HomeHome Metamath Proof Explorer
Theorem List (p. 282 of 311)
< 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
(1-21328)
  Hilbert Space Explorer  Hilbert Space Explorer
(21329-22851)
  Users' Mathboxes  Users' Mathboxes
(22852-31058)
 

Theorem List for Metamath Proof Explorer - 28101-28200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlsat2el 28101 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 28102* Convert comparison of atom with sum of subspaces to a comparison to sum with atom. (elpaddatiN 28898 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 28103* Show equality with the span of the sum of two vectors, one of which ( X) is fixed in advance. Compare lspfixed 15716. (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 28104 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 22079 analog.) Explicit atom version of lsmcv 15729. (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 28105* The lattice of subspaces is atomic, i.e. any nonzero element is greater than or equal to some atom. (shatomici 22768 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 28106* 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 22771 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 28107* Two subspaces in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (chpssati 22773 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 28108* Subspaces are relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 22774 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 28109* The ordering of two subspaces is determined by the atoms under them. (chrelat3 22781 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 28110* 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 22773 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 28111* Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 28074. (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 28112 Extend class notation with the covering relation for a left module or left vector space.
 class  <oLL
 
Definitiondf-lcv 28113* 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 28115 for binary relation. (df-cv 22689 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 28114* 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 28115* The covers relation for a left vector space (or a left module). (cvbr 22692 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 28116* The covers relation for a left vector space (or a left module). (cvbr2 22693 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 28117* 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 28118 The covers relation implies proper subset. (cvpss 22695 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 28119 The covers relation implies no in-betweenness. (cvnbtwn 22696 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 28120 The covers relation is not transitive. (cvntr 22702 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 28121 The covers relation implies no in-betweenness. (cvnbtwn2 22697 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 28122 The covers relation implies no in-betweenness. (cvnbtwn3 22698 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 28123 Subspace sum has the covering property (using spans of singletons to represent atoms). Proposition 1(ii) of [Kalmbach] p. 153. (spansncv2 22703 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 28124* If a subspace covers another, it equals the other joined with some atom. This is a consequence of relative atomicity. (cvati 22776 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 28125 An atom covers the zero subspace. (atcv0 22752 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 28126 A subspace covered by an atom must be the zero subspace. (atcveq0 22758 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 28127 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 28128 Lemma for lcvexch 28133. (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 28129 Lemma for lcvexch 28133. (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 28130 Lemma for lcvexch 28133. (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 28131 Lemma for lcvexch 28133. (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 28132 Lemma for lcvexch 28133. (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 28133 Subspaces satisfy the exchange axiom. Lemma 7.5 of [MaedaMaeda] p. 31. (cvexchi 22779 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 28134 Covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22785 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 28135 Covering property of a subspace plus an atom. (chcv1 22765 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 28136 Covering property of a subspace plus an atom. (chcv2 22766 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 28137 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 22791 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 28138 The meet of a subspace and an incomparable atom is the zero subspace. (atnssm0 22786 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 28139 The meet of distinct atoms is the zero subspace. (atnemeq0 22787 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 28140 The atom exch1ange property. (hlatexch1 28488 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 28141 If the sum of two atoms cover the zero subspace, they are equal.. (atcv0eq 22789 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 28142 Two atoms covering the zero subspace are equal. (atcv1 22790 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 28143 Lemma for lsatcvat 28144. (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 28144 A nonzero subspace less than the sum of two atoms is an atom. (atcvati 22796 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 28145 A subspace covered by the sum of two distinct atoms is an atom. (atcvat2i 22797 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 28146 A condition implying that a certain subspace is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 22806 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 28147 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 28148 A subspace covered by the set of all vectors, when summed with an atom not under it, equals the set of all vectors. (1cvrjat 28568 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 28149 Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (1cvrat 28569 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 28150 Create an atom under a hyperplane. Part of proof of Lemma B in [Crawley] p. 112. (lhpat 29136 analog.) TODO: This changes  U C V in l1cvpat 28148 and l1cvat 28149 to  U  e.  H, which in turn change  U  e.  H in islshpcv 28147 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 )
 
