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Theorem List for Metamath Proof Explorer - 30801-30900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-lpolN 30801* Define the set of all polarities of a left module or left vector space. A polarity is a kind of complementation operation on a subspace. The double polarity of a subspace is a closure operation. Based on Definition 3.2 of [Holland95] p. 214 for projective geometry polarities. For convenience, we open up the domain to include all vector subsets and not just subspaces, but any more restricted polarity can be converted to this one by taking the span of its argument. (Contributed by NM, 24-Nov-2014.)
 |- LPol  =  ( w  e.  _V  |->  { o  e.  ( (
 LSubSp `  w )  ^m  ~P ( Base `  w )
 )  |  ( ( o `  ( Base `  w ) )  =  { ( 0g `  w ) }  /\  A. x A. y ( ( x  C_  ( Base `  w )  /\  y  C_  ( Base `  w )  /\  x  C_  y
 )  ->  ( o `  y )  C_  (
 o `  x )
 )  /\  A. x  e.  (LSAtoms `  w )
 ( ( o `  x )  e.  (LSHyp `  w )  /\  (
 o `  ( o `  x ) )  =  x ) ) }
 )
 
TheoremlpolsetN 30802* The set of polarities of a left module or left vector space. (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  P  =  (LPol `  W )   =>    |-  ( W  e.  X  ->  P  =  { o  e.  ( S  ^m  ~P V )  |  (
 ( o `  V )  =  {  .0.  } 
 /\  A. x A. y
 ( ( x  C_  V  /\  y  C_  V  /\  x  C_  y ) 
 ->  ( o `  y
 )  C_  ( o `  x ) )  /\  A. x  e.  A  ( ( o `  x )  e.  H  /\  ( o `  (
 o `  x )
 )  =  x ) ) } )
 
TheoremislpolN 30803* The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  P  =  (LPol `  W )   =>    |-  ( W  e.  X  ->  (  ._|_  e.  P  <->  ( 
 ._|_  : ~P V --> S  /\  ( (  ._|_  `  V )  =  {  .0.  } 
 /\  A. x A. y
 ( ( x  C_  V  /\  y  C_  V  /\  x  C_  y ) 
 ->  (  ._|_  `  y
 )  C_  (  ._|_  `  x ) )  /\  A. x  e.  A  ( (  ._|_  `  x )  e.  H  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) ) ) ) )
 
TheoremislpoldN 30804* Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  A  =  (LSAtoms `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  P  =  (LPol `  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  ._|_  : ~P V --> S )   &    |-  ( ph  ->  ( 
 ._|_  `  V )  =  {  .0.  } )   &    |-  (
 ( ph  /\  ( x 
 C_  V  /\  y  C_  V  /\  x  C_  y ) )  ->  (  ._|_  `  y )  C_  (  ._|_  `  x ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  (  ._|_  `  x )  e.  H )   &    |-  (
 ( ph  /\  x  e.  A )  ->  (  ._|_  `  (  ._|_  `  x ) )  =  x )   =>    |-  ( ph  ->  ._|_  e.  P )
 
TheoremlpolfN 30805 Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  P  =  (LPol `  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  ._|_  e.  P )   =>    |-  ( ph  ->  ._|_  : ~P V
 --> S )
 
TheoremlpolvN 30806 The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  P  =  (LPol `  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  ._|_  e.  P )   =>    |-  ( ph  ->  (  ._|_  `  V )  =  {  .0.  } )
 
TheoremlpolconN 30807 Contraposition property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
 |-  V  =  ( Base `  W )   &    |-  P  =  (LPol `  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  ._|_  e.  P )   &    |-  ( ph  ->  X  C_  V )   &    |-  ( ph  ->  Y 
 C_  V )   &    |-  ( ph  ->  X  C_  Y )   =>    |-  ( ph  ->  (  ._|_  `  Y )  C_  (  ._|_  `  X )
 )
 
TheoremlpolsatN 30808 The polarity of an atomic subspace is a hyperplane. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
 |-  A  =  (LSAtoms `  W )   &    |-  H  =  (LSHyp `  W )   &    |-  P  =  (LPol `  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  ._|_  e.  P )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  (  ._|_  `  Q )  e.  H )
 
