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Theorem List for Metamath Proof Explorer - 34401-34500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
21.30.3  Atoms, hyperplanes, and covering in a left vector space (or module)
 
Syntaxclsa 34401 Extend class notation with all 1-dim subspaces (atoms) of a left module or left vector space.
 class LSAtoms
 
Syntaxclsh 34402 Extend class notation with all subspaces of a left module or left vector space that are hyperplanes.
 class LSHyp
 
Definitiondf-lsatoms 34403* Define the set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.)
 |- LSAtoms  =  ( w  e.  _V  |->  ran  ( v  e.  (
 ( Base `  w )  \  { ( 0g `  w ) } )  |->  ( ( LSpan `  w ) `  { v }
 ) ) )
 
Definitiondf-lshyp 34404* Define the set of all hyperplanes of a left module or left vector space. Also called co-atoms, these are subspaces that are one dimension less that the full space. (Contributed by NM, 29-Jun-2014.)
 |- LSHyp  =  ( w  e.  _V  |->  { s  e.  ( LSubSp `  w )  |  (
 s  =/=  ( Base `  w )  /\  E. v  e.  ( Base `  w ) ( (
 LSpan `  w ) `  ( s  u.  { v } ) )  =  ( Base `  w )
 ) } )
 
Theoremlshpset 34405* The set of all hyperplanes of a left module or left vector space. The vector  v is called a generating vector for the hyperplane. (Contributed by NM, 29-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  H  =  (LSHyp `  W )   =>    |-  ( W  e.  X  ->  H  =  { s  e.  S  |  ( s  =/=  V  /\  E. v  e.  V  ( N `  ( s  u. 
 { v } )
 )  =  V ) } )
 
Theoremislshp 34406* The predicate "is a hyperplane" (of a left module or left vector space). (Contributed by NM, 29-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  H  =  (LSHyp `  W )   =>    |-  ( W  e.  X  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( N `  ( U  u.  { v }
 ) )  =  V ) ) )
 
Theoremislshpsm 34407* Hyperplane properties expressed with subspace sum. (Contributed by NM, 3-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( U  .(+)  ( N `
  { v }
 ) )  =  V ) ) )
 
Theoremlshplss 34408 A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  H )   =>    |-  ( ph  ->  U  e.  S )
 
Theoremlshpne 34409 A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  H )   =>    |-  ( ph  ->  U  =/=  V )
 
Theoremlshpnel 34410 A hyperplane's generating vector does not belong to the hyperplane. (Contributed by NM, 3-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { X } ) )  =  V )   =>    |-  ( ph  ->  -.  X  e.  U )
 
Theoremlshpnelb 34411 The subspace sum of a hyperplane and the span of an element equals the vector space iff the element is not in the hyperplane. (Contributed by NM, 2-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( -.  X  e.  U  <->  ( U  .(+)  ( N ` 
 { X } )
 )  =  V ) )
 
Theoremlshpnel2N 34412 Condition that determines a hyperplane. (Contributed by NM, 3-Oct-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  U  =/=  V )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  U )   =>    |-  ( ph  ->  ( U  e.  H  <->  ( U  .(+)  ( N `  { X } ) )  =  V ) )
 
Theoremlshpne0 34413 The member of the span in the hyperplane definition does not belong to the hyperplane. (Contributed by NM, 14-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { X } ) )  =  V )   =>    |-  ( ph  ->  X  =/=  .0.  )
 
Theoremlshpdisj 34414 A hyperplane and the span in the hyperplane definition are disjoint. (Contributed by NM, 3-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( U  .(+)  ( N `  { X } ) )  =  V )   =>    |-  ( ph  ->  ( U  i^i  ( N `
  { X }
 ) )  =  {  .0.  } )
 
Theoremlshpcmp 34415 If two hyperplanes are comparable, they are equal. (Contributed by NM, 9-Oct-2014.)
 |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  T  e.  H )   &    |-  ( ph  ->  U  e.  H )   =>    |-  ( ph  ->  ( T  C_  U  <->  T  =  U ) )
 
TheoremlshpinN 34416 The intersection of two different hyperplanes is not a hyperplane. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
 |-  H  =  (LSHyp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  T  e.  H )   &    |-  ( ph  ->  U  e.  H )   =>    |-  ( ph  ->  ( ( T  i^i  U )  e.  H  <->  T  =  U ) )
 
