HomeHome Metamath Proof Explorer
Theorem List (p. 162 of 325)
< 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
(1-22374)
  Hilbert Space Explorer  Hilbert Space Explorer
(22375-23897)
  Users' Mathboxes  Users' Mathboxes
(23898-32447)
 

Theorem List for Metamath Proof Explorer - 16101-16200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
10.6.4  Subspace sum; bases for a left module
 
Syntaxclbs 16101 Extend class notation with the set of bases for a vector space.
 class LBasis
 
Definitiondf-lbs 16102* Define the set of bases to a left module or left vector space. (Contributed by Mario Carneiro, 24-Jun-2014.)
 |- LBasis  =  ( w  e.  _V  |->  { b  e.  ~P ( Base `  w )  | 
 [. ( LSpan `  w )  /  n ]. [. (Scalar `  w )  /  s ]. ( ( n `  b )  =  ( Base `  w )  /\  A. x  e.  b  A. y  e.  ( ( Base `  s )  \  { ( 0g `  s ) } )  -.  ( y ( .s
 `  w ) x )  e.  ( n `
  ( b  \  { x } ) ) ) } )
 
Theoremislbs 16103* The predicate " B is a basis for the left module or vector space  W". A subset of the base set is a basis if zero is not in the set, it spans the set, and no nonzero multiple of an element of the basis is in the span of the rest of the family. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 14-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .0.  =  ( 0g `  F )   =>    |-  ( W  e.  X  ->  ( B  e.  J  <->  ( B  C_  V  /\  ( N `  B )  =  V  /\  A. x  e.  B  A. y  e.  ( K  \  {  .0.  } )  -.  (
 y  .x.  x )  e.  ( N `  ( B  \  { x }
 ) ) ) ) )
 
Theoremlbsss 16104 A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   =>    |-  ( B  e.  J  ->  B  C_  V )
 
Theoremlbsel 16105 An element of a basis is a vector. (Contributed by Mario Carneiro, 24-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   =>    |-  ( ( B  e.  J  /\  E  e.  B )  ->  E  e.  V )
 
Theoremlbssp 16106 The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( B  e.  J  ->  ( N `  B )  =  V )
 
Theoremlbsind 16107 A basis is linearly independent; that is, every element has a span which trivially intersects the span of the remainder of the basis. (Contributed by Mario Carneiro, 12-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  .0.  =  ( 0g `  F )   =>    |-  ( ( ( B  e.  J  /\  E  e.  B )  /\  ( A  e.  K  /\  A  =/=  .0.  ) ) 
 ->  -.  ( A  .x.  E )  e.  ( N `
  ( B  \  { E } ) ) )
 
Theoremlbsind2 16108 A basis is linearly independent; that is, every element is not in the span of the remainder of the basis. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 12-Jan-2015.)
 |-  J  =  (LBasis `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .1.  =  ( 1r `  F )   &    |-  .0.  =  ( 0g `  F )   =>    |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  E  e.  B )  ->  -.  E  e.  ( N `  ( B 
 \  { E }
 ) ) )
 
Theoremlbspss 16109 No proper subset of a basis spans the space. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  J  =  (LBasis `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .1.  =  ( 1r `  F )   &    |-  .0.  =  ( 0g `  F )   &    |-  V  =  (
 Base `  W )   =>    |-  ( ( ( W  e.  LMod  /\  .1.  =/=  .0.  )  /\  B  e.  J  /\  C  C.  B )  ->  ( N `
  C )  =/= 
 V )
 
Theoremlsmcl 16110 The sum of two subspaces is a subspace. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  S  =  ( LSubSp `  W )   &    |-  .(+)  =  ( LSSum `  W )   =>    |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  e.  S )
 
Theoremlsmspsn 16111* Member of subspace sum of spans of singletons. (Contributed by NM, 8-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .(+) 
 =  ( LSSum `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( U  e.  ( ( N `  { X }
 )  .(+)  ( N `  { Y } ) )  <->  E. j  e.  K  E. k  e.  K  U  =  ( (
 j  .x.  X )  .+  ( k  .x.  Y ) ) ) )
 
Theoremlsmelval2 16112* Subspace sum membership in terms of a sum of 1-dim subspaces (atoms), which can be useful for treating subspaces as projective lattice elements. (Contributed by NM, 9-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( X  e.  ( T 
 .(+)  U )  <->  ( X  e.  V  /\  E. y  e.  T  E. z  e.  U  ( N `  { X } )  C_  ( ( N `  { y } )  .(+) 
 ( N `  { z } ) ) ) ) )
 
Theoremlsmsp 16113 Subspace sum in terms of span. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   =>    |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  .(+)  U )  =  ( N `
  ( T  u.  U ) ) )
 