TheoremlpolpolsatN 30809 Property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
 |-  A  =  (LSAtoms `  W )   &    |-  P  =  (LPol `  W )   &    |-  ( ph  ->  W  e.  X )   &    |-  ( ph  ->  ._|_  e.  P )   &    |-  ( ph  ->  Q  e.  A )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  Q ) )  =  Q )
 
TheoremdochpolN 30810 The subspace orthocomplement for the  DVecH vector space is a polarity. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  P  =  (LPol `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  ._|_  e.  P )
 
Theoremlcfl1lem 30811* Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.)
 |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   =>    |-  ( G  e.  C 
 <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) ) )
 
Theoremlcfl1 30812* Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.)
 |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) ) )
 
Theoremlcfl2 30813* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  (  ._|_  `  ( L `
  G ) ) )  =/=  V  \/  ( L `  G )  =  V ) ) )
 
Theoremlcfl3 30814* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  ( L `  G ) )  e.  A  \/  ( L `  G )  =  V )
 ) )
 
Theoremlcfl4N 30815* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  Y  =  (LSHyp `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  (  ._|_  `  ( L `
  G ) ) )  e.  Y  \/  ( L `  G )  =  V ) ) )
 
Theoremlcfl5 30816* Property of a functional with a closed kernel. (Contributed by NM, 1-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( L `  G )  e.  ran  I ) )
 
Theoremlcfl5a 30817 Property of a functional with a closed kernel. TODO: Make lcfl5 30816 etc. obsolete and rewrite w/out 
C hypothesis? (Contributed by NM, 29-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) 
 <->  ( L `  G )  e.  ran  I ) )
 
Theoremlcfl6lem 30818* Lemma for lcfl6 30820. A functional  G (whose kernel is closed by dochsnkr 30792) is comletely determined by a vector  X in the orthocomplement in its kernel at which the functional value is 1. Note that the  \  {  .0.  } in the  X hypothesis is redundant by the last hypothesis but allows easier use of other theorems. (Contributed by NM, 3-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .1.  =  ( 1r `  S )   &    |-  R  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  ( (  ._|_  `  ( L `  G ) )  \  {  .0.  } ) )   &    |-  ( ph  ->  ( G `  X )  =  .1.  )   =>    |-  ( ph  ->  G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
 k  .x.  X )
 ) ) ) )
 
Theoremlcfl7lem 30819* Lemma for lcfl7N 30821. If two functionals  G and  J are equal, they are determined by the same vector. (Contributed by NM, 4-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
 k  .x.  X )
 ) ) )   &    |-  J  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { Y }
 ) v  =  ( w  .+  ( k 
 .x.  Y ) ) ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  G  =  J )   =>    |-  ( ph  ->  X  =  Y )
 
Theoremlcfl6 30820* Property of a functional with a closed kernel. Note that  ( L `  G )  =  V means the functional is zero by lkr0f 28414. (Contributed by NM, 3-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  {
 f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `
  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( ( L `
  G )  =  V  \/  E. x  e.  ( V  \  {  .0.  } ) G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x }
 ) v  =  ( w  .+  ( k 
 .x.  x ) ) ) ) ) ) )
 
Theoremlcfl7N 30821* Property of a functional with a closed kernel. Every nonzero functional is determined by a unique nonzero vector. Note that  ( L `  G )  =  V means the functional is zero by lkr0f 28414. (Contributed by NM, 4-Jan-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  {
 f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `
  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( ( L `
  G )  =  V  \/  E! x  e.  ( V  \  {  .0.  } ) G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x }
 ) v  =  ( w  .+  ( k 
 .x.  x ) ) ) ) ) ) )
 
Theoremlcfl8 30822* Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  E. x  e.  V  ( L `  G )  =  (  ._|_  `  { x } ) ) )
 
Theoremlcfl8a 30823* Property of a functional with a closed kernel. (Contributed by NM, 17-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G ) 
 <-> 
 E. x  e.  V  ( L `  G )  =  (  ._|_  `  { x } ) ) )
 