Theoremlsatset 34417* The set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  ( W  e.  X  ->  A  =  ran  (
 v  e.  ( V 
 \  {  .0.  }
 )  |->  ( N `  { v } )
 ) )
 
Theoremislsat 34418* The predicate "is a 1-dim subspace (atom)" (of a left module or left vector space). (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  ( W  e.  X  ->  ( U  e.  A  <->  E. x  e.  ( V 
 \  {  .0.  }
 ) U  =  ( N `  { x } ) ) )
 
Theoremlsatlspsn2 34419 The span of a non-zero singleton is an atom. TODO: make this obsolete and use lsatlspsn 34420 instead? (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V  /\  X  =/=  .0.  )  ->  ( N `  { X } )  e.  A )
 
Theoremlsatlspsn 34420 The span of a non-zero singleton is an atom. (Contributed by NM, 16-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( N ` 
 { X } )  e.  A )
 
Theoremislsati 34421* A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  (
 ( W  e.  X  /\  U  e.  A ) 
 ->  E. v  e.  V  U  =  ( N ` 
 { v } )
 )
 
Theoremlsateln0 34422* A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  A )   =>    |-  ( ph  ->  E. v  e.  U  v  =/=  .0.  )
 
Theoremlsatlss 34423 The set of 1-dim subspaces is a set of subspaces. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  ( W  e.  LMod  ->  A  C_  S )
 
Theoremlsatlssel 34424 An atom is a subspace. (Contributed by NM, 25-Aug-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  A )   =>    |-  ( ph  ->  U  e.  S )
 
Theoremlsatssv 34425 An atom is a set of vectors. (Contributed by NM, 27-Feb-2015.)
 |-  V  =  ( Base `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  Q  C_  V )
 
Theoremlsatn0 34426 A 1-dim subspace (atom) of a left module or left vector space is nonzero. (atne0 27129 analog.) (Contributed by NM, 25-Aug-2014.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  A )   =>    |-  ( ph  ->  U  =/=  {  .0.  }
 )
 
Theoremlsatspn0 34427 The span of a vector is an atom iff the vector is nonzero. (Contributed by NM, 4-Feb-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( ( N `  { X } )  e.  A  <->  X  =/=  .0.  ) )
 
Theoremlsator0sp 34428 The span of a vector is either an atom or the zero subspace. (Contributed by NM, 15-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( ( N `  { X } )  e.  A  \/  ( N `  { X } )  =  {  .0.  } ) )
 
Theoremlsatssn0 34429 A subspace (or any class) including an atom is nonzero. (Contributed by NM, 3-Feb-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  Q  C_  U )   =>    |-  ( ph  ->  U  =/=  {  .0.  } )
 
Theoremlsatcmp 34430 If two atoms are comparable, they are equal. (atsseq 27131 analog.) TODO: can lspsncmp 17630 shorten this? (Contributed by NM, 25-Aug-2014.)
 |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  T  e.  A )   &    |-  ( ph  ->  U  e.  A )   =>    |-  ( ph  ->  ( T  C_  U  <->  T  =  U ) )
 
Theoremlsatcmp2 34431 If an atom is included in at-most an atom, they are equal. More general version of lsatcmp 34430. TODO: can lspsncmp 17630 shorten this? (Contributed by NM, 3-Feb-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  T  e.  A )   &    |-  ( ph  ->  ( U  e.  A  \/  U  =  {  .0.  } ) )   =>    |-  ( ph  ->  ( T  C_  U  <->  T  =  U ) )
 
Theoremlsatel 34432 A nonzero vector in an atom determines the atom. (Contributed by NM, 25-Aug-2014.)
 |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  A )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  X  =/=  .0.  )   =>    |-  ( ph  ->  U  =  ( N `  { X } ) )
 
TheoremlsatelbN 34433 A nonzero vector in an atom determines the atom. (Contributed by NM, 3-Feb-2015.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  U  e.  A )   =>    |-  ( ph  ->  ( X  e.  U  <->  U  =  ( N `  { X }
 ) ) )
 