Theoremlsmsp2 16114 Subspace sum of spans of subsets is the span of their union. (spanuni 22999 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   =>    |-  ( ( W  e.  LMod  /\  T  C_  V  /\  U  C_  V )  ->  ( ( N `
  T )  .(+)  ( N `  U ) )  =  ( N `
  ( T  u.  U ) ) )
 
Theoremlsmssspx 16115 Subspace sum (in its extended domain) is a subset of the span of the union of its arguments. (Contributed by NM, 6-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  ( ph  ->  T  C_  V )   &    |-  ( ph  ->  U  C_  V )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  ( T  .(+)  U )  C_  ( N `  ( T  u.  U ) ) )
 
Theoremlsmpr 16116 The span of a pair of vectors equals the sum of the spans of their singletons. (Contributed by NM, 13-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  =  ( ( N `  { X } )  .(+)  ( N `  { Y } ) ) )
 
Theoremlsppreli 16117 A vector expressed as a sum belongs to the span of its components. (Contributed by NM, 9-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (
 ( A  .x.  X )  .+  ( B  .x.  Y ) )  e.  ( N `  { X ,  Y } ) )
 
Theoremlsmelpr 16118 Two ways to say that a vector belongs to the span of a pair of vectors. (Contributed by NM, 14-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   =>    |-  ( ph  ->  ( X  e.  ( N ` 
 { Y ,  Z } )  <->  ( N `  { X } )  C_  ( ( N `  { Y } )  .(+)  ( N `  { Z } ) ) ) )
 
Theoremlsppr0 16119 The span of a vector paired with zero equals the span of the singleton of the vector. (Contributed by NM, 29-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( N `  { X ,  .0.  } )  =  ( N `  { X } ) )
 
Theoremlsppr 16120* Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  =  {
 v  |  E. k  e.  K  E. l  e.  K  v  =  ( ( k  .x.  X )  .+  ( l  .x.  Y ) ) } )
 
Theoremlspprel 16121* Member of the span of a pair of vectors. (Contributed by NM, 10-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( Z  e.  ( N ` 
 { X ,  Y } )  <->  E. k  e.  K  E. l  e.  K  Z  =  ( (
 k  .x.  X )  .+  ( l  .x.  Y ) ) ) )
 
Theoremlspprabs 16122 Absorption of vector sum into span of pair. (Contributed by NM, 27-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( N `  { X ,  ( X  .+  Y ) } )  =  ( N `  { X ,  Y } ) )
 
Theoremlspvadd 16123 The span of a vector sum is included in the span of its arguments. (Contributed by NM, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  (
 ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( N `  { ( X  .+  Y ) }
 )  C_  ( N ` 
 { X ,  Y } ) )
 
Theoremlspsntri 16124 Triangle-type inequality for span of a singleton. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( N `  { ( X  .+  Y ) } )  C_  ( ( N `  { X } )  .(+)  ( N `  { Y } ) ) )
 
Theoremlspsntrim 16125 Triangle-type inequality for span of a singleton of vector difference. (Contributed by NM, 25-Apr-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  (
 ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( N `  { ( X  .-  Y ) }
 )  C_  ( ( N `  { X }
 )  .(+)  ( N `  { Y } ) ) )
 
Theoremlbspropd 16126* If two structures have the same components (properties), they have the same set of bases. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  B  C_  W )   &    |-  ( ( ph  /\  ( x  e.  W  /\  y  e.  W ) )  ->  ( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  K ) y )  e.  W )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  K ) y )  =  ( x ( .s
 `  L ) y ) )   &    |-  F  =  (Scalar `  K )   &    |-  G  =  (Scalar `  L )   &    |-  ( ph  ->  P  =  ( Base `  F ) )   &    |-  ( ph  ->  P  =  ( Base `  G ) )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  P ) )  ->  ( x ( +g  `  F ) y )  =  ( x ( +g  `  G ) y ) )   &    |-  ( ph  ->  K  e.  _V )   &    |-  ( ph  ->  L  e.  _V )   =>    |-  ( ph  ->  (LBasis `  K )  =  (LBasis `  L ) )
 
Theorempj1lmhm 16127 The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  L  =  ( LSubSp `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  P  =  ( proj 1 `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  L )   &    |-  ( ph  ->  U  e.  L )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   =>    |-  ( ph  ->  ( T P U )  e.  ( ( Ws  ( T 
 .(+)  U ) ) LMHom  W ) )
 