Theoremlcfl8b 30824* Property of a nonzero functional with a closed kernel. (Contributed by NM, 4-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Y  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  ( C  \  { Y } ) )   =>    |-  ( ph  ->  E. x  e.  ( V  \  {  .0.  } ) (  ._|_  `  ( L `  G ) )  =  ( N `  { x }
 ) )
 
Theoremlcfl9a 30825 Property implying that a functional has a closed kernel. (Contributed by NM, 16-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  (  ._|_  `  { X }
 )  C_  ( L `  G ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `
  G ) )
 
Theoremlclkrlem1 30826* The set of functionals having closed kernels is closed under scalar product. (Contributed by NM, 28-Dec-2014.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .s `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  G  e.  C )   =>    |-  ( ph  ->  ( X  .x.  G )  e.  C )
 
Theoremlclkrlem2a 30827 Lemma for lclkr 30853. Use lshpat 28376 to show that the intersection of a hyperplane with a noncomparable sum of atoms is an atom. (Contributed by NM, 16-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  (  ._|_  ` 
 { X } )  =/=  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  -.  X  e.  (  ._|_  `  { B }
 ) )   =>    |-  ( ph  ->  (
 ( ( N `  { X } )  .(+)  ( N `  { Y } ) )  i^i  (  ._|_  `  { B } ) )  e.  A )
 
Theoremlclkrlem2b 30828 Lemma for lclkr 30853. (Contributed by NM, 17-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  (  ._|_  ` 
 { X } )  =/=  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  `  { B } )  \/  -.  Y  e.  (  ._|_  `  { B } ) ) )   =>    |-  ( ph  ->  ( (
 ( N `  { X } )  .(+)  ( N `
  { Y }
 ) )  i^i  (  ._|_  `  { B }
 ) )  e.  A )
 
Theoremlclkrlem2c 30829 Lemma for lclkr 30853. (Contributed by NM, 16-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  (  ._|_  ` 
 { X } )  =/=  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  `  { B } )  \/  -.  Y  e.  (  ._|_  `  { B } ) ) )   &    |-  J  =  (LSHyp `  U )   =>    |-  ( ph  ->  (
 ( (  ._|_  `  { X } )  i^i  (  ._|_  ` 
 { Y } )
 )  .(+)  ( N `  { B } ) )  e.  J )
 
Theoremlclkrlem2d 30830 Lemma for lclkr 30853. (Contributed by NM, 16-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  (  ._|_  ` 
 { X } )  =/=  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  `  { B } )  \/  -.  Y  e.  (  ._|_  `  { B } ) ) )   &    |-  I  =  ( ( DIsoH `  K ) `  W )   =>    |-  ( ph  ->  (
 ( (  ._|_  `  { X } )  i^i  (  ._|_  ` 
 { Y } )
 )  .(+)  ( N `  { B } ) )  e.  ran  I )
 
Theoremlclkrlem2e 30831 Lemma for lclkr 30853. The kernel of the sum is closed when the kernels of the summands are equal and closed. (Contributed by NM, 17-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  E )  =  ( L `  G ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2f 30832 Lemma for lclkr 30853. Construct a closed hyperplane under the kernel of the sum. (Contributed by NM, 16-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( L `  E )  =/=  ( L `  G ) )   &    |-  ( ph  ->  ( L `  ( E  .+  G ) )  e.  J )   =>    |-  ( ph  ->  (
 ( ( L `  E )  i^i  ( L `
  G ) ) 
 .(+)  ( N `  { B } ) )  C_  ( L `  ( E 
 .+  G ) ) )
 
