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Theorem List for Metamath Proof Explorer - 15801-15900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremislmhm3 15801* Property of a module homomorphism, similar to ismhm 14433. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  K  =  (Scalar `  S )   &    |-  L  =  (Scalar `  T )   &    |-  B  =  ( Base `  K )   &    |-  E  =  (
 Base `  S )   &    |-  .x.  =  ( .s `  S )   &    |-  .X. 
 =  ( .s `  T )   =>    |-  ( ( S  e.  LMod  /\  T  e.  LMod )  ->  ( F  e.  ( S LMHom  T )  <->  ( F  e.  ( S  GrpHom  T ) 
 /\  L  =  K  /\  A. x  e.  B  A. y  e.  E  ( F `  ( x 
 .x.  y ) )  =  ( x  .X.  ( F `  y ) ) ) ) )
 
Theoremlmhmlem 15802 Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  K  =  (Scalar `  S )   &    |-  L  =  (Scalar `  T )   =>    |-  ( F  e.  ( S LMHom  T )  ->  (
 ( S  e.  LMod  /\  T  e.  LMod )  /\  ( F  e.  ( S  GrpHom  T )  /\  L  =  K )
 ) )
 
Theoremlmhmsca 15803 A homomorphism of left modules constrains both modules to the same ring of scalars. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  K  =  (Scalar `  S )   &    |-  L  =  (Scalar `  T )   =>    |-  ( F  e.  ( S LMHom  T )  ->  L  =  K )
 
Theoremlmghm 15804 A homomorphism of left modules is a homomorphism of groups. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  ( F  e.  ( S LMHom  T )  ->  F  e.  ( S  GrpHom  T ) )
 
Theoremlmhmlmod2 15805 A homomorphism of left modules has a left module as codomain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  ( F  e.  ( S LMHom  T )  ->  T  e.  LMod )
 
Theoremlmhmlmod1 15806 A homomorphism of left modules has a left module as domain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  ( F  e.  ( S LMHom  T )  ->  S  e.  LMod )
 
Theoremlmhmf 15807 A homomorphism of left modules is a function. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  (
 Base `  T )   =>    |-  ( F  e.  ( S LMHom  T )  ->  F : B --> C )
 
Theoremlmhmlin 15808 A homomorphism of left modules is 
K-linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  K  =  (Scalar `  S )   &    |-  B  =  ( Base `  K )   &    |-  E  =  (
 Base `  S )   &    |-  .x.  =  ( .s `  S )   &    |-  .X. 
 =  ( .s `  T )   =>    |-  ( ( F  e.  ( S LMHom  T )  /\  X  e.  B  /\  Y  e.  E )  ->  ( F `  ( X  .x.  Y ) )  =  ( X  .X.  ( F `  Y ) ) )
 
Theoremlmodvsinv 15809 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( inv g `  W )   &    |-  M  =  ( inv g `  F )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( ( M `  R )  .x.  X )  =  ( N `  ( R  .x.  X ) ) )
 
Theoremlmodvsinv2 15810 Multiplying a negated vector by a scalar. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( inv g `  W )   &    |-  K  =  (
 Base `  F )   =>    |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  B )  ->  ( R  .x.  ( N `  X ) )  =  ( N `
  ( R  .x.  X ) ) )
 
Theoremislmhm2 15811* A one-equation proof of linearity of a left module homomorphism, similar to df-lss 15706. (Contributed by Mario Carneiro, 7-Oct-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  (
 Base `  T )   &    |-  K  =  (Scalar `  S )   &    |-  L  =  (Scalar `  T )   &    |-  E  =  ( Base `  K )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   &    |-  .x.  =  ( .s `  S )   &    |-  .X. 
 =  ( .s `  T )   =>    |-  ( ( S  e.  LMod  /\  T  e.  LMod )  ->  ( F  e.  ( S LMHom  T )  <->  ( F : B
 --> C  /\  L  =  K  /\  A. x  e.  E  A. y  e.  B  A. z  e.  B  ( F `  ( ( x  .x.  y )  .+  z ) )  =  ( ( x  .X.  ( F `  y ) )  .+^  ( F `  z ) ) ) ) )
 
Theoremislmhmd 15812* Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
 |-  X  =  ( Base `  S )   &    |-  .x.  =  ( .s `  S )   &    |-  .X.  =  ( .s `  T )   &    |-  K  =  (Scalar `  S )   &    |-  J  =  (Scalar `  T )   &    |-  N  =  ( Base `  K )   &    |-  ( ph  ->  S  e.  LMod )   &    |-  ( ph  ->  T  e.  LMod )   &    |-  ( ph  ->  J  =  K )   &    |-  ( ph  ->  F  e.  ( S  GrpHom  T ) )   &    |-  ( ( ph  /\  ( x  e.  N  /\  y  e.  X )
 )  ->  ( F `  ( x  .x.  y
 ) )  =  ( x  .X.  ( F `  y ) ) )   =>    |-  ( ph  ->  F  e.  ( S LMHom  T ) )
 