Theorempj1lmhm2 16128 The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  L  =  ( LSubSp `  W )   &    |-  .(+)  =  ( LSSum `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  P  =  ( proj 1 `
  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T  e.  L )   &    |-  ( ph  ->  U  e.  L )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   =>    |-  ( ph  ->  ( T P U )  e.  ( ( Ws  ( T 
 .(+)  U ) ) LMHom  ( Ws  T ) ) )
 
10.7  Vector spaces
 
10.7.1  Definition and basic properties
 
Syntaxclvec 16129 Extend class notation with class of all left vector spaces.
 class  LVec
 
Definitiondf-lvec 16130 Define the class of all left vector spaces. A left vector space over a division ring is an Abelian group (vectors) together with a division ring (scalars) and a left scalar product connecting them. Some authors call this a "left module over a division ring", reserving "vector space" for those where the division ring multiplication is commutative i.e. a field. (Contributed by NM, 11-Nov-2013.)
 |- 
 LVec  =  { f  e.  LMod  |  (Scalar `  f
 )  e.  DivRing }
 
Theoremislvec 16131 The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  LVec  <->  ( W  e.  LMod  /\  F  e. 
 DivRing ) )
 
Theoremlvecdrng 16132 The set of scalars of a left vector space is a division ring. (Contributed by NM, 17-Apr-2014.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e.  LVec  ->  F  e.  DivRing )
 
Theoremlveclmod 16133 A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)
 |-  ( W  e.  LVec  ->  W  e.  LMod )
 
Theoremlsslvec 16134 A vector subspace is a vector space. (Contributed by NM, 14-Mar-2015.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LVec  /\  U  e.  S )  ->  X  e.  LVec
 )
 
Theoremlvecvs0or 16135 If a scalar product is zero, one of its factors must be zero. (hvmul0or 22480 analog.) (Contributed by NM, 2-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  O  =  ( 0g `  F )   &    |- 
 .0.  =  ( 0g `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  (
 ( A  .x.  X )  =  .0.  <->  ( A  =  O  \/  X  =  .0.  ) ) )
 
Theoremlvecvsn0 16136 A scalar product is nonzero iff both of its factors are nonzero. (Contributed by NM, 3-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  O  =  ( 0g `  F )   &    |- 
 .0.  =  ( 0g `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  (
 ( A  .x.  X )  =/=  .0.  <->  ( A  =/=  O 
 /\  X  =/=  .0.  ) ) )
 
Theoremlssvs0or 16137 If a scalar product belongs to a subspace, either the scalar component is zero or the vector component also belongs. (Contributed by NM, 5-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .0.  =  ( 0g `  F )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  A  e.  K )   =>    |-  ( ph  ->  (
 ( A  .x.  X )  e.  U  <->  ( A  =  .0.  \/  X  e.  U ) ) )
 
Theoremlvecvscan 16138 Cancellation law for scalar multiplication. (hvmulcan 22527 analog.) (Contributed by NM, 2-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .0.  =  ( 0g `  F )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  A  =/=  .0.  )   =>    |-  ( ph  ->  ( ( A  .x.  X )  =  ( A  .x.  Y )  <->  X  =  Y ) )
 
Theoremlvecvscan2 16139 Cancellation law for scalar multiplication. (hvmulcan2 22528 analog.) (Contributed by NM, 2-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .0.  =  ( 0g `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  X  =/=  .0.  )   =>    |-  ( ph  ->  ( ( A  .x.  X )  =  ( B  .x.  X )  <->  A  =  B ) )
 
Theoremlvecinv 16140 Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .0.  =  ( 0g `  F )   &    |-  I  =  ( invr `  F )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  A  e.  ( K  \  {  .0.  } ) )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( X  =  ( A 
 .x.  Y )  <->  Y  =  (
 ( I `  A )  .x.  X ) ) )
 
Theoremlspsnvs 16141 A non-zero scalar product does not change the span of a singleton. (spansncol 23023 analog.) (Contributed by NM, 23-Apr-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  .0.  =  ( 0g `  F )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( ( W  e.  LVec  /\  ( R  e.  K  /\  R  =/=  .0.  )  /\  X  e.  V ) 
 ->  ( N `  { ( R  .x.  X ) }
 )  =  ( N `
  { X }
 ) )
 
Theoremlspsneleq 16142 Membership relation that implies equality of spans. (spansneleq 23025 analog.) (Contributed by NM, 4-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( N `  { X }
 ) )   &    |-  ( ph  ->  Y  =/=  .0.  )   =>    |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { X }
 ) )
 
Theoremlspsncmp 16143 Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (
 ( N `  { X } )  C_  ( N `
  { Y }
 ) 
 <->  ( N `  { X } )  =  ( N `  { Y }
 ) ) )
 
Theoremlspsnne1 16144 Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )
 
Theoremlspsnne2 16145 Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y } ) )   =>    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )
 