Theoremlclkrlem2g 30833 Lemma for lclkr 30853. Comparable hyperplanes are equal, so the kernel of the sum is closed. (Contributed by NM, 16-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( L `  E )  =/=  ( L `  G ) )   &    |-  ( ph  ->  ( L `  ( E  .+  G ) )  e.  J )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2h 30834 Lemma for lclkr 30853. Eliminate the  ( L `  ( E 
.+  G ) )  e.  J hypothesis. (Contributed by NM, 16-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  ( L `  E )  =/=  ( L `  G ) )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `
  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2i 30835 Lemma for lclkr 30853. Eliminate the  ( L `  E )  =/=  ( L `  G ) hypothesis. (Contributed by NM, 17-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `
  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2j 30836 Lemma for lclkr 30853. Kernel closure when  Y is zero. (Contributed by NM, 18-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  =  .0.  )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2k 30837 Lemma for lclkr 30853. Kernel closure when  X is zero. (Contributed by NM, 18-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  =  .0.  )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2l 30838 Lemma for lclkr 30853. Eliminate the  X  =/=  .0.,  Y  =/=  .0. hypotheses. (Contributed by NM, 18-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  Q  =  ( 0g `  S )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  .(+)  =  ( LSSum `  U )   &    |-  N  =  (
 LSpan `  U )   &    |-  F  =  (LFnl `  U )   &    |-  J  =  (LSHyp `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  B  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   &    |-  ( ph  ->  ( ( E 
 .+  G ) `  B )  =  Q )   &    |-  ( ph  ->  ( -.  X  e.  (  ._|_  ` 
 { B } )  \/  -.  Y  e.  (  ._|_  `  { B }
 ) ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2m 30839 Lemma for lclkr 30853. Construct a vector  B that makes the sum of functionals zero. Combine with  B  e.  V to shorten overall proof. (Contributed by NM, 17-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  U  e.  LVec )   &    |-  B  =  ( X  .-  ( (
 ( ( E  .+  G ) `  X )  .X.  ( I `  ( ( E  .+  G ) `  Y ) ) )  .x.  Y ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   =>    |-  ( ph  ->  ( B  e.  V  /\  ( ( E  .+  G ) `  B )  =  .0.  )
 )
 
Theoremlclkrlem2n 30840 Lemma for lclkr 30853. (Contributed by NM, 12-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  ( ph  ->  U  e.  LVec )   &    |-  ( ph  ->  ( ( E  .+  G ) `  X )  =  .0.  )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =  .0.  )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  C_  ( L `  ( E  .+  G ) ) )
 
Theoremlclkrlem2o 30841 Lemma for lclkr 30853. When  B is nonzero, the vectors  X and  Y can't both belong to the hyperplane generated by  B. (Contributed by NM, 17-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  B  =  ( X  .-  ( (
 ( ( E  .+  G ) `  X )  .X.  ( I `  ( ( E  .+  G ) `  Y ) ) )  .x.  Y ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   &    |-  ( ph  ->  B  =/=  ( 0g `  U ) )   =>    |-  ( ph  ->  ( -.  X  e.  (  ._|_  `  { B }
 )  \/  -.  Y  e.  (  ._|_  `  { B } ) ) )
 
Theoremlclkrlem2p 30842 Lemma for lclkr 30853. When  B is zero,  X and  Y must colinear, so their orthocomplements must be comparable. (Contributed by NM, 17-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  B  =  ( X  .-  ( (
 ( ( E  .+  G ) `  X )  .X.  ( I `  ( ( E  .+  G ) `  Y ) ) )  .x.  Y ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   &    |-  ( ph  ->  B  =  ( 0g `  U ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  { Y }
 )  C_  (  ._|_  ` 
 { X } )
 )
 
Theoremlclkrlem2q 30843 Lemma for lclkr 30853. The sum has a closed kernel when  B is nonzero. (Contributed by NM, 18-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  B  =  ( X  .-  ( (
 ( ( E  .+  G ) `  X )  .X.  ( I `  ( ( E  .+  G ) `  Y ) ) )  .x.  Y ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   &    |-  ( ph  ->  B  =/=  ( 0g `  U ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2r 30844 Lemma for lclkr 30853. When  B is zero, i.e. when  X and  Y are colinear, the intersection of the kernels of  E and  G equal the kernel of  G, so the kernels of  G and the sum are comparable. (Contributed by NM, 18-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  B  =  ( X  .-  ( (
 ( ( E  .+  G ) `  X )  .X.  ( I `  ( ( E  .+  G ) `  Y ) ) )  .x.  Y ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   &    |-  ( ph  ->  B  =  ( 0g `  U ) )   =>    |-  ( ph  ->  ( L `  G ) 
 C_  ( L `  ( E  .+  G ) ) )
 
