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Theorem List for Metamath Proof Explorer - 15501-15600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlmodvnegid 15501 Addition of a vector with its negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( inv
 g `  W )   =>    |-  (
 ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( X  .+  ( N `  X ) )  =  .0.  )
 
Theoremlmodvneg1 15502 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( inv g `  W )   &    |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  .1.  =  ( 1r `  F )   &    |-  M  =  ( inv g `
  F )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( ( M `  .1.  )  .x.  X )  =  ( N `  X ) )
 
TheoremlmodvsnegOLD 15503 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  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 ) ) )
 
Theoremlmodvsneg 15504 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 )   &    |-  K  =  (
 Base `  F )   &    |-  M  =  ( inv g `  F )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  R  e.  K )   =>    |-  ( ph  ->  ( N `  ( R  .x.  X ) )  =  ( ( M `  R )  .x.  X ) )
 
Theoremlmodvsubcl 15505 Closure of vector subtraction. (hvsubcl 21427 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .-  Y )  e.  V )
 
Theoremlmodcom 15506 Left module vector sum is commutative. (Contributed by Gérard Lang, 25-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremlmodabl 15507 A left module is an abelian group (of vectors, under addition). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)
 |-  ( W  e.  LMod  ->  W  e.  Abel )
 
Theoremlmodcmn 15508 A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
 |-  ( W  e.  LMod  ->  W  e. CMnd )
 
Theoremlmodnegadd 15509 Distribute negation through addition of scalar products. (Contributed by NM, 9-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( inv g `
  W )   &    |-  R  =  (Scalar `  W )   &    |-  K  =  ( Base `  R )   &    |-  I  =  ( inv g `  R )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( N `  ( ( A  .x.  X )  .+  ( B  .x.  Y ) ) )  =  ( ( ( I `
  A )  .x.  X )  .+  ( ( I `  B ) 
 .x.  Y ) ) )
 
Theoremlmod4 15510 Commutative/associative law for left module vector sum. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   =>    |-  ( ( W  e.  LMod  /\  ( X  e.  V  /\  Y  e.  V )  /\  ( Z  e.  V  /\  U  e.  V )
 )  ->  ( ( X  .+  Y )  .+  ( Z  .+  U ) )  =  ( ( X  .+  Z ) 
 .+  ( Y  .+  U ) ) )
 
Theoremlmodvsubadd 15511 Relationship between vector subtraction and addition. (hvsubadd 21486 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( ( A  .-  B )  =  C  <->  ( B  .+  C )  =  A ) )
 
Theoremlmodvaddsub4 15512 Vector addition/subtraction law. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) )  ->  ( ( A  .+  B )  =  ( C  .+  D )  <->  ( A  .-  C )  =  ( D  .-  B ) ) )
 
Theoremlmodvpncan 15513 Addition/subtraction cancellation law for vectors. (hvpncan 21448 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .+  B )  .-  B )  =  A )
 
Theoremlmodvnpcan 15514 Cancellation law for vector subtraction (npcan 8940 analog). (Contributed by NM, 19-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .-  B )  .+  B )  =  A )
 
Theoremlmodvsubval2 15515 Value of vector subtraction in terms of addition. (hvsubval 21426 analog.) (Contributed by NM, 31-Mar-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   &    |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  N  =  ( inv g `  F )   &    |- 
 .1.  =  ( 1r `  F )   =>    |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .-  B )  =  ( A  .+  ( ( N `  .1.  )  .x.  B )
 ) )
 
Theoremlmodsubvs 15516 Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .-  =  ( -g `  W )   &    |-  .x. 
 =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  (
 Base `  F )   &    |-  N  =  ( inv g `  F )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( X  .-  ( A  .x.  Y ) )  =  ( X  .+  ( ( N `  A ) 
 .x.  Y ) ) )
 
Theoremlmodsubdi 15517 Scalar multiplication distributive law for subtraction. (hvsubdistr1 21458 analog, with longer proof since our scalar multiplication is not commutative.) (Contributed by NM, 2-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .-  =  ( -g `  W )   &    |-  ( ph  ->  W  e.  LMod
 )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( A  .x.  ( X  .-  Y ) )  =  ( ( A  .x.  X )  .-  ( A  .x.  Y ) ) )
 
Theoremlmodsubdir 15518 Scalar multiplication distributive law for subtraction. (hvsubdistr2 21459 analog.) (Contributed by NM, 2-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .-  =  ( -g `  W )   &    |-  S  =  ( -g `  F )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  (
 ( A S B )  .x.  X )  =  ( ( A  .x.  X )  .-  ( B  .x.  X ) ) )
 