Theorem0lmhm 15813 The constant zero linear function between two modules. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- 
 .0.  =  ( 0g `  N )   &    |-  B  =  (
 Base `  M )   &    |-  S  =  (Scalar `  M )   &    |-  T  =  (Scalar `  N )   =>    |-  (
 ( M  e.  LMod  /\  N  e.  LMod  /\  S  =  T )  ->  ( B  X.  {  .0.  }
 )  e.  ( M LMHom  N ) )
 
Theoremidlmhm 15814 The identity function on a module is linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  B  =  ( Base `  M )   =>    |-  ( M  e.  LMod  ->  (  _I  |`  B )  e.  ( M LMHom  M ) )
 
Theoreminvlmhm 15815 The negative function on a module is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  I  =  ( inv
 g `  M )   =>    |-  ( M  e.  LMod  ->  I  e.  ( M LMHom  M ) )
 
Theoremlmhmco 15816 The composition of two module-linear functions is module-linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  ( ( F  e.  ( N LMHom  O )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o.  G )  e.  ( M LMHom  O ) )
 
Theoremlmhmplusg 15817 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- 
 .+  =  ( +g  `  N )   =>    |-  ( ( F  e.  ( M LMHom  N )  /\  G  e.  ( M LMHom  N ) )  ->  ( F  o F  .+  G )  e.  ( M LMHom  N ) )
 
Theoremlmhmvsca 15818 The pointwise scalar product of a linear function and a constant is linear, over a commutative ring. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  V  =  ( Base `  M )   &    |-  .x.  =  ( .s `  N )   &    |-  J  =  (Scalar `  N )   &    |-  K  =  ( Base `  J )   =>    |-  (
 ( J  e.  CRing  /\  A  e.  K  /\  F  e.  ( M LMHom  N ) )  ->  (
 ( V  X.  { A } )  o F  .x.  F )  e.  ( M LMHom  N ) )
 
Theoremlmhmf1o 15819 A bijective module homomorphism is also converse homomorphic. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  X  =  ( Base `  S )   &    |-  Y  =  (
 Base `  T )   =>    |-  ( F  e.  ( S LMHom  T )  ->  ( F : X -1-1-onto-> Y  <->  `' F  e.  ( T LMHom  S ) ) )
 
Theoremlmhmima 15820 The image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  X  =  ( LSubSp `  S )   &    |-  Y  =  (
 LSubSp `  T )   =>    |-  ( ( F  e.  ( S LMHom  T )  /\  U  e.  X )  ->  ( F " U )  e.  Y )
 
Theoremlmhmpreima 15821 The inverse image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  X  =  ( LSubSp `  S )   &    |-  Y  =  (
 LSubSp `  T )   =>    |-  ( ( F  e.  ( S LMHom  T )  /\  U  e.  Y )  ->  ( `' F " U )  e.  X )
 
Theoremlmhmlsp 15822 Homomorphisms preserve spans. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  V  =  ( Base `  S )   &    |-  K  =  (
 LSpan `  S )   &    |-  L  =  ( LSpan `  T )   =>    |-  (
 ( F  e.  ( S LMHom  T )  /\  U  C_  V )  ->  ( F " ( K `  U ) )  =  ( L `  ( F " U ) ) )
 
Theoremlmhmrnlss 15823 The range of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  ( F  e.  ( S LMHom  T )  ->  ran  F  e.  ( LSubSp `  T )
 )
 
Theoremlmhmkerlss 15824 The kernel of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  K  =  ( `' F " {  .0.  } )   &    |-  .0.  =  ( 0g `  T )   &    |-  U  =  ( LSubSp `  S )   =>    |-  ( F  e.  ( S LMHom  T )  ->  K  e.  U )
 
Theoremreslmhm 15825 Restriction of a homomorphism to a subspace. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  U  =  ( LSubSp `  S )   &    |-  R  =  ( Ss  X )   =>    |-  ( ( F  e.  ( S LMHom  T )  /\  X  e.  U )  ->  ( F  |`  X )  e.  ( R LMHom  T ) )
 
Theoremreslmhm2 15826 Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  U  =  ( Ts  X )   &    |-  L  =  (
 LSubSp `  T )   =>    |-  ( ( F  e.  ( S LMHom  U )  /\  T  e.  LMod  /\  X  e.  L ) 
 ->  F  e.  ( S LMHom  T ) )
 
Theoremreslmhm2b 15827 Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  U  =  ( Ts  X )   &    |-  L  =  (
 LSubSp `  T )   =>    |-  ( ( T  e.  LMod  /\  X  e.  L  /\  ran  F  C_  X )  ->  ( F  e.  ( S LMHom  T )  <->  F  e.  ( S LMHom  U ) ) )
 
Theoremlmhmeql 15828 The equalizer of two module homomorphisms is a subspace. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  U  =  ( LSubSp `  S )   =>    |-  ( ( F  e.  ( S LMHom  T )  /\  G  e.  ( S LMHom  T ) )  ->  dom  ( F  i^i  G )  e.  U )
 