Theoremlclkrlem2s 30845 Lemma for lclkr 30853. Thus the sum has a closed kernel when  B is zero. (Contributed by NM, 18-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  B  =  ( X  .-  ( (
 ( ( E  .+  G ) `  X )  .X.  ( I `  ( ( E  .+  G ) `  Y ) ) )  .x.  Y ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   &    |-  ( ph  ->  B  =  ( 0g `  U ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2t 30846 Lemma for lclkr 30853. We eliminate all hypotheses with  B here. (Contributed by NM, 18-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =/= 
 .0.  )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2u 30847 Lemma for lclkr 30853. lclkrlem2t 30846 with  X and  Y swapped. (Contributed by NM, 18-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  X )  =/= 
 .0.  )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2v 30848 Lemma for lclkr 30853. When the hypotheses of lclkrlem2u 30847 and lclkrlem2u 30847 are negated, the functional sum must be zero, so the kernel is the vector space. We make use of the law of excluded middle, dochexmid 30788, which requires the orthomodular law dihoml4 30697 (Lemma 3.3 of [Holland95] p. 214). (Contributed by NM, 16-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  X )  =  .0.  )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =  .0.  )   =>    |-  ( ph  ->  ( L `  ( E  .+  G ) )  =  V )
 
Theoremlclkrlem2w 30849 Lemma for lclkr 30853. This is the same as lclkrlem2u 30847 and lclkrlem2u 30847 with the inequality hypotheses negated. When the sum of two functionals is zero at each generating vector, the kernel is the vector space and therefore closed. (Contributed by NM, 16-Jan-2015.)
 |-  V  =  ( Base `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |- 
 .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  ( invr `  S )   &    |-  .-  =  ( -g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  H  =  (
 LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .(+)  =  (
 LSSum `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X } ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y }
 ) )   &    |-  ( ph  ->  ( ( E  .+  G ) `  X )  =  .0.  )   &    |-  ( ph  ->  ( ( E  .+  G ) `  Y )  =  .0.  )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2x 30850 Lemma for lclkr 30853. Eliminate by cases the hypotheses of lclkrlem2u 30847, lclkrlem2u 30847 and lclkrlem2w 30849. (Contributed by NM, 18-Jan-2015.)
 |-  L  =  (LKer `  U )   &    |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  ( L `  E )  =  (  ._|_  `  { X }
 ) )   &    |-  ( ph  ->  ( L `  G )  =  (  ._|_  `  { Y } ) )   =>    |-  ( ph  ->  ( 
 ._|_  `  (  ._|_  `  ( L `  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2y 30851 Lemma for lclkr 30853. Restate the hypotheses for  E and  G to say their kernels are closed, in order to eliminate the generating vectors  X and  Y. (Contributed by NM, 18-Jan-2015.)
 |-  L  =  (LKer `  U )   &    |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  E ) ) )  =  ( L `
  E ) )   &    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `
  G ) ) )  =  ( L `
  G ) )   =>    |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `
  ( E  .+  G ) ) ) )  =  ( L `
  ( E  .+  G ) ) )
 
Theoremlclkrlem2 30852* The set of functionals having closed kernels is closed under vector (functional) addition. Lemmas lclkrlem2a 30827 through lclkrlem2y 30851 are used for the proof. Here we express lclkrlem2y 30851 in terms of membership in the set  C of functionals with closed kernels. (Contributed by NM, 18-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  .+  =  ( +g  `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  E  e.  C )   &    |-  ( ph  ->  G  e.  C )   =>    |-  ( ph  ->  ( E  .+  G )  e.  C )
 
Theoremlclkr 30853* The set of functionals with closed kernels is a subspace. Part of proof of Theorem 3.6 of [Holland95] p. 218, line 20, stating "The fM that arise this way generate a subspace F of E'". Our proof was suggested by Mario Carneiro, 5-Jan-2015. (Contributed by NM, 18-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  S  =  ( LSubSp `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  C  e.  S )
 