Theoremlmodsubeq0 15519 If the difference between two vectors is zero, they are equal. (hvsubeq0 21477 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( ( A  .-  B )  =  .0.  <->  A  =  B ) )
 
Theoremlmodsubid 15520 Subtraction of a vector from itself. (hvsubid 21435 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  .-  =  ( -g `  W )   =>    |-  ( ( W  e.  LMod  /\  A  e.  V ) 
 ->  ( A  .-  A )  =  .0.  )
 
Theoremlmodvsghm 15521* Scalar multiplication of the vector space by a fixed scalar is an automorphism of the addiive group of vectors. (Contributed by Mario Carneiro, 5-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   =>    |-  (
 ( W  e.  LMod  /\  R  e.  K ) 
 ->  ( x  e.  V  |->  ( R  .x.  x ) )  e.  ( W 
 GrpHom  W ) )
 
Theoremlmodprop2d 15522* If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 15523 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.  LMod  <->  L  e.  LMod )
 )
 
Theoremlmodpropd 15523* If two structures have the same components (properties), one is a left module iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised 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.  LMod  <->  L  e.  LMod )
 )
 
10.6.2  Subspaces and spans in a left module
 
Syntaxclss 15524 Extend class notation with linear subspaces of a left module or left vector space.
 class  LSubSp
 
Definitiondf-lss 15525* Define the set of linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
 |-  LSubSp  =  ( w  e. 
 _V  |->  { s  e.  ( ~P ( Base `  w )  \  { (/) } )  | 
 A. x  e.  ( Base `  (Scalar `  w ) ) A. a  e.  s  A. b  e.  s  ( ( x ( .s `  w ) a ) (
 +g  `  w )
 b )  e.  s } )
 
Theoremlssset 15526* The set of all (not necessarily closed) linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 15-Jul-2014.)
 |-  F  =  (Scalar `  W )   &    |-  B  =  ( Base `  F )   &    |-  V  =  (
 Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  ( W  e.  X  ->  S  =  { s  e.  ( ~P V  \  { (/) } )  | 
 A. x  e.  B  A. a  e.  s  A. b  e.  s  (
 ( x  .x.  a
 )  .+  b )  e.  s } )
 
Theoremislss 15527* The predicate "is a subspace" (of a left module or left vector space). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  F  =  (Scalar `  W )   &    |-  B  =  ( Base `  F )   &    |-  V  =  (
 Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  ( U  e.  S  <->  ( U  C_  V  /\  U  =/=  (/)  /\  A. x  e.  B  A. a  e.  U  A. b  e.  U  ( ( x 
 .x.  a )  .+  b )  e.  U ) )
 
Theoremislssd 15528* Properties that determine a subspace of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  ( ph  ->  F  =  (Scalar `  W )
 )   &    |-  ( ph  ->  B  =  ( Base `  F )
 )   &    |-  ( ph  ->  V  =  ( Base `  W )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  W )
 )   &    |-  ( ph  ->  .x.  =  ( .s `  W ) )   &    |-  ( ph  ->  S  =  ( LSubSp `  W ) )   &    |-  ( ph  ->  U 
 C_  V )   &    |-  ( ph  ->  U  =/=  (/) )   &    |-  (
 ( ph  /\  ( x  e.  B  /\  a  e.  U  /\  b  e.  U ) )  ->  ( ( x  .x.  a )  .+  b )  e.  U )   =>    |-  ( ph  ->  U  e.  S )
 
Theoremlssss 15529 A subspace is a set of vectors. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( U  e.  S  ->  U  C_  V )
 
Theoremlssel 15530 A subspace member is a vector. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( U  e.  S  /\  X  e.  U )  ->  X  e.  V )
 
Theoremlss1 15531 The set of vectors in a left module is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( W  e.  LMod 
 ->  V  e.  S )
 
Theoremlssuni 15532 The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   =>    |-  ( ph  ->  U. S  =  V )
 
Theoremlssn0 15533 A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( U  e.  S  ->  U  =/=  (/) )
 
Theorem00lss 15534 The empty structure has no subspaces (for use with fvco4i 5449). (Contributed by Stefan O'Rear, 31-Mar-2015.)
 |-  (/)  =  ( LSubSp `  (/) )
 