Theoremlspsnnecom 16146 Swap two vectors with different spans. (Contributed by NM, 20-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y }
 ) )   =>    |-  ( ph  ->  -.  Y  e.  ( N `  { X } ) )
 
Theoremlspabs2 16147 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =  ( N `  { ( X 
 .+  Y ) }
 ) )   =>    |-  ( ph  ->  ( N `  { X }
 )  =  ( N `
  { Y }
 ) )
 
Theoremlspabs3 16148 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( X  .+  Y )  =/= 
 .0.  )   &    |-  ( ph  ->  ( N `  { X } )  =  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  ( N `  { X }
 )  =  ( N `
  { ( X 
 .+  Y ) }
 ) )
 
Theoremlspsneq 16149* Equal spans of singletons must have proportional vectors. See lspsnss2 16036 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (Scalar `  W )   &    |-  K  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  S )   &    |- 
 .x.  =  ( .s `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y }
 ) 
 <-> 
 E. k  e.  ( K  \  {  .0.  }
 ) X  =  ( k  .x.  Y )
 ) )
 
Theoremlspsneu 16150* Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (Scalar `  W )   &    |-  K  =  (
 Base `  S )   &    |-  O  =  ( 0g `  S )   &    |- 
 .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   =>    |-  ( ph  ->  (
 ( N `  { X } )  =  ( N `  { Y }
 ) 
 <->  E! k  e.  ( K  \  { O }
 ) X  =  ( k  .x.  Y )
 ) )
 
Theoremlspsnel4 16151 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 23028 analog.) (Contributed by NM, 4-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( N `  { X }
 ) )   &    |-  ( ph  ->  Y  =/=  .0.  )   =>    |-  ( ph  ->  ( X  e.  U  <->  Y  e.  U ) )
 
Theoremlspdisj 16152 The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  U )   =>    |-  ( ph  ->  (
 ( N `  { X } )  i^i  U )  =  {  .0.  }
 )
 
Theoremlspdisjb 16153 A nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   =>    |-  ( ph  ->  ( -.  X  e.  U  <->  ( ( N `
  { X }
 )  i^i  U )  =  {  .0.  } )
 )
 
Theoremlspdisj2 16154 Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   =>    |-  ( ph  ->  (
 ( N `  { X } )  i^i  ( N `
  { Y }
 ) )  =  {  .0.  } )
 
Theoremlspfixed 16155* Show membership in the span of the sum of two vectors, one of which ( Y) is fixed in advance. (Contributed by NM, 27-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Y }
 ) )   &    |-  ( ph  ->  -.  X  e.  ( N `
  { Z }
 ) )   &    |-  ( ph  ->  X  e.  ( N `  { Y ,  Z }
 ) )   =>    |-  ( ph  ->  E. z  e.  ( ( N `  { Z } )  \  {  .0.  } ) X  e.  ( N `  { ( Y  .+  z ) } )
 )
 
Theoremlspexch 16156 Exchange property for span of a pair. TODO: see if a version with Y,Z and X,Z reversed will shorten proofs (analogous to lspexchn1 16157 vs. lspexchn2 16158); look for lspexch 16156 and prcom 3842 in same proof. TODO: would a hypothesis of  -.  X  e.  ( N `  { Z } ) instead of  ( N `  { X } )  =/=  ( N { Z } ) ` be better overall? This would be shorter and also satisfy the 
X  =/=  .0. condition. Here and also lspindp* and all proofs affected by them (all in NM's mathbox); there are 58 hypotheses with the 
=/= pattern as of 24-May-2015. (Contributed by NM, 11-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Z } ) )   &    |-  ( ph  ->  X  e.  ( N `  { Y ,  Z } ) )   =>    |-  ( ph  ->  Y  e.  ( N `  { X ,  Z }
 ) )
 
Theoremlspexchn1 16157 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 16156 to see if this will shorten proofs. (Contributed by NM, 20-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  Y  e.  ( N `  { Z } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   =>    |-  ( ph  ->  -.  Y  e.  ( N `  { X ,  Z } ) )
 
Theoremlspexchn2 16158 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 16156 to see if this will shorten proofs. (Contributed by NM, 24-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  Y  e.  ( N `  { Z } ) )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Z ,  Y } ) )   =>    |-  ( ph  ->  -.  Y  e.  ( N `  { Z ,  X } ) )
 
Theoremlspindpi 16159 Partial independence property. (Contributed by NM, 23-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )   =>    |-  ( ph  ->  ( ( N `  { X }
 )  =/=  ( N ` 
 { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z }
 ) ) )
 