Theoremlcfls1lem 30854* Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.)
 |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f )  /\  (  ._|_  `  ( L `  f
 ) )  C_  Q ) }   =>    |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `
  G )  /\  (  ._|_  `  ( L `  G ) )  C_  Q ) )
 
Theoremlcfls1N 30855* Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)
 |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f )  /\  (  ._|_  `  ( L `  f
 ) )  C_  Q ) }   &    |-  ( ph  ->  G  e.  F )   =>    |-  ( ph  ->  ( G  e.  C  <->  ( (  ._|_  `  (  ._|_  `  ( L `
  G ) ) )  =  ( L `
  G )  /\  (  ._|_  `  ( L `  G ) )  C_  Q ) ) )
 
Theoremlcfls1c 30856* Property of a functional with a closed kernel. (Contributed by NM, 28-Jan-2015.)
 |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f )  /\  (  ._|_  `  ( L `  f
 ) )  C_  Q ) }   &    |-  D  =  {
 f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `
  f ) }   =>    |-  ( G  e.  C  <->  ( G  e.  D  /\  (  ._|_  `  ( L `  G ) ) 
 C_  Q ) )
 
Theoremlclkrslem1 30857* The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace  Q is closed under scalar product. (Contributed by NM, 27-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .s `  D )   &    |-  C  =  { f  e.  F  |  ( ( 
 ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `
  f )  /\  (  ._|_  `  ( L `  f ) )  C_  Q ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  Q  e.  S )   &    |-  ( ph  ->  G  e.  C )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( X  .x.  G )  e.  C )
 
Theoremlclkrslem2 30858* The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace  Q is closed under scalar product. (Contributed by NM, 28-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  R  =  (Scalar `  U )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .s `  D )   &    |-  C  =  { f  e.  F  |  ( ( 
 ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `
  f )  /\  (  ._|_  `  ( L `  f ) )  C_  Q ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  Q  e.  S )   &    |-  ( ph  ->  G  e.  C )   &    |- 
 .+  =  ( +g  `  D )   &    |-  ( ph  ->  E  e.  C )   =>    |-  ( ph  ->  ( E  .+  G )  e.  C )
 
Theoremlclkrs 30859* The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace  R is a subspace of the dual space. TODO: This proof repeats large parts of the lclkr 30853 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 30853 a special case of this? (Contributed by NM, 29-Jan-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  T  =  ( LSubSp `  D )   &    |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f )  /\  (  ._|_  `  ( L `  f
 ) )  C_  R ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  R  e.  S )   =>    |-  ( ph  ->  C  e.  T )
 
Theoremlclkrs2 30860* The set of functionals with closed kernels and majorizing the orthocomplement of a given subspace  Q is a subspace of the dual space containing functionals with closed kernels. Note that  R is the value given by mapdval 30948. (Contributed by NM, 12-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  S  =  ( LSubSp `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  T  =  ( LSubSp `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }   &    |-  R  =  { g  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `  g ) ) )  =  ( L `  g )  /\  (  ._|_  `  ( L `  g
 ) )  C_  Q ) }   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  Q  e.  S )   =>    |-  ( ph  ->  ( R  e.  T  /\  R  C_  C ) )
 
TheoremlcfrvalsnN 30861* Reconstruction from the dual space span of a singleton. (Contributed by NM, 19-Feb-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  N  =  ( LSpan `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  F )   &    |-  Q  =  U_ f  e.  R  (  ._|_  `  ( L `  f ) )   &    |-  R  =  ( N `  { G } )   =>    |-  ( ph  ->  Q  =  (  ._|_  `  ( L `  G ) ) )
 
Theoremlcfrlem1 30862 Lemma for lcfr 30905. Note that  X is z in Mario's notes. (Contributed by NM, 27-Feb-2015.)
 |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  (
 invr `  S )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .x.  =  ( .s `  D )   &    |-  .-  =  ( -g `  D )   &    |-  ( ph  ->  U  e.  LVec )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( G `  X )  =/=  .0.  )   &    |-  H  =  ( E  .-  (
 ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )   =>    |-  ( ph  ->  ( H `  X )  =  .0.  )
 