Theoremlsscl 15535 Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  F  =  (Scalar `  W )   &    |-  B  =  ( Base `  F )   &    |-  .+  =  ( +g  `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  ( ( U  e.  S  /\  ( Z  e.  B  /\  X  e.  U  /\  Y  e.  U ) )  ->  ( ( Z  .x.  X )  .+  Y )  e.  U )
 
Theoremlssvsubcl 15536 Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  .-  =  ( -g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( X  e.  U  /\  Y  e.  U )
 )  ->  ( X  .-  Y )  e.  U )
 
Theoremlssvancl1 15537 Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. TODO: notice similarity to lspindp3 15724. Can it be used along with lspsnne1 15705, lspsnne2 15706 to shorten this proof? (Contributed by NM, 14-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  -.  Y  e.  U )   =>    |-  ( ph  ->  -.  ( X  .+  Y )  e.  U )
 
Theoremlssvancl2 15538 Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. (Contributed by NM, 20-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  -.  Y  e.  U )   =>    |-  ( ph  ->  -.  ( Y  .+  X )  e.  U )
 
Theoremlss0cl 15539 The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  .0.  e.  U )
 
Theoremlsssn0 15540 The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( W  e.  LMod 
 ->  {  .0.  }  e.  S )
 
Theoremlss0ss 15541 The zero subspace is included in every subspace. (sh0le 21849 analog.) (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  S )  ->  {  .0.  } 
 C_  X )
 
Theoremlssle0 15542 No subspace is smaller than the zero subspace. (shle0 21851 analog.) (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  S )  ->  ( X 
 C_  {  .0.  }  <->  X  =  {  .0.  } ) )
 
Theoremlssne0 15543* A nonzero subspace has a nonzero vector. (shne0i 21857 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( X  e.  S  ->  ( X  =/=  {  .0.  }  <->  E. y  e.  X  y  =/=  .0.  ) )
 
Theoremlssneln0 15544 A vector which doesn't belong to a subspace is nonzero. (Contributed by NM, 14-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  -.  X  e.  U )   =>    |-  ( ph  ->  X  e.  ( V  \  {  .0.  } ) )
 
Theoremlssssr 15545* Conclude subspace ordering from nonzero vector membership. (ssrdv 3106 analog.) (Contributed by NM, 17-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  S  =  ( LSubSp `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  T 
 C_  V )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ( ph  /\  x  e.  ( V  \  {  .0.  } ) )  ->  ( x  e.  T  ->  x  e.  U ) )   =>    |-  ( ph  ->  T  C_  U )
 
Theoremlssvacl 15546 Closure of vector addition in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |- 
 .+  =  ( +g  `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  ( X  e.  U  /\  Y  e.  U )
 )  ->  ( X  .+  Y )  e.  U )
 
Theoremlssvscl 15547 Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   &    |-  B  =  (
 Base `  F )   &    |-  S  =  ( LSubSp `  W )   =>    |-  (
 ( ( W  e.  LMod  /\  U  e.  S ) 
 /\  ( X  e.  B  /\  Y  e.  U ) )  ->  ( X 
 .x.  Y )  e.  U )
 
Theoremlssvnegcl 15548 Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  ( inv g `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  X )  e.  U )
 
Theoremlsssubg 15549 All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S ) 
 ->  U  e.  (SubGrp `  W ) )
 
Theoremlsssssubg 15550 All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( W  e.  LMod  ->  S  C_  (SubGrp `  W ) )
 
Theoremislss3 15551 A linear subspace of a module is a subset which is a module in its own right. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  X  =  ( Ws  U )   &    |-  V  =  (
 Base `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  ( W  e.  LMod  ->  ( U  e.  S  <->  ( U  C_  V  /\  X  e.  LMod ) ) )
 
Theoremlsslmod 15552 A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod
 )
 
Theoremlsslss 15553 The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   &    |-  T  =  ( LSubSp `  X )   =>    |-  (
 ( W  e.  LMod  /\  U  e.  S ) 
 ->  ( V  e.  T  <->  ( V  e.  S  /\  V  C_  U ) ) )
 
Theoremislss4 15554* A linear subspace is a subgroup which respects scalar multiplication. (Contributed by Stefan O'Rear, 11-Dec-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  F  =  (Scalar `  W )   &    |-  B  =  ( Base `  F )   &    |-  V  =  (
 Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  S  =  ( LSubSp `  W )   =>    |-  ( W  e.  LMod  ->  ( U  e.  S  <->  ( U  e.  (SubGrp `  W )  /\  A. a  e.  B  A. b  e.  U  ( a  .x.  b )  e.  U ) ) )
 