Theoremlsat2el 34434 Two atoms sharing a nonzero vector are equal. (Contributed by NM, 8-Mar-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  P  e.  A )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  X  =/=  .0.  )   &    |-  ( ph  ->  X  e.  P )   &    |-  ( ph  ->  X  e.  Q )   =>    |-  ( ph  ->  P  =  Q )
 
Theoremlsmsat 34435* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 35231 analog.) TODO: any way to shorten this? (Contributed by NM, 15-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  T  =/=  {  .0.  }
 )   &    |-  ( ph  ->  Q  C_  ( T  .(+)  U ) )   =>    |-  ( ph  ->  E. p  e.  A  ( p  C_  T  /\  Q  C_  ( p  .(+)  U ) ) )
 
TheoremlsatfixedN 34436* Show equality with the span of the sum of two vectors, one of which ( X) is fixed in advance. Compare lspfixed 17642. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Q  =/=  ( N `  { X } ) )   &    |-  ( ph  ->  Q  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  Q  C_  ( N `  { X ,  Y } ) )   =>    |-  ( ph  ->  E. z  e.  ( ( N `  { Y } )  \  {  .0.  } ) Q  =  ( N `  { ( X  .+  z ) }
 ) )
 
Theoremlsmsatcv 34437 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 26435 analog.) Explicit atom version of lsmcv 17655. (Contributed by NM, 29-Oct-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ( ph  /\  T  C.  U  /\  U  C_  ( T  .(+)  Q ) )  ->  U  =  ( T  .(+)  Q ) )
 
Theoremlssatomic 34438* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 27142 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  U  =/=  {  .0.  } )   =>    |-  ( ph  ->  E. q  e.  A  q  C_  U )
 
Theoremlssats 34439* The lattice of subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. Hypothesis (shatomistici 27145 analog.) (Contributed by NM, 9-Apr-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  (
 ( W  e.  LMod  /\  U  e.  S ) 
 ->  U  =  ( N `
  U. { x  e.  A  |  x  C_  U } ) )
 
Theoremlpssat 34440* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 27147 analog.) (Contributed by NM, 11-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  T  C.  U )   =>    |-  ( ph  ->  E. q  e.  A  ( q  C_  U  /\  -.  q  C_  T ) )
 
Theoremlrelat 34441* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 27148 analog.) (Contributed by NM, 11-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  T  C.  U )   =>    |-  ( ph  ->  E. q  e.  A  ( T  C.  ( T  .(+)  q ) 
 /\  ( T  .(+)  q )  C_  U )
 )
 
Theoremlssatle 34442* The ordering of two subspaces is determined by the atoms under them. (chrelat3 27155 analog.) (Contributed by NM, 29-Oct-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( T  C_  U  <->  A. p  e.  A  ( p  C_  T  ->  p 
 C_  U ) ) )
 
Theoremlssat 34443* Two subspaces in a proper subset relationship imply the existence of a 1-dim subspace less than or equal to one but not the other. (chpssati 27147 analog.) (Contributed by NM, 9-Apr-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   =>    |-  (
 ( ( W  e.  LMod  /\  U  e.  S  /\  V  e.  S )  /\  U  C.  V )  ->  E. p  e.  A  ( p  C_  V  /\  -.  p  C_  U )
 )
 
Theoremislshpat 34444* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 34407. (Contributed by NM, 11-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  V  /\  E. q  e.  A  ( U  .(+)  q )  =  V ) ) )
 
Syntaxclcv 34445 Extend class notation with the covering relation for a left module or left vector space.
 class  <oLL
 
Definitiondf-lcv 34446* Define the covering relation for subspaces of a left vector space. Similar to Definition 3.2.18 of [PtakPulmannova] p. 68. Ptak/Pulmannova's notation 
A (  <oLL  `  W ) B is read " B covers  A " or " A is covered by  B " , and it means that  B is larger than  A and there is nothing in between. See lcvbr 34448 for binary relation. (df-cv 27063 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  <oLL  =  ( w  e.  _V  |->  { <. t ,  u >.  |  ( ( t  e.  ( LSubSp `
  w )  /\  u  e.  ( LSubSp `  w ) )  /\  ( t  C.  u  /\  -. 
 E. s  e.  ( LSubSp `
  w ) ( t  C.  s  /\  s  C.  u ) ) ) } )
 