Theoremlspextmo 15829* A linear function is completely determined (or overdetermined) by its values on a spanning subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by NM, 17-Jun-2017.)
 |-  B  =  ( Base `  S )   &    |-  K  =  (
 LSpan `  S )   =>    |-  ( ( X 
 C_  B  /\  ( K `  X )  =  B )  ->  E* g  e.  ( S LMHom  T ) ( g  |`  X )  =  F )
 
Theorempwsdiaglmhm 15830* Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  R )   &    |-  F  =  ( x  e.  B  |->  ( I  X.  { x } ) )   =>    |-  ( ( R  e.  LMod  /\  I  e.  W )  ->  F  e.  ( R LMHom  Y ) )
 
Theoremislmim 15831 An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  C  =  (
 Base `  S )   =>    |-  ( F  e.  ( R LMIso  S )  <->  ( F  e.  ( R LMHom  S )  /\  F : B -1-1-onto-> C ) )
 
Theoremlmimf1o 15832 An isomorphism of left modules is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  C  =  (
 Base `  S )   =>    |-  ( F  e.  ( R LMIso  S )  ->  F : B -1-1-onto-> C )
 
Theoremlmimlmhm 15833 An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  ( F  e.  ( R LMIso  S )  ->  F  e.  ( R LMHom  S ) )
 
Theoremlmimgim 15834 An isomorphism of modules is an isomorphism of groups. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( F  e.  ( R LMIso  S )  ->  F  e.  ( R GrpIso  S )
 )
 
Theoremislmim2 15835 An isomorphism of left modules is a homomorphism whose converse is a homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( F  e.  ( R LMIso  S )  <->  ( F  e.  ( R LMHom  S )  /\  `' F  e.  ( S LMHom  R ) ) )
 
Theoremlmimcnv 15836 The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( F  e.  ( S LMIso  T )  ->  `' F  e.  ( T LMIso  S ) )
 
Theorembrlmic 15837 The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑚 
 S 
 <->  ( R LMIso  S )  =/=  (/) )
 
Theorembrlmici 15838 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( F  e.  ( R LMIso  S )  ->  R  ~=ph𝑚  S )
 
Theoremlmiclcl 15839 Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑚 
 S  ->  R  e.  LMod
 )
 
Theoremlmicrcl 15840 Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( R  ~=ph𝑚 
 S  ->  S  e.  LMod
 )
 
Theoremlmicsym 15841 Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  ( R  ~=ph𝑚 
 S  ->  S  ~=ph𝑚  R )
 
Theoremlmhmpropd 15842* Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  J )
 )   &    |-  ( ph  ->  C  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  C  =  ( Base `  M )
 )   &    |-  ( ph  ->  F  =  (Scalar `  J )
 )   &    |-  ( ph  ->  G  =  (Scalar `  K )
 )   &    |-  ( ph  ->  F  =  (Scalar `  L )
 )   &    |-  ( ph  ->  G  =  (Scalar `  M )
 )   &    |-  P  =  ( Base `  F )   &    |-  Q  =  (
 Base `  G )   &    |-  (
 ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( x ( +g  `  J ) y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  C ) )  ->  ( x ( +g  `  K ) y )  =  ( x ( +g  `  M ) y ) )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  B ) )  ->  ( x ( .s `  J ) y )  =  ( x ( .s
 `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  Q  /\  y  e.  C ) )  ->  ( x ( .s `  K ) y )  =  ( x ( .s
 `  M ) y ) )   =>    |-  ( ph  ->  ( J LMHom  K )  =  ( L LMHom  M ) )
 
10.6.4  Subspace sum; bases for a left module
 
Syntaxclbs 15843 Extend class notation with the set of bases for a vector space.
 class LBasis
 
Definitiondf-lbs 15844* 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 15845* 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 15846 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 15847 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 15848 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 15849 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 15850 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 15851 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 15852 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 15853* 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 15854* 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 15855 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 15856 Subspace sum of spans of subsets is the span of their union. (spanuni 22139 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 15857 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 15858 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 15859 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 15860 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 15861 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 15862* 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 15863* 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 15864 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 15865 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 15866 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 15867 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 15868* 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 15869 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 15870 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 15871 Extend class notation with class of all left vector spaces.
 class  LVec
 
Definitiondf-lvec 15872 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 15873 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 15874 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 15875 A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)
 |-  ( W  e.  LVec  ->  W  e.  LMod )
 
Theoremlsslvec 15876 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 15877 If a scalar product is zero, one of its factors must be zero. (hvmul0or 21620 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 15878 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 15879 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 15880 Cancellation law for scalar multiplication. (hvmulcan 21667 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 15881 Cancellation law for scalar multiplication. (hvmulcan2 21668 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 15882 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 15883 A non-zero scalar product does not change the span of a singleton. (spansncol 22163 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 15884 Membership relation that implies equality of spans. (spansneleq 22165 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 15885 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 15886 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 15887 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 15888 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 15889 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 15890 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 15891* Equal spans of singletons must have proportional vectors. See lspsnss2 15778 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 15892* 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 15893 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 22168 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 15894 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 15895 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 15896 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 15897* 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 15898 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 15899 vs. lspexchn2 15900); look for lspexch 15898 and prcom 3718 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 15899 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 15898 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 15900 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 15898 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 } ) )
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