Theoremlss1d 15555* One-dimensional subspace (or zero-dimensional if  X is the zero vector). (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  S  =  ( LSubSp `  W )   =>    |-  (
 ( W  e.  LMod  /\  X  e.  V ) 
 ->  { v  |  E. k  e.  K  v  =  ( k  .x.  X ) }  e.  S )
 
Theoremlssintcl 15556 The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  A  C_  S  /\  A  =/=  (/) )  ->  |^| A  e.  S )
 
Theoremlssincl 15557 The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  S  =  ( LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  T  e.  S  /\  U  e.  S )  ->  ( T  i^i  U )  e.  S )
 
Theoremlssmre 15558 The subspaces of a module comprise a Moore system on the vectors of the module. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  B  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( W  e.  LMod 
 ->  S  e.  (Moore `  B ) )
 
Theoremlssacs 15559 Submodules are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  B  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   =>    |-  ( W  e.  LMod 
 ->  S  e.  (ACS `  B ) )
 
Theoremprdsvscacl 15560* Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  .x.  =  ( .s `  Y )   &    |-  K  =  ( Base `  S )   &    |-  ( ph  ->  S  e.  Ring )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R : I --> LMod )   &    |-  ( ph  ->  F  e.  K )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ( ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x ) )  =  S )   =>    |-  ( ph  ->  ( F  .x.  G )  e.  B )
 
Theoremprdslmodd 15561* The product of a family of left modules is a left module. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  S  e.  Ring )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R : I --> LMod )   &    |-  ( ( ph  /\  y  e.  I ) 
 ->  (Scalar `  ( R `  y ) )  =  S )   =>    |-  ( ph  ->  Y  e.  LMod )
 
Theorempwslmod 15562 The product of a family of left modules is a left module. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  LMod  /\  I  e.  V )  ->  Y  e.  LMod )
 
Syntaxclspn 15563 Extend class notation with span of a set of vectors.
 class  LSpan
 
Definitiondf-lsp 15564* Define span of a set of vectors of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
 |- 
 LSpan  =  ( w  e.  _V  |->  ( s  e. 
 ~P ( Base `  w )  |->  |^| { t  e.  ( LSubSp `  w )  |  s  C_  t }
 ) )
 
Theoremlspfval 15565* The span function for a left vector space (or a left module). (df-span 21718 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( W  e.  X  ->  N  =  ( s  e. 
 ~P V  |->  |^| { t  e.  S  |  s  C_  t } ) )
 
Theoremlspf 15566 The span operator on a left module maps subsets to subsets. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( W  e.  LMod  ->  N : ~P V --> S )
 
Theoremlspval 15567* The span of a set of vectors (in a left module). (spanval 21742 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  (
 ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  =  |^| { t  e.  S  |  U  C_  t } )
 
Theoremlspcl 15568 The span of a set of vectors is a subspace. (spancl 21745 analog.) (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  (
 ( W  e.  LMod  /\  U  C_  V )  ->  ( N `  U )  e.  S )
 
Theoremlspsncl 15569 The span of a singleton is a subspace (frequently used special case of lspcl 15568). (Contributed by NM, 17-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  (
 ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( N `  { X } )  e.  S )
 
Theoremlspprcl 15570 The span of a pair is a subspace (frequently used special case of lspcl 15568). (Contributed by NM, 11-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  e.  S )
 
Theoremlsptpcl 15571 The span of an unordered triple is a subspace (frequently used special case of lspcl 15568). (Contributed by NM, 22-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  Z  e.  V )   =>    |-  ( ph  ->  ( N `  { X ,  Y ,  Z }
 )  e.  S )
 
Theoremlspsnsubg 15572 The span of a singleton is an additive subgroup (frequently used special case of lspcl 15568). (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `
  { X }
 )  e.  (SubGrp `  W ) )
 
Theorem00lsp 15573 fvco4i 5449 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)
 |-  (/)  =  ( LSpan `  (/) )
 
Theoremlspid 15574 The span of a subspace is itself. (spanid 21756 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `
  U )  =  U )
 
Theoremlspssv 15575 A span is a set of vectors. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `
  U )  C_  V )
 
Theoremlspss 15576 Span preserves subset ordering. (spanss 21757 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  C_  V  /\  T  C_  U )  ->  ( N `  T )  C_  ( N `
  U ) )
 
Theoremlspssid 15577 A set of vectors is a subset of its span. (spanss2 21754 analog.) (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  U  C_  ( N `  U ) )
 