Theoremlspindp1 16160 Alternate way to say 3 vectors are mutually independent (swap 1st and 2nd). (Contributed by NM, 11-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  -.  Z  e.  ( N `  { X ,  Y } ) )   =>    |-  ( ph  ->  ( ( N `  { Z }
 )  =/=  ( N ` 
 { Y } )  /\  -.  X  e.  ( N `  { Z ,  Y } ) ) )
 
Theoremlspindp2l 16161 Alternate way to say 3 vectors are mutually independent (rotate left). (Contributed by NM, 10-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  -.  Z  e.  ( N `  { X ,  Y } ) )   =>    |-  ( ph  ->  ( ( N `  { Y }
 )  =/=  ( N ` 
 { Z } )  /\  -.  X  e.  ( N `  { Y ,  Z } ) ) )
 
Theoremlspindp2 16162 Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )   &    |-  ( ph  ->  -.  Z  e.  ( N `  { X ,  Y } ) )   =>    |-  ( ph  ->  ( ( N `  { Z }
 )  =/=  ( N ` 
 { X } )  /\  -.  Y  e.  ( N `  { Z ,  X } ) ) )
 
Theoremlspindp3 16163 Independence of 2 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  }
 ) )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { ( X  .+  Y ) } )
 )
 
Theoremlspindp4 16164 (Partial) independence of 3 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   &    |-  ( ph  ->  -.  Z  e.  ( N `  { X ,  Y } ) )   =>    |-  ( ph  ->  -.  Z  e.  ( N `  { X ,  ( X  .+  Y ) } ) )
 
Theoremlvecindp 16165 Compute the  X coefficient in a sum with an independent vector  X (first conjunct), which can then be removed to continue with the remaining vectors summed in expressions  Y and 
Z (second conjunct). Typically,  U is the span of the remaining vectors. (Contributed by NM, 5-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  ( ( A 
 .x.  X )  .+  Y )  =  ( ( B  .x.  X )  .+  Z ) )   =>    |-  ( ph  ->  ( A  =  B  /\  Y  =  Z )
 )
 
Theoremlvecindp2 16166 Sums of independent vectors must have equal coefficients. (Contributed by NM, 22-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  Y  e.  ( V  \  {  .0.  } ) )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  C  e.  K )   &    |-  ( ph  ->  D  e.  K )   &    |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y }
 ) )   &    |-  ( ph  ->  ( ( A  .x.  X )  .+  ( B  .x.  Y ) )  =  ( ( C  .x.  X )  .+  ( D  .x.  Y ) ) )   =>    |-  ( ph  ->  ( A  =  C  /\  B  =  D )
 )
 
Theoremlspsnsubn0 16167 Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .-  =  ( -g `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   =>    |-  ( ph  ->  ( X  .-  Y )  =/= 
 .0.  )
 
Theoremlsmcv 16168 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 23107 analog.) TODO: ugly proof; can it be shortened? (Contributed by NM, 2-Oct-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  T  e.  S )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ( ph  /\  T  C.  U  /\  U  C_  ( T  .(+)  ( N `
  { X }
 ) ) )  ->  U  =  ( T  .(+) 
 ( N `  { X } ) ) )
 
Theoremlspsolvlem 16169* Lemma for lspsolv 16170. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  F  =  (Scalar `  W )   &    |-  B  =  ( Base `  F )   &    |-  .+  =  ( +g  `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  Q  =  { z  e.  V  |  E. r  e.  B  ( z  .+  ( r  .x.  Y ) )  e.  ( N `
  A ) }   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  C_  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  e.  ( N `  ( A  u.  { Y } ) ) )   =>    |-  ( ph  ->  E. r  e.  B  ( X  .+  ( r  .x.  Y ) )  e.  ( N `
  A ) )
 
Theoremlspsolv 16170 If  X is in the span of  A  u.  { Y } but not  A, then  Y is in the span of  A  u.  { X }. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  (
 ( W  e.  LVec  /\  ( A  C_  V  /\  Y  e.  V  /\  X  e.  ( ( N `  ( A  u.  { Y } ) ) 
 \  ( N `  A ) ) ) )  ->  Y  e.  ( N `  ( A  u.  { X }
 ) ) )
 
Theoremlssacsex 16171* In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 15998 by lspsolv 16170. (Contributed by David Moews, 1-May-2017.)
 |-  A  =  ( LSubSp `  W )   &    |-  N  =  (mrCls `  A )   &    |-  X  =  (
 Base `  W )   =>    |-  ( W  e.  LVec 
 ->  ( A  e.  (ACS `  X )  /\  A. s  e.  ~P  X A. y  e.  X  A. z  e.  ( ( N `  ( s  u.  { y } ) )  \  ( N `  s ) ) y  e.  ( N `  ( s  u. 
 { z } )
 ) ) )
 
