i^i  ( I `  Y ) )  =  ( ( I `  X )  i^i  ( I `
  Y ) ) )
 
Theoremdihmeetlem12N 30787 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
 |-  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  /\  Y  e.  B ) 
 /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  p  .<_  X  /\  ( X  ./\  Y )  .<_  W ) )  ->  (
 ( I `  ( X  ./\  Y ) ) 
 .(+)  ( ( I `  p )  i^i  ( I `
  Y ) ) )  =  ( ( I `  X )  i^i  ( I `  Y ) ) )
 
Theoremdihmeetlem13N 30788* Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( (
 LTrn `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  I  =  ( (
 DIsoH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  ( iota_ h  e.  T ( h `  P )  =  Q )   &    |-  G  =  ( iota_ h  e.  T ( h `
  P )  =  R )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  Q  =/=  R )  ->  (
 ( I `  Q )  i^i  ( I `  R ) )  =  {  .0.  } )
 
Theoremdihmeetlem14N 30789 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 ) 
 /\  Y  e.  B  /\  p  e.  B )  /\  ( ( r  e.  A  /\  -.  r  .<_  W )  /\  r  .<_  Y  /\  ( Y  ./\  p )  .<_  W ) )  ->  (
 ( I `  ( Y  ./\  p ) ) 
 .(+)  ( ( I `  r )  i^i  ( I `
  p ) ) )  =  ( ( I `  Y )  i^i  ( I `  p ) ) )
 
Theoremdihmeetlem15N 30790 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 )   &    |-  .0.  =  ( 0g `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  e.  B  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( ( r  e.  A  /\  -.  r  .<_  W )  /\  r  .<_  Y  /\  ( Y  ./\  p )  .<_  W ) )  ->  (
 ( I `  r
 )  i^i  ( I `  p ) )  =  {  .0.  } )
 
Theoremdihmeetlem16N 30791 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 ) 
 /\  Y  e.  B  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( ( r  e.  A  /\  -.  r  .<_  W )  /\  r  .<_  Y  /\  ( Y  ./\  p )  .<_  W ) )  ->  ( I `  ( Y  ./\  p ) )  =  ( ( I `  Y )  i^i  ( I `  p ) ) )
 
Theoremdihmeetlem17N 30792 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 )   &    |-  .0.  =  ( 0. `  K )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y ) 
 .<_  W  /\  p  .<_  X ) )  ->  ( Y  ./\  p )  =  .0.  )
 
Theoremdihmeetlem18N 30793 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 )   &    |-  .0.  =  ( 0g `  U )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) 
 /\  Y  e.  B )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  ( r  e.  A  /\  -.  r  .<_  W ) 
 /\  ( p  .<_  X 
 /\  r  .<_  Y  /\  ( X  ./\  Y ) 
 .<_  W ) ) ) 
 ->  ( ( I `  Y )  i^i  ( I `
  p ) )  =  {  .0.  }
 )
 
Theoremdihmeetlem19N 30794 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 )  /\  ( ( p  e.  A  /\  -.  p  .<_  W )  /\  ( r  e.  A  /\  -.  r  .<_  W ) 
 /\  ( p  .<_  X 
 /\  r  .<_  Y  /\  ( X  ./\  Y ) 
 .<_  W ) ) ) 
 ->  ( I `  ( X  ./\  Y ) )  =  ( ( I `
  X )  i^i  ( I `  Y ) ) )
 
Theoremdihmeetlem20N 30795 Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
 |-  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 )  /\  ( X  ./\  Y ) 
 .<_  W ) )  ->  ( I `  ( X 
 ./\  Y ) )  =  ( ( I `  X )  i^i  ( I `
  Y ) ) )
 
TheoremdihmeetALTN 30796 Isomorphism H of a lattice meet. This version does not depend on the atomisticity of the constructed vector space. TODO: Delete? (Contributed by NM, 7-Apr-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  /\  Y  e.  B )  ->  ( I `  ( X  ./\  Y ) )  =  ( ( I `
  X )  i^i  ( I `  Y ) ) )
 
Theoremdih1dimatlem0 30797* Lemma for dih1dimat 30799. (Contributed by NM, 11-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  A  =  (LSAtoms `  U )   &    |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  ( Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  F  =  (Scalar `  U )   &    |-  J  =  ( invr `  F )   &    |-  V  =  (
 Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `
  P )  =  ( ( ( J `
  s ) `  f ) `  P ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 f  e.  T  /\  s  e.  E )  /\  s  =/=  O ) 
 ->  ( ( i  =  ( p `  G )  /\  p  e.  E ) 
 <->  ( ( i  e.  T  /\  p  e.  E )  /\  E. t  e.  E  (
 i  =  ( t `
  f )  /\  p  =  ( t  o.  s ) ) ) ) )
 
Theoremdih1dimatlem 30798* Lemma for dih1dimat 30799. (Contributed by NM, 10-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  A  =  (LSAtoms `  U )   &    |-  B  =  ( Base `  K )   &    |-  .<_  =  ( le `  K )   &    |-  C  =  ( Atoms `  K )   &    |-  P  =  ( ( oc `  K ) `  W )   &    |-  T  =  ( ( LTrn `  K ) `  W )   &    |-  R  =  ( ( trL `  K ) `  W )   &    |-  E  =  ( ( TEndo `  K ) `  W )   &    |-  O  =  ( h  e.  T  |->  (  _I  |`  B )
 )   &    |-  F  =  (Scalar `  U )   &    |-  J  =  ( invr `  F )   &    |-  V  =  (
 Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  ( LSubSp `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  G  =  ( iota_ h  e.  T ( h `
  P )  =  ( ( ( J `
  s ) `  f ) `  P ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  A )  ->  D  e.  ran  I )
 
Theoremdih1dimat 30799 Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  A  =  (LSAtoms `  U )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A )  ->  P  e.  ran  I )
 
Theoremdihlsprn 30800 The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  V ) 
 ->  ( N `  { X } )  e.  ran  I )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31433
  Copyright terms: Public domain < Previous  Next >