16.23.8  Functionals and kernels of a left vector space (or module)
 
Syntaxclfn 28151 Extend class notation with all linear functionals of a left module or left vector space.
 class LFnl
 
Definitiondf-lfl 28152* Define the set of all linear functionals (maps from vectors to 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 28153* 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 28154* 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 28155 Property of a linear functional. (lnfnli 22450 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 28156* Properties that determine a linear functional. TODO: use this in place of islfl 28154 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 28157 A linear functional is a function from vectors to scalars. (lnfnfi 22451 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 28158 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 28159 A linear functional is zero at the zero vector. (lnfn0i 22452 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 28160 Property of a linear functional. (lnfnaddi 22453 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 28161 Property of a linear functional. (lnfnaddi 22453 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 28162 Property of a linear functional. (lnfnmuli 22454 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 28163 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 28164* 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 28165 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  o F  .+  H )  e.  F )
 
Theoremlfladdcom 28166 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  o F  .+  H )  =  ( H  o F  .+  G ) )
 
Theoremlfladdass 28167 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  o F  .+  H )  o F  .+  I )  =  ( G  o F  .+  ( H  o F  .+  I ) ) )
 
Theoremlfladd0l 28168 Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  .+  =  ( +g  `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( ( V  X.  {  .0.  } )  o F  .+  G )  =  G )
 
Theoremlflnegcl 28169* Closure of the negative of a functional. (This is specialized for the purpose of proving ldualgrp 28240, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  I  =  ( inv g `  R )   &    |-  N  =  ( x  e.  V  |->  ( I `  ( G `
  x ) ) )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  N  e.  F )
 
Theoremlflnegl 28170* A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 28240, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  I  =  ( inv g `  R )   &    |-  N  =  ( x  e.  V  |->  ( I `  ( G `
  x ) ) )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  .+  =  ( +g  `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ph  ->  ( N  o F  .+  G )  =  ( V  X.  {  .0.  } )
 )
 
Theoremlflvscl 28171 Closure of a scalar product with a functional. Note that this is the scalar product for a right vector space with the scalar after the vector; reversing these fails closure. (Contributed by NM, 9-Oct-2014.) (Revised by Mario Carneiro, 22-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  R  e.  K )   =>    |-  ( ph  ->  ( G  o F  .x.  ( V  X.  { R }
 ) )  e.  F )
 
Theoremlflvsdi1 28172 Distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   =>    |-  ( ph  ->  (
 ( G  o F  .+  H )  o F  .x.  ( V  X.  { X } ) )  =  ( ( G  o F  .x.  ( V  X.  { X } ) )  o F  .+  ( H  o F  .x.  ( V  X.  { X }
 ) ) ) )
 
Theoremlflvsdi2 28173 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  o F  .x.  (
 ( V  X.  { X } )  o F  .+  ( V  X.  { Y } ) ) )  =  ( ( G  o F  .x.  ( V  X.  { X }
 ) )  o F  .+  ( G  o F  .x.  ( V  X.  { Y } ) ) ) )
 
Theoremlflvsdi2a 28174 Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  o F  .x.  ( V  X.  { ( X 
 .+  Y ) }
 ) )  =  ( ( G  o F  .x.  ( V  X.  { X } ) )  o F  .+  ( G  o F  .x.  ( V  X.  { Y }
 ) ) ) )
 
Theoremlflvsass 28175 Associative law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  F  =  (LFnl `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  Y  e.  K )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  o F  .x.  ( V  X.  { ( X 
 .x.  Y ) } )
 )  =  ( ( G  o F  .x.  ( V  X.  { X } ) )  o F  .x.  ( V  X.  { Y } )
 ) )
 