Theoremlcvfbr 34447* The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   =>    |-  ( ph  ->  C  =  { <. t ,  u >.  |  ( ( t  e.  S  /\  u  e.  S )  /\  (
 t  C.  u  /\  -.  E. s  e.  S  ( t  C.  s  /\  s  C.  u ) ) ) } )
 
Theoremlcvbr 34448* The covers relation for a left vector space (or a left module). (cvbr 27066 analog.) (Contributed by NM, 9-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( T C U  <->  ( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) ) ) )
 
Theoremlcvbr2 34449* The covers relation for a left vector space (or a left module). (cvbr2 27067 analog.) (Contributed by NM, 9-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( T C U  <->  ( T  C.  U  /\  A. s  e.  S  ( ( T  C.  s  /\  s  C_  U )  ->  s  =  U ) ) ) )
 
Theoremlcvbr3 34450* The covers relation for a left vector space (or a left module). (Contributed by NM, 9-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( T C U  <->  ( T  C.  U  /\  A. s  e.  S  ( ( T 
 C_  s  /\  s  C_  U )  ->  (
 s  =  T  \/  s  =  U )
 ) ) ) )
 
Theoremlcvpss 34451 The covers relation implies proper subset. (cvpss 27069 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  T C U )   =>    |-  ( ph  ->  T  C.  U )
 
Theoremlcvnbtwn 34452 The covers relation implies no in-betweenness. (cvnbtwn 27070 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R C T )   =>    |-  ( ph  ->  -.  ( R  C.  U  /\  U  C.  T ) )
 
Theoremlcvntr 34453 The covers relation is not transitive. (cvntr 27076 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R C T )   &    |-  ( ph  ->  T C U )   =>    |-  ( ph  ->  -.  R C U )
 
Theoremlcvnbtwn2 34454 The covers relation implies no in-betweenness. (cvnbtwn2 27071 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R C T )   &    |-  ( ph  ->  R  C.  U )   &    |-  ( ph  ->  U 
 C_  T )   =>    |-  ( ph  ->  U  =  T )
 
Theoremlcvnbtwn3 34455 The covers relation implies no in-betweenness. (cvnbtwn3 27072 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R C T )   &    |-  ( ph  ->  R 
 C_  U )   &    |-  ( ph  ->  U  C.  T )   =>    |-  ( ph  ->  U  =  R )
 
Theoremlsmcv2 34456 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 27077 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  ( N ` 
 { X } )  C_  U )   =>    |-  ( ph  ->  U C ( U  .(+)  ( N `  { X } ) ) )
 
Theoremlcvat 34457* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 27150 analog.) (Contributed by NM, 11-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  T C U )   =>    |-  ( ph  ->  E. q  e.  A  ( T  .(+)  q )  =  U )
 
Theoremlsatcv0 34458 An atom covers the zero subspace. (atcv0 27126 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  {  .0.  } C Q )
 
Theoremlsatcveq0 34459 A subspace covered by an atom must be the zero subspace. (atcveq0 27132 analog.) (Contributed by NM, 7-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  ( 
 <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( U C Q  <->  U  =  {  .0.  } ) )
 
Theoremlsat0cv 34460 A subspace is an atom iff it covers the zero subspace. This could serve as an alternate definition of an atom. TODO: this is a quick-and-dirty proof that could probably be more efficient. (Contributed by NM, 14-Mar-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  ( 
 <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( U  e.  A  <->  {  .0.  } C U ) )
 
Theoremlcvexchlem1 34461 Lemma for lcvexch 34466. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( T  C.  ( T  .(+)  U )  <->  ( T  i^i  U )  C.  U )
 )
 
Theoremlcvexchlem2 34462 Lemma for lcvexch 34466. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  ( T  i^i  U )  C_  R )   &    |-  ( ph  ->  R 
 C_  U )   =>    |-  ( ph  ->  ( ( R  .(+)  T )  i^i  U )  =  R )
 
Theoremlcvexchlem3 34463 Lemma for lcvexch 34466. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  R  e.  S )   &    |-  ( ph  ->  T  C_  R )   &    |-  ( ph  ->  R 
 C_  ( T  .(+)  U ) )   =>    |-  ( ph  ->  (
 ( R  i^i  U )  .(+)  T )  =  R )
 