Theoremlspsnat 16172 There is no subspace strictly between the zero subspace and the span of a vector (i.e. a 1-dimensional subspace is an atom). (h1datomi 23036 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( ( W  e.  LVec  /\  U  e.  S  /\  X  e.  V )  /\  U  C_  ( N `  { X } ) )  ->  ( U  =  ( N `  { X }
 )  \/  U  =  {  .0.  } ) )
 
Theoremlspsncv0 16173* The span of a singleton covers the zero subspace, using Definition 3.2.18 of [PtakPulmannova] p. 68 for "covers".) (Contributed by NM, 12-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  X  =/=  .0.  )   =>    |-  ( ph  ->  -. 
 E. y  e.  S  ( {  .0.  }  C.  y  /\  y  C.  ( N `  { X }
 ) ) )
 
Theoremlsppratlem1 16174 Lemma for lspprat 16180. Let  x  e.  ( U  \  { 0 } ) (if there is no such  x then  U is the zero subspace), and let  y  e.  ( U  \  ( N `
 { x }
) ) (assuming the conclusion is false). The goal is to write  X,  Y in terms of  x,  y, which would normally be done by solving the system of linear equations. The span equivalent of this process is lspsolv 16170 (hence the name), which we use extensively below. In this lemma, we show that since  x  e.  ( N `  { X ,  Y } ), either  x  e.  ( N `  { Y } ) or  X  e.  ( N `  { x ,  Y } ). (Contributed by NM, 29-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   &    |-  .0.  =  ( 0g `  W )   &    |-  ( ph  ->  x  e.  ( U  \  {  .0.  } ) )   &    |-  ( ph  ->  y  e.  ( U  \  ( N `
  { x }
 ) ) )   =>    |-  ( ph  ->  ( x  e.  ( N `
  { Y }
 )  \/  X  e.  ( N `  { x ,  Y } ) ) )
 
Theoremlsppratlem2 16175 Lemma for lspprat 16180. Show that if  X and 
Y are both in  ( N `  { x ,  y } ) (which will be our goal for each of the two cases above), then  ( N `  { X ,  Y }
)  C_  U, contradicting the hypothesis for  U. (Contributed by NM, 29-Aug-2014.) (Revised by Mario Carneiro, 5-Sep-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   &    |-  .0.  =  ( 0g `  W )   &    |-  ( ph  ->  x  e.  ( U  \  {  .0.  } ) )   &    |-  ( ph  ->  y  e.  ( U  \  ( N `
  { x }
 ) ) )   &    |-  ( ph  ->  X  e.  ( N `  { x ,  y } ) )   &    |-  ( ph  ->  Y  e.  ( N `  { x ,  y } ) )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  C_  U )
 
Theoremlsppratlem3 16176 Lemma for lspprat 16180. In the first case of lsppratlem1 16174, since  x  e/  ( N `  (/) ), also  Y  e.  ( N `  {
x } ), and since  y  e.  ( N `  { X ,  Y } )  C_  ( N `  { X ,  x } ) and  y  e/  ( N `  { x } ), we have  X  e.  ( N `  { x ,  y } ) as desired. (Contributed by NM, 29-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   &    |-  .0.  =  ( 0g `  W )   &    |-  ( ph  ->  x  e.  ( U  \  {  .0.  } ) )   &    |-  ( ph  ->  y  e.  ( U  \  ( N `
  { x }
 ) ) )   &    |-  ( ph  ->  x  e.  ( N `  { Y }
 ) )   =>    |-  ( ph  ->  ( X  e.  ( N ` 
 { x ,  y } )  /\  Y  e.  ( N `  { x ,  y } ) ) )
 
Theoremlsppratlem4 16177 Lemma for lspprat 16180. In the second case of lsppratlem1 16174,  y  e.  ( N `  { X ,  Y } )  C_  ( N `  { x ,  Y } ) and  y  e/  ( N `  { x } ) implies  Y  e.  ( N `  { x ,  y } ) and thus  X  e.  ( N `  { x ,  Y } )  C_  ( N `  { x ,  y } ) as well. (Contributed by NM, 29-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   &    |-  .0.  =  ( 0g `  W )   &    |-  ( ph  ->  x  e.  ( U  \  {  .0.  } ) )   &    |-  ( ph  ->  y  e.  ( U  \  ( N `
  { x }
 ) ) )   &    |-  ( ph  ->  X  e.  ( N `  { x ,  Y } ) )   =>    |-  ( ph  ->  ( X  e.  ( N `
  { x ,  y } )  /\  Y  e.  ( N `  { x ,  y } ) ) )
 