Theoremlfl0sc 28176 The (right vector space) scalar product of a functional with zero is the zero functional. Note that the first occurrence of  ( V  X.  {  .0.  }
) represents the zero scalar, and the second is the zero functional. (Contributed by NM, 7-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  .0.  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  o F  .x.  ( V  X.  {  .0.  } ) )  =  ( V  X.  {  .0.  } ) )
 
Theoremlflsc0N 28177 The scalar product with the zero functional is the zero functional. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  .0.  =  ( 0g `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  K )   =>    |-  ( ph  ->  ( ( V  X.  {  .0.  } )  o F  .x.  ( V  X.  { X } ) )  =  ( V  X.  {  .0.  } ) )
 
Theoremlfl1sc 28178 The (right vector space) scalar product of a functional with one is the functional. (Contributed by NM, 21-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  .1.  =  ( 1r `  D )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  o F  .x.  ( V  X.  {  .1.  } ) )  =  G )
 
Syntaxclk 28179 Extend class notation with the kernel of a functional (set of vectors whose functional value is zero) on a left module or left vector space.
 class LKer
 
Definitiondf-lkr 28180* Define the kernel of a functional (set of vectors whose functional value is zero) on a left module or left vector space. (Contributed by NM, 15-Apr-2014.)
 |- LKer  =  ( w  e.  _V  |->  ( f  e.  (LFnl `  w )  |->  ( `' f " { ( 0g `  (Scalar `  w ) ) } )
 ) )
 
Theoremlkrfval 28181* The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( W  e.  X  ->  K  =  ( f  e.  F  |->  ( `' f " {  .0.  } ) ) )
 
Theoremlkrval 28182 Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( ( W  e.  X  /\  G  e.  F )  ->  ( K `  G )  =  ( `' G " {  .0.  } ) )
 
Theoremellkr 28183 Membership in the kernel of a functional. (elnlfn 22338 analog.) (Contributed by NM, 16-Apr-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( ( W  e.  Y  /\  G  e.  F )  ->  ( X  e.  ( K `  G )  <-> 
 ( X  e.  V  /\  ( G `  X )  =  .0.  )
 ) )
 
Theoremlkrval2 28184* Value of the kernel of a functional. (Contributed by NM, 15-Apr-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( ( W  e.  X  /\  G  e.  F )  ->  ( K `  G )  =  { x  e.  V  |  ( G `  x )  =  .0.  } )
 
Theoremellkr2 28185 Membership in the kernel of a functional. (Contributed by NM, 12-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  Y )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( X  e.  ( K `  G )  <->  ( G `  X )  =  .0.  ) )
 
Theoremlkrcl 28186 A member of the kernel of a functional is a vector. (Contributed by NM, 16-Apr-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  (
 ( W  e.  Y  /\  G  e.  F  /\  X  e.  ( K `  G ) )  ->  X  e.  V )
 
Theoremlkrf0 28187 The value of a functional at a member of its kernel is zero. (Contributed by NM, 16-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( ( W  e.  Y  /\  G  e.  F  /\  X  e.  ( K `
  G ) ) 
 ->  ( G `  X )  =  .0.  )
 
Theoremlkr0f 28188 The kernel of the zero functional is the set of all vectors. (Contributed by NM, 17-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( ( W  e.  LMod  /\  G  e.  F ) 
 ->  ( ( K `  G )  =  V  <->  G  =  ( V  X.  {  .0.  } ) ) )
 
Theoremlkrlss 28189 The kernel of a linear functional is a subspace. (nlelshi 22470 analog.) (Contributed by NM, 16-Apr-2014.)
 |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  (
 ( W  e.  LMod  /\  G  e.  F ) 
 ->  ( K `  G )  e.  S )
 
Theoremlkrssv 28190 The kernel of a linear functional is a set of vectors. (Contributed by NM, 1-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( K `  G )  C_  V )
 