Theoremlcvexchlem4 34464 Lemma for lcvexch 34466. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  T C ( T  .(+)  U )
 )   =>    |-  ( ph  ->  ( T  i^i  U ) C U )
 
Theoremlcvexchlem5 34465 Lemma for lcvexch 34466. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  ( T  i^i  U ) C U )   =>    |-  ( ph  ->  T C ( T  .(+)  U ) )
 
Theoremlcvexch 34466 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 27153 analog.) TODO: combine some lemmas. (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( ( T  i^i  U ) C U  <->  T C ( T 
 .(+)  U ) ) )
 
Theoremlcvp 34467 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 27159 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  (
 ( U  i^i  Q )  =  {  .0.  }  <->  U C ( U  .(+)  Q ) ) )
 
Theoremlcv1 34468 Covering property of a subspace plus an atom. (chcv1 27139 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( -.  Q  C_  U  <->  U C ( U  .(+)  Q ) ) )
 
Theoremlcv2 34469 Covering property of a subspace plus an atom. (chcv2 27140 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( U  C.  ( U  .(+)  Q )  <->  U C ( U 
 .(+)  Q ) ) )
 
Theoremlsatexch 34470 The atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem was originally proved by Hermann Grassmann in 1862. (atexch 27165 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  Q 
 C_  ( U  .(+)  R ) )   &    |-  ( ph  ->  ( U  i^i  Q )  =  {  .0.  }
 )   =>    |-  ( ph  ->  R  C_  ( U  .(+)  Q ) )
 
Theoremlsatnle 34471 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 27160 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  ( -.  Q  C_  U  <->  ( U  i^i  Q )  =  {  .0.  } ) )
 
Theoremlsatnem0 34472 The meet of distinct atoms is the zero subspace. (atnemeq0 27161 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   =>    |-  ( ph  ->  ( Q  =/=  R  <->  ( Q  i^i  R )  =  {  .0.  } ) )
 
Theoremlsatexch1 34473 The atom exch1ange property. (hlatexch1 34821 analog.) (Contributed by NM, 14-Jan-2015.)
 |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  S  e.  A )   &    |-  ( ph  ->  Q  C_  ( S  .(+)  R ) )   &    |-  ( ph  ->  Q  =/=  S )   =>    |-  ( ph  ->  R 
 C_  ( S  .(+)  Q ) )
 
Theoremlsatcv0eq 34474 If the sum of two atoms cover the zero subspace, they are equal. (atcv0eq 27163 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   =>    |-  ( ph  ->  ( {  .0.  } C ( Q  .(+)  R )  <->  Q  =  R ) )
 
Theoremlsatcv1 34475 Two atoms covering the zero subspace are equal. (atcv1 27164 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  ( 
 <oLL  `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  U C ( Q  .(+)  R ) )   =>    |-  ( ph  ->  ( U  =  {  .0.  }  <->  Q  =  R )
 )
 
Theoremlsatcvatlem 34476 Lemma for lsatcvat 34477. (Contributed by NM, 10-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  U  =/=  {  .0.  }
 )   &    |-  ( ph  ->  U  C.  ( Q  .(+)  R ) )   &    |-  ( ph  ->  -.  Q  C_  U )   =>    |-  ( ph  ->  U  e.  A )
 
Theoremlsatcvat 34477 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 27170 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  U  =/=  {  .0.  }
 )   &    |-  ( ph  ->  U  C.  ( Q  .(+)  R ) )   =>    |-  ( ph  ->  U  e.  A )
 
Theoremlsatcvat2 34478 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 27171 analog.) (Contributed by NM, 10-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  Q  =/=  R )   &    |-  ( ph  ->  U C ( Q  .(+)  R ) )   =>    |-  ( ph  ->  U  e.  A )
 
Theoremlsatcvat3 34479 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 27180 analog.) (Contributed by NM, 11-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  Q  =/=  R )   &    |-  ( ph  ->  -.  R  C_  U )   &    |-  ( ph  ->  Q  C_  ( U  .(+)  R ) )   =>    |-  ( ph  ->  ( U  i^i  ( Q  .(+)  R ) )  e.  A )
 