Theoremlspidm 15578 The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  C_  V )  ->  ( N `
  ( N `  U ) )  =  ( N `  U ) )
 
Theoremlspun 15579 The span of union is the span of the union of spans. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  T  C_  V  /\  U  C_  V )  ->  ( N `  ( T  u.  U ) )  =  ( N `  ( ( N `
  T )  u.  ( N `  U ) ) ) )
 
Theoremlspssp 15580 If a set of vectors is a subset of a subspace, then the span of those vectors is also contained in the subspace. (Contributed by Mario Carneiro, 4-Sep-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S  /\  T  C_  U )  ->  ( N `  T )  C_  U )
 
Theoremmrclsp 15581 Moore closure generalizes module span. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |-  U  =  ( LSubSp `  W )   &    |-  K  =  (
 LSpan `  W )   &    |-  F  =  (mrCls `  U )   =>    |-  ( W  e.  LMod  ->  K  =  F )
 
Theoremlspsnss 15582 The span of the singleton of a subspace member is included in the subspace. (spansnss 21980 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  S  /\  X  e.  U )  ->  ( N `  { X } )  C_  U )
 
Theoremlspsnel3 15583 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 21981 analog.) (Contributed by NM, 4-Jul-2014.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  ( N `  { X }
 ) )   =>    |-  ( ph  ->  Y  e.  U )
 
Theoremlspprss 15584 The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( N `  { X ,  Y } )  C_  U )
 
Theoremlspsnid 15585 A vector belongs to the span of its singleton. (spansnid 21972 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  X  e.  ( N `  { X } ) )
 
Theoremlspsnel6 15586 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  ( X  e.  U  <->  ( X  e.  V  /\  ( N `  { X } )  C_  U ) ) )
 
Theoremlspsnel5 15587 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( X  e.  U  <->  ( N `  { X } )  C_  U ) )
 
Theoremlspsnel5a 15588 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   &    |-  ( ph  ->  X  e.  U )   =>    |-  ( ph  ->  ( N `  { X } )  C_  U )
 
Theoremlspprid1 15589 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  X  e.  ( N `  { X ,  Y }
 ) )
 
Theoremlspprid2 15590 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  Y  e.  ( N `  { X ,  Y }
 ) )
 
Theoremlspprvacl 15591 The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  ( N `  { X ,  Y }
 ) )
 
Theoremlssats2 15592* A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.)
 |-  S  =  ( LSubSp `  W )   &    |-  N  =  (
 LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  U  e.  S )   =>    |-  ( ph  ->  U  =  U_ x  e.  U  ( N `  { x } ) )
 
Theoremlspsneli 15593 A scalar product with a vector belongs to the span of its singleton. (spansnmul 21973 analog.) (Contributed by NM, 2-Jul-2014.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  ( A  .x.  X )  e.  ( N `  { X } ) )
 
Theoremlspsn 15594* Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  V  =  (
 Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( N `  { X } )  =  {
 v  |  E. k  e.  K  v  =  ( k  .x.  X ) } )
 
Theoremlspsnel 15595* Member of span of the singleton of a vector. (elspansn 21975 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  V  =  (
 Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( U  e.  ( N `  { X }
 ) 
 <-> 
 E. k  e.  K  U  =  ( k  .x.  X ) ) )
 
Theoremlspsnvsi 15596 Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  V  =  (
 Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  R  e.  K  /\  X  e.  V )  ->  ( N `  { ( R  .x.  X ) }
 )  C_  ( N ` 
 { X } )
 )
 
Theoremlspsnss2 15597* Comparable spans of singletons must have proportional vectors. See lspsneq 15710 for equal span version. (Contributed by NM, 7-Jun-2015.)
 |-  V  =  ( Base `  W )   &    |-  S  =  (Scalar `  W )   &    |-  K  =  (
 Base `  S )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  (
 ( N `  { X } )  C_  ( N `
  { Y }
 ) 
 <-> 
 E. k  e.  K  X  =  ( k  .x.  Y ) ) )
 
Theoremlspsnneg 15598 Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  M  =  ( inv g `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( N `  { ( M `  X ) }
 )  =  ( N `
  { X }
 ) )
 
Theoremlspsnsub 15599 Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-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  .-  Y ) }
 )  =  ( N `
  { ( Y 
 .-  X ) }
 ) )
 
Theoremlspsn0 15600 Span of the singleton of the zero vector. (spansn0 21950 analog.) (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( W  e.  LMod 
 ->  ( N `  {  .0.  } )  =  {  .0.  } )
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