Theoremlkrsc 28191 The kernel of a non-zero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 9-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  R  e.  K )   &    |- 
 .0.  =  ( 0g `  D )   &    |-  ( ph  ->  R  =/=  .0.  )   =>    |-  ( ph  ->  ( L `  ( G  o F  .x.  ( V  X.  { R }
 ) ) )  =  ( L `  G ) )
 
Theoremlkrscss 28192 The kernel of a scalar product of a functional includes the kernel of the functional. (The inclusion is proper for the zero product and equality otherwise.) (Contributed by NM, 9-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  R  e.  K )   =>    |-  ( ph  ->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { R }
 ) ) ) )
 
Theoremeqlkr 28193* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 18-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   =>    |-  ( ( W  e.  LVec  /\  ( G  e.  F  /\  H  e.  F ) 
 /\  ( L `  G )  =  ( L `  H ) ) 
 ->  E. r  e.  K  A. x  e.  V  ( H `  x )  =  ( ( G `
  x )  .x.  r ) )
 
Theoremeqlkr2 28194* Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 10-Oct-2014.)
 |-  D  =  (Scalar `  W )   &    |-  K  =  ( Base `  D )   &    |-  .x.  =  ( .r `  D )   &    |-  V  =  ( Base `  W )   &    |-  F  =  (LFnl `  W )   &    |-  L  =  (LKer `  W )   =>    |-  ( ( W  e.  LVec  /\  ( G  e.  F  /\  H  e.  F ) 
 /\  ( L `  G )  =  ( L `  H ) ) 
 ->  E. r  e.  K  H  =  ( G  o F  .x.  ( V  X.  { r }
 ) ) )
 
Theoremeqlkr3 28195 Two functionals with the same kernel are equal if they are equal at any nonzero value. (Contributed by NM, 2-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (Scalar `  W )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  H  e.  F )   &    |-  ( ph  ->  ( K `  G )  =  ( K `  H ) )   &    |-  ( ph  ->  ( G `  X )  =  ( H `  X ) )   &    |-  ( ph  ->  ( G `  X )  =/=  .0.  )   =>    |-  ( ph  ->  G  =  H )
 
Theoremlkrlsp 28196 The subspace sum of a kernel and the span of a vector not in the kernel (by ellkr 28183) is the whole vector space. (Contributed by NM, 19-Apr-2014.)
 |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  (
 ( W  e.  LVec  /\  ( X  e.  V  /\  G  e.  F ) 
 /\  ( G `  X )  =/=  .0.  )  ->  ( ( K `  G )  .(+)  ( N `
  { X }
 ) )  =  V )
 
Theoremlkrlsp2 28197 The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 12-May-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  (
 ( W  e.  LVec  /\  ( X  e.  V  /\  G  e.  F ) 
 /\  -.  X  e.  ( K `  G ) )  ->  ( ( K `  G )  .(+)  ( N `  { X } ) )  =  V )
 
Theoremlkrlsp3 28198 The subspace sum of a kernel and the span of a vector not in the kernel is the whole vector space. (Contributed by NM, 29-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  (
 ( W  e.  LVec  /\  ( X  e.  V  /\  G  e.  F ) 
 /\  -.  X  e.  ( K `  G ) )  ->  ( N `  ( ( K `  G )  u.  { X } ) )  =  V )
 
Theoremlkrshp 28199 The kernel of a nonzero functional is a hyperplane. (Contributed by NM, 29-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  H  =  (LSHyp `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   =>    |-  ( ( W  e.  LVec  /\  G  e.  F  /\  G  =/=  ( V  X.  {  .0.  } ) ) 
 ->  ( K `  G )  e.  H )
 
Theoremlkrshp3 28200 The kernels of nonzero functionals are hyperplanes. (Contributed by NM, 17-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  D  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  D )   &    |-  H  =  (LSHyp `  W )   &    |-  F  =  (LFnl `  W )   &    |-  K  =  (LKer `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( ( K `  G )  e.  H  <->  G  =/=  ( V  X.  {  .0.  }
 ) ) )
    < 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 31001-31058
  Copyright terms: Public domain < Previous  Next >