Theoremislshpcv 34480 Hyperplane properties expressed with covers relation. (Contributed by NM, 11-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   =>    |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U C V ) ) )
 
Theoreml1cvpat 34481 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 34901 analog.) (Contributed by NM, 11-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  U C V )   &    |-  ( ph  ->  -.  Q  C_  U )   =>    |-  ( ph  ->  ( U  .(+)  Q )  =  V )
 
Theoreml1cvat 34482 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 34902 analog.) (Contributed by NM, 11-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  C  =  (  <oLL  `
  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  Q  =/=  R )   &    |-  ( ph  ->  U C V )   &    |-  ( ph  ->  -.  Q  C_  U )   =>    |-  ( ph  ->  (
 ( Q  .(+)  R )  i^i  U )  e.  A )
 
Theoremlshpat 34483 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 35469 analog.) TODO: This changes  U C V in l1cvpat 34481 and l1cvat 34482 to  U  e.  H, which in turn change  U  e.  H in islshpcv 34480 to  U C V, with a couple of conversions of span to atom. Seems convoluted. Would a direct proof be better? (Contributed by NM, 11-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  H )   &    |-  ( ph  ->  Q  e.  A )   &    |-  ( ph  ->  R  e.  A )   &    |-  ( ph  ->  Q  =/=  R )   &    |-  ( ph  ->  -.  Q  C_  U )   =>    |-  ( ph  ->  ( ( Q 
 .(+)  R )  i^i  U )  e.  A )
 
21.30.4  Functionals and kernels of a left vector space (or module)
 
Syntaxclfn 34484 Extend class notation with all linear functionals of a left module or left vector space.
 class LFnl
 
Definitiondf-lfl 34485* Define the set of all linear functionals (maps from vectors to the ring) of a left module or left vector space. (Contributed by NM, 15-Apr-2014.)
 |- LFnl  =  ( w  e.  _V  |->  { f  e.  ( (
 Base `  (Scalar `  w ) )  ^m  ( Base `  w ) )  | 
 A. r  e.  ( Base `  (Scalar `  w ) ) A. x  e.  ( Base `  w ) A. y  e.  ( Base `  w ) ( f `  ( ( r ( .s `  w ) x ) ( +g  `  w ) y ) )  =  ( ( r ( .r `  (Scalar `  w ) ) ( f `  x ) ) ( +g  `  (Scalar `  w ) ) ( f `  y ) ) } )
 
Theoremlflset 34486* The set of linear functionals in a left module or left vector space. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  D )   &    |-  .+^  =  ( +g  `  D )   &    |-  .X.  =  ( .r `  D )   &    |-  F  =  (LFnl `  W )   =>    |-  ( W  e.  X  ->  F  =  { f  e.  ( K  ^m  V )  |  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( f `  ( ( r  .x.  x )  .+  y ) )  =  ( ( r  .X.  ( f `  x ) )  .+^  ( f `  y
 ) ) } )
 
Theoremislfl 34487* The predicate "is a linear functional". (Contributed by NM, 15-Apr-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  D )   &    |-  .+^  =  ( +g  `  D )   &    |-  .X.  =  ( .r `  D )   &    |-  F  =  (LFnl `  W )   =>    |-  ( W  e.  X  ->  ( G  e.  F  <->  ( G : V
 --> K  /\  A. r  e.  K  A. x  e.  V  A. y  e.  V  ( G `  ( ( r  .x.  x )  .+  y ) )  =  ( ( r  .X.  ( G `  x ) )  .+^  ( G `  y ) ) ) ) )
 
Theoremlfli 34488 Property of a linear functional. (lnfnli 26824 analog.) (Contributed by NM, 16-Apr-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  D )   &    |-  .+^  =  ( +g  `  D )   &    |-  .X.  =  ( .r `  D )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  Z  /\  G  e.  F  /\  ( R  e.  K  /\  X  e.  V  /\  Y  e.  V )
 )  ->  ( G `  ( ( R  .x.  X )  .+  Y ) )  =  ( ( R  .X.  ( G `  X ) )  .+^  ( G `  Y ) ) )
 