Theoremlcfrlem2 30863 Lemma for lcfr 30905. (Contributed by NM, 27-Feb-2015.)
 |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  (
 invr `  S )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .x.  =  ( .s `  D )   &    |-  .-  =  ( -g `  D )   &    |-  ( ph  ->  U  e.  LVec )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( G `  X )  =/=  .0.  )   &    |-  H  =  ( E  .-  (
 ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )   &    |-  L  =  (LKer `  U )   =>    |-  ( ph  ->  (
 ( L `  E )  i^i  ( L `  G ) )  C_  ( L `  H ) )
 
Theoremlcfrlem3 30864 Lemma for lcfr 30905. (Contributed by NM, 27-Feb-2015.)
 |-  V  =  ( Base `  U )   &    |-  S  =  (Scalar `  U )   &    |-  .X.  =  ( .r `  S )   &    |-  .0.  =  ( 0g `  S )   &    |-  I  =  (
 invr `  S )   &    |-  F  =  (LFnl `  U )   &    |-  D  =  (LDual `  U )   &    |-  .x.  =  ( .s `  D )   &    |-  .-  =  ( -g `  D )   &    |-  ( ph  ->  U  e.  LVec )   &    |-  ( ph  ->  E  e.  F )   &    |-  ( ph  ->  G  e.  F )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  ( G `  X )  =/=  .0.  )   &    |-  H  =  ( E  .-  (
 ( ( I `  ( G `  X ) )  .X.  ( E `  X ) )  .x.  G ) )   &    |-  L  =  (LKer `  U )   =>    |-  ( ph  ->  X  e.  ( L `  H ) )
 
Theoremlcfrlem4 30865* Lemma for lcfr 30905. (Contributed by NM, 10-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( LSubSp `  D )   &    |-  E  =  U_ g  e.  G  (  ._|_  `  ( L `  g ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  Q )   &    |-  ( ph  ->  X  e.  E )   =>    |-  ( ph  ->  X  e.  V )
 
Theoremlcfrlem5 30866* Lemma for lcfr 30905. The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace  Q is closed under scalar product. TODO: share hypotheses with others. Use more consistent variable names here or elsewhere when possible. (Contributed by NM, 5-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  S  =  ( LSubSp `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  R  e.  S )   &    |-  Q  =  U_ f  e.  R  (  ._|_  `  ( L `  f ) )   &    |-  ( ph  ->  X  e.  Q )   &    |-  C  =  (Scalar `  U )   &    |-  B  =  ( Base `  C )   &    |-  .x.  =  ( .s `  U )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  ( A  .x.  X )  e.  Q )
 
Theoremlcfrlem6 30867* Lemma for lcfr 30905. Closure of vector sum with colinear vectors. TODO: Move down  N definition so top hypotheses can be shared. (Contributed by NM, 10-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( LSubSp `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  Q )   &    |-  E  =  U_ g  e.  G  (  ._|_  `  ( L `  g ) )   &    |-  ( ph  ->  X  e.  E )   &    |-  ( ph  ->  Y  e.  E )   &    |-  ( ph  ->  ( N `  { X } )  =  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  E )
 
Theoremlcfrlem7 30868* Lemma for lcfr 30905. Closure of vector sum when one vector is zero. TODO: share hypotheses with others. (Contributed by NM, 11-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  .+  =  ( +g  `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( LSubSp `  D )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  G  e.  Q )   &    |-  E  =  U_ g  e.  G  (  ._|_  `  ( L `  g ) )   &    |-  ( ph  ->  X  e.  E )   &    |- 
 .0.  =  ( 0g `  U )   &    |-  ( ph  ->  Y  =  .0.  )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  E )
 
Theoremlcfrlem8 30869* Lemma for lcf1o 30871 and lcfr 30905. (Contributed by NM, 21-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( J `  X )  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X }
 ) v  =  ( w  .+  ( k 
 .x.  X ) ) ) ) )
 