Theoremlsppratlem5 16178 Lemma for lspprat 16180. Combine the two cases and show a contradiction to  U  C.  ( N `  { X ,  Y } ) under the assumptions on  x and  y. (Contributed by NM, 29-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   &    |-  .0.  =  ( 0g `  W )   &    |-  ( ph  ->  x  e.  ( U  \  {  .0.  } ) )   &    |-  ( ph  ->  y  e.  ( U  \  ( N `
  { x }
 ) ) )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  C_  U )
 
Theoremlsppratlem6 16179 Lemma for lspprat 16180. Negating the assumption on  y, we arrive close to the desired conclusion. (Contributed by NM, 29-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( ph  ->  ( x  e.  ( U  \  {  .0.  } )  ->  U  =  ( N `
  { x }
 ) ) )
 
Theoremlspprat 16180* A proper subspace of the span of a pair of vectors is the span of a singleton (an atom) or the zero subspace (if  z is zero). Proof suggested by Mario Carneiro, 28-Aug-2014. (Contributed by NM, 29-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  U  C.  ( N `  { X ,  Y } ) )   =>    |-  ( ph  ->  E. z  e.  V  U  =  ( N `  { z } ) )
 
Theoremislbs2 16181* An equivalent formulation of the basis predicate in a vector space: a subset is a basis iff no element is in the span of the rest of the set. (Contributed by Mario Carneiro, 14-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( W  e.  LVec 
 ->  ( B  e.  J  <->  ( B  C_  V  /\  ( N `  B )  =  V  /\  A. x  e.  B  -.  x  e.  ( N `  ( B  \  { x } ) ) ) ) )
 
Theoremislbs3 16182* An equivalent formulation of the basis predicate: a subset is a basis iff it is a minimal spanning set. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( W  e.  LVec 
 ->  ( B  e.  J  <->  ( B  C_  V  /\  ( N `  B )  =  V  /\  A. s ( s  C.  B  ->  ( N `  s )  C.  V ) ) ) )
 
Theoremlbsacsbs 16183 Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 16181. (Contributed by David Moews, 1-May-2017.)
 |-  A  =  ( LSubSp `  W )   &    |-  N  =  (mrCls `  A )   &    |-  X  =  (
 Base `  W )   &    |-  I  =  (mrInd `  A )   &    |-  J  =  (LBasis `  W )   =>    |-  ( W  e.  LVec  ->  ( S  e.  J  <->  ( S  e.  I  /\  ( N `  S )  =  X ) ) )
 
Theoremlvecdim 16184 The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 16171 and lbsacsbs 16183 to show that being a basis for a vector space is equivalent to being a basis for the associated algebraic closure system, and then using acsexdimd 14564. (Contributed by David Moews, 1-May-2017.)
 |-  J  =  (LBasis `  W )   =>    |-  ( ( W  e.  LVec  /\  S  e.  J  /\  T  e.  J )  ->  S  ~~  T )
 
Theoremlbsextlem1 16185* Lemma for lbsext 16190. The set  S is the set of all linearly independent sets containing 
C; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  C  C_  V )   &    |-  ( ph  ->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x }
 ) ) )   &    |-  S  =  { z  e.  ~P V  |  ( C  C_  z  /\  A. x  e.  z  -.  x  e.  ( N `  (
 z  \  { x } ) ) ) }   =>    |-  ( ph  ->  S  =/= 
 (/) )
 
Theoremlbsextlem2 16186* Lemma for lbsext 16190. Since  A is a chain (actually, we only need it to be closed under binary union), the union  T of the spans of each individual element of 
A is a subspace, and it contains all of  U. A (except for our target vector  x- we are trying to make  x a linear combination of all the other vectors in some set from  A). (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  C  C_  V )   &    |-  ( ph  ->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x }
 ) ) )   &    |-  S  =  { z  e.  ~P V  |  ( C  C_  z  /\  A. x  e.  z  -.  x  e.  ( N `  (
 z  \  { x } ) ) ) }   &    |-  P  =  (
 LSubSp `  W )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  -> [ C.] 
 Or  A )   &    |-  T  =  U_ u  e.  A  ( N `  ( u 
 \  { x }
 ) )   =>    |-  ( ph  ->  ( T  e.  P  /\  ( U. A  \  { x } )  C_  T ) )
 
Theoremlbsextlem3 16187* Lemma for lbsext 16190. A chain in  S has an upper bound in  S. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  C  C_  V )   &    |-  ( ph  ->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x }
 ) ) )   &    |-  S  =  { z  e.  ~P V  |  ( C  C_  z  /\  A. x  e.  z  -.  x  e.  ( N `  (
 z  \  { x } ) ) ) }   &    |-  P  =  (
 LSubSp `  W )   &    |-  ( ph  ->  A  C_  S )   &    |-  ( ph  ->  A  =/= 
 (/) )   &    |-  ( ph  -> [ C.] 
 Or  A )   &    |-  T  =  U_ u  e.  A  ( N `  ( u 
 \  { x }
 ) )   =>    |-  ( ph  ->  U. A  e.  S )
 
Theoremlbsextlem4 16188* Lemma for lbsext 16190. lbsextlem3 16187 satisfies the conditions for the application of Zorn's lemma zorn 8343 (thus invoking AC), and so there is a maximal linearly independent set extending  C. Here we prove that such a set is a basis. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  J  =  (LBasis `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LVec )   &    |-  ( ph  ->  C  C_  V )   &    |-  ( ph  ->  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x }
 ) ) )   &    |-  S  =  { z  e.  ~P V  |  ( C  C_  z  /\  A. x  e.  z  -.  x  e.  ( N `  (
 z  \  { x } ) ) ) }   &    |-  ( ph  ->  ~P V  e.  dom  card )   =>    |-  ( ph  ->  E. s  e.  J  C  C_  s
 )
 
Theoremlbsextg 16189* For any linearly independent subset 
C of  V, there is a basis containing the vectors in 
C. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  J  =  (LBasis `  W )   &    |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( ( W  e.  LVec  /\  ~P V  e.  dom  card )  /\  C  C_  V  /\  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x } ) ) ) 
 ->  E. s  e.  J  C  C_  s )
 
Theoremlbsext 16190* For any linearly independent subset 
C of  V, there is a basis containing the vectors in 
C. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 17-May-2015.)
 |-  J  =  (LBasis `  W )   &    |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LVec  /\  C  C_  V  /\  A. x  e.  C  -.  x  e.  ( N `  ( C  \  { x }
 ) ) )  ->  E. s  e.  J  C  C_  s )
 
Theoremlbsexg 16191 Every vector space has a basis. This theorem is an AC equivalent; this is the forward implication. (Contributed by Mario Carneiro, 17-May-2015.)
 |-  J  =  (LBasis `  W )   =>    |-  ( (CHOICE 
 /\  W  e.  LVec ) 
 ->  J  =/=  (/) )
 
Theoremlbsex 16192 Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014.)
 |-  J  =  (LBasis `  W )   =>    |-  ( W  e.  LVec  ->  J  =/=  (/) )
 
Theoremlvecprop2d 16193* If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 16194 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  F  =  (Scalar `  K )   &    |-  G  =  (Scalar `  L )   &    |-  ( ph  ->  P  =  ( Base `  F )
 )   &    |-  ( ph  ->  P  =  ( Base `  G )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  P )
 )  ->  ( x ( +g  `  F )
 y )  =  ( x ( +g  `  G ) y ) )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  P )
 )  ->  ( x ( .r `  F ) y )  =  ( x ( .r `  G ) y ) )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  K ) y )  =  ( x ( .s
 `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  LVec  <->  L  e.  LVec )
 )
 
Theoremlvecpropd 16194* If two structures have the same components (properties), one is a left vector space iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ph  ->  F  =  (Scalar `  K ) )   &    |-  ( ph  ->  F  =  (Scalar `  L ) )   &    |-  P  =  ( Base `  F )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  K ) y )  =  ( x ( .s
 `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  LVec  <->  L  e.  LVec )
 )
 
10.8  Ideals
 
10.8.1  The subring algebra; ideals
 
Syntaxcsra 16195 Extend class notation with the subring algebra generator.
 class subringAlg
 
Syntaxcrglmod 16196 Extend class notation with the left module induced by a ring over itself.
 class ringLMod
 
Syntaxclidl 16197 Ring left-ideal function.
 class LIdeal
 
Syntaxcrsp 16198 Ring span function.
 class RSpan
 
Definitiondf-sra 16199* Given any subring of a ring, we can construct a left-algebra by regarding the elements of the subring as scalars and the ring itself as a set of vectors. (Contributed by Mario Carneiro, 27-Nov-2014.)
 |- subringAlg  =  ( w  e.  _V  |->  ( s  e.  ~P ( Base `  w )  |->  ( ( w sSet  <. (Scalar `  ndx ) ,  ( ws  s ) >. ) sSet  <. ( .s `  ndx ) ,  ( .r `  w ) >. ) ) )
 
Definitiondf-rgmod 16200 Every ring can be viewed as a left module over itself. (Contributed by Stefan O'Rear, 6-Dec-2014.)
 |- ringLMod  =  ( w  e.  _V  |->  ( ( subringAlg  `  w ) `
  ( Base `  w ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32447
  Copyright terms: Public domain < Previous  Next >