Theoremislfld 34489* Properties that determine a linear functional. TODO: use this in place of islfl 34487 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)
 |-  ( ph  ->  V  =  (
 Base `  W ) )   &    |-  ( ph  ->  .+  =  (
 +g  `  W )
 )   &    |-  ( ph  ->  D  =  (Scalar `  W )
 )   &    |-  ( ph  ->  .x.  =  ( .s `  W ) )   &    |-  ( ph  ->  K  =  ( Base `  D ) )   &    |-  ( ph  ->  .+^  =  ( +g  `  D ) )   &    |-  ( ph  ->  .X. 
 =  ( .r `  D ) )   &    |-  ( ph  ->  F  =  (LFnl `  W ) )   &    |-  ( ph  ->  G : V --> K )   &    |-  ( ( ph  /\  ( r  e.  K  /\  x  e.  V  /\  y  e.  V ) )  ->  ( G `
  ( ( r 
 .x.  x )  .+  y ) )  =  ( ( r  .X.  ( G `  x ) )  .+^  ( G `  y ) ) )   &    |-  ( ph  ->  W  e.  X )   =>    |-  ( ph  ->  G  e.  F )
 
Theoremlflf 34490 A linear functional is a function from vectors to scalars. (lnfnfi 26825 analog.) (Contributed by NM, 15-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  X  /\  G  e.  F ) 
 ->  G : V --> K )
 
Theoremlflcl 34491 A linear functional value is a scalar. (Contributed by NM, 15-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  Y  /\  G  e.  F  /\  X  e.  V )  ->  ( G `  X )  e.  K )
 
Theoremlfl0 34492 A linear functional is zero at the zero vector. (lnfn0i 26826 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  Z  =  ( 0g
 `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  LMod  /\  G  e.  F ) 
 ->  ( G `  Z )  =  .0.  )
 
Theoremlfladd 34493 Property of a linear functional. (lnfnaddi 26827 analog.) (Contributed by NM, 18-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .+^  =  (
 +g  `  D )   &    |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  LMod  /\  G  e.  F  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  ( G `  ( X  .+  Y ) )  =  (
 ( G `  X )  .+^  ( G `  Y ) ) )
 
Theoremlflsub 34494 Property of a linear functional. (lnfnaddi 26827 analog.) (Contributed by NM, 18-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  M  =  ( -g `  D )   &    |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  LMod  /\  G  e.  F  /\  ( X  e.  V  /\  Y  e.  V ) )  ->  ( G `  ( X  .-  Y ) )  =  (
 ( G `  X ) M ( G `  Y ) ) )
 
Theoremlflmul 34495 Property of a linear functional. (lnfnmuli 26828 analog.) (Contributed by NM, 16-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .X.  =  ( .r `  D )   &    |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  LMod  /\  G  e.  F  /\  ( R  e.  K  /\  X  e.  V ) )  ->  ( G `  ( R  .x.  X ) )  =  ( R  .X.  ( G `  X ) ) )
 
Theoremlfl0f 34496 The zero function is a functional. (Contributed by NM, 16-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  ( W  e.  LMod  ->  ( V  X.  {  .0.  } )  e.  F )
 
Theoremlfl1 34497* A non-zero functional has a value of 1 at some argument. (Contributed by NM, 16-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |- 
 .1.  =  ( 1r `  D )   &    |-  V  =  (
 Base `  W )   &    |-  F  =  (LFnl `  W )   =>    |-  (
 ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) ) 
 ->  E. x  e.  V  ( G `  x )  =  .1.  )
 
Theoremlfladdcl 34498 Closure of addition of two functionals. (Contributed by NM, 19-Oct-2014.)
 |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( G  oF  .+  H )  e.  F )
 
Theoremlfladdcom 34499 Commutativity of functional addition. (Contributed by NM, 19-Oct-2014.)
 |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  ( G  oF  .+  H )  =  ( H  oF  .+  G ) )
 
Theoremlfladdass 34500 Associativity of functional addition. (Contributed by NM, 19-Oct-2014.)
 |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  I  e.  F )   =>    |-  ( ph  ->  (
 ( G  oF  .+  H )  oF  .+  I )  =  ( G  oF  .+  ( H  oF  .+  I ) ) )
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