Theoremlcfrlem9 30870* Lemma for lcf1o 30871. (This part has undesirable $d's on  J and  ph that we remove in lcf1o 30871.) TODO: ugly proof; maybe have better subtheorems or abbreviate some  iota_
k expansions with  J `  z? TODO: Some redundant $d's? (Contributed by NM, 22-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  J : ( V  \  {  .0.  } ) -1-1-onto-> ( C 
 \  { Q }
 ) )
 
Theoremlcf1o 30871* Define a function  J that provides a bijection from nonzero vectors  V to nonzero functionals with closed kernels  C. (Contributed by NM, 22-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   =>    |-  ( ph  ->  J : ( V  \  {  .0.  } ) -1-1-onto-> ( C 
 \  { Q }
 ) )
 
Theoremlcfrlem10 30872* Lemma for lcfr 30905. (Contributed by NM, 23-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( J `  X )  e.  F )
 
Theoremlcfrlem11 30873* Lemma for lcfr 30905. (Contributed by NM, 23-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( L `  ( J `  X ) )  =  (  ._|_  `  { X } ) )
 
Theoremlcfrlem12N 30874* Lemma for lcfr 30905. (Contributed by NM, 23-Feb-2015.) (New usage is discouraged.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  B  =  ( 0g `  S )   &    |-  ( ph  ->  Y  e.  (  ._|_  `  { X }
 ) )   =>    |-  ( ph  ->  (
 ( J `  X ) `  Y )  =  B )
 
Theoremlcfrlem13 30875* Lemma for lcfr 30905. (Contributed by NM, 8-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( J `  X )  e.  ( C  \  { Q } ) )
 
Theoremlcfrlem14 30876* Lemma for lcfr 30905. (Contributed by NM, 10-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   &    |-  N  =  (
 LSpan `  U )   =>    |-  ( ph  ->  ( 
 ._|_  `  ( L `  ( J `  X ) ) )  =  ( N `  { X } ) )
 
Theoremlcfrlem15 30877* Lemma for lcfr 30905. (Contributed by NM, 9-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  X  e.  (  ._|_  `  ( L `  ( J `  X ) ) ) )
 
Theoremlcfrlem16 30878* Lemma for lcfr 30905. (Contributed by NM, 8-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .x.  =  ( .s `  U )   &    |-  S  =  (Scalar `  U )   &    |-  R  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  U )   &    |-  F  =  (LFnl `  U )   &    |-  L  =  (LKer `  U )   &    |-  D  =  (LDual `  U )   &    |-  Q  =  ( 0g `  D )   &    |-  C  =  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `
  f ) ) )  =  ( L `
  f ) }   &    |-  J  =  ( x  e.  ( V  \  {  .0.  }
 )  |->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { x } ) v  =  ( w  .+  (
 k  .x.  x )
 ) ) ) )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )   &    |-  P  =  ( LSubSp `  D )   &    |-  ( ph  ->  G  e.  P )   &    |-  ( ph  ->  G  C_  C )   &    |-  E  =  U_ g  e.  G  (  ._|_  `  ( L `  g ) )   &    |-  ( ph  ->  X  e.  ( E  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  ( J `  X )  e.  G )
 
Theoremlcfrlem17 30879 Lemma for lcfr 30905. Condition needed more than once. (Contributed by NM, 11-Mar-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  ( V  \  {  .0.  } ) )
 
Theoremlcfrlem18 30880 Lemma for lcfr 30905. (Contributed by NM, 24-Feb-2015.)
 |-  H  =  ( LHyp `  K )   &    |-  ._|_  =  ( ( ocH `  K ) `  W )   &    |-  U  =  ( ( DVecH `  K ) `  W )   &    |-  V  =  ( Base `  U )   &    |-  .+  =  ( +g  `  U )   &    |-  .0.  =  ( 0g `  U )   &    |-  N  =  ( LSpan `  U )   &    |-  A  =  (LSAtoms `  U )   &    |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H )
 )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph