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Theorem List for Metamath Proof Explorer - 15601-15700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlsp0 15601 Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( W  e.  LMod 
 ->  ( N `  (/) )  =  {  .0.  } )
 
Theoremlspuni0 15602 Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.)
 |- 
 .0.  =  ( 0g `  W )   &    |-  N  =  (
 LSpan `  W )   =>    |-  ( W  e.  LMod 
 ->  U. ( N `  (/) )  =  .0.  )
 
Theoremlspun0 15603 The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X 
 C_  V )   =>    |-  ( ph  ->  ( N `  ( X  u.  {  .0.  }
 ) )  =  ( N `  X ) )
 
Theoremlspsneq0 15604 Span of the singleton is the zero subspace iff the vector is zero. (Contributed by NM, 27-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   =>    |-  ( ( W  e.  LMod  /\  X  e.  V ) 
 ->  ( ( N `  { X } )  =  {  .0.  }  <->  X  =  .0.  ) )
 
Theoremlspsneq0b 15605 Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.)
 |-  V  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( N `  { X }
 )  =  ( N `
  { Y }
 ) )   =>    |-  ( ph  ->  ( X  =  .0.  <->  Y  =  .0.  ) )
 
Theoremlmodindp1 15606 Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.)
 |-  V  =  ( Base `  W )   &    |-  .+  =  ( +g  `  W )   &    |-  .0.  =  ( 0g `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  ( N `  { X }
 )  =/=  ( N ` 
 { Y } )
 )   =>    |-  ( ph  ->  ( X  .+  Y )  =/= 
 .0.  )
 
Theoremlsslsp 15607 Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) TODO: Shouldn't we swap  M `  G and  N `
 G since we are computing a property of  N `  G? (Like we say sin 0 = 0 and not 0 = sin 0.) - NM 15-Mar-2015.
 |-  X  =  ( Ws  U )   &    |-  M  =  (
 LSpan `  W )   &    |-  N  =  ( LSpan `  X )   &    |-  L  =  ( LSubSp `  W )   =>    |-  (
 ( W  e.  LMod  /\  U  e.  L  /\  G  C_  U )  ->  ( M `  G )  =  ( N `  G ) )
 
Theoremlss0v 15608 The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.)
 |-  X  =  ( Ws  U )   &    |-  .0.  =  ( 0g `  W )   &    |-  Z  =  ( 0g `  X )   &    |-  L  =  (
 LSubSp `  W )   =>    |-  ( ( W  e.  LMod  /\  U  e.  L )  ->  Z  =  .0.  )
 
Theoremlsspropd 15609* If two structures have the same components (properties), they have the same subspace structure. (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 ) )   &    |-  ( ph  ->  P  =  ( Base `  (Scalar `  K ) ) )   &    |-  ( ph  ->  P  =  ( Base `  (Scalar `  L ) ) )   =>    |-  ( ph  ->  (
 LSubSp `  K )  =  ( LSubSp `  L )
 )
 
Theoremlsppropd 15610* If two structures have the same components (properties), they have the same span function. (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 ) )   &    |-  ( ph  ->  P  =  ( Base `  (Scalar `  K ) ) )   &    |-  ( ph  ->  P  =  ( Base `  (Scalar `  L ) ) )   &    |-  ( ph  ->  K  e.  _V )   &    |-  ( ph  ->  L  e.  _V )   =>    |-  ( ph  ->  ( LSpan `  K )  =  ( LSpan `  L )
 )
 
10.6.3  Homomorphisms and isomorphisms of left modules
 
Syntaxclmhm 15611 Extend class notation with the generator of left module hom-sets.
 class LMHom
 
Syntaxclmim 15612 The class of left module isomorphism sets.
 class LMIso
 
Syntaxclmic 15613 The class of the left module isomorphism relation.
 class  ~=ph𝑚
 
Definitiondf-lmhm 15614* A homomorphism of left modules is a group homomorphism which additionally preserves the scalar product. This requires both structures to be left modules over the same ring. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |- LMHom  =  ( s  e.  LMod ,  t  e.  LMod  |->  { f  e.  ( s  GrpHom  t )  |  [. (Scalar `  s
 )  /  w ]. (
 (Scalar `  t )  =  w  /\  A. x  e.  ( Base `  w ) A. y  e.  ( Base `  s ) ( f `  ( x ( .s `  s
 ) y ) )  =  ( x ( .s `  t ) ( f `  y
 ) ) ) }
 )
 
Definitiondf-lmim 15615* An isomorphism of modules is a homomorphism which is also a bijection, i.e. it preserves equality as well as the group and scalar operations. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |- LMIso  =  ( s  e.  LMod ,  t  e.  LMod  |->  { g  e.  ( s LMHom  t )  |  g : (
 Base `  s ) -1-1-onto-> ( Base `  t ) } )
 
Definitiondf-lmic 15616 Two modules are said to be isomorphic iff they are connected by at least one isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |- 
 ~=ph𝑚  =  ( `' LMIso  " ( _V  \  1o ) )
 
Theoremreldmlmhm 15617 Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |- 
 Rel  dom LMHom
 
Theoremlmimfn 15618 Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |- LMIso  Fn  ( LMod  X.  LMod )
 
Theoremislmhm 15619* Property of being a homomorphism of left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
 |-  K  =  (Scalar `  S )   &    |-  L  =  (Scalar `  T )   &    |-  B  =  ( Base `  K )   &    |-  E  =  (
 Base `  S )   &    |-  .x.  =  ( .s `  S )   &    |-  .X. 
 =  ( .s `  T )   =>    |-  ( F  e.  ( S LMHom  T )  <->  ( ( S  e.  LMod  /\  T  e.  LMod
 )  /\  ( F  e.  ( S  GrpHom  T ) 
 /\  L  =  K  /\  A. x  e.  B  A. y  e.  E  ( F `  ( x 
 .x.  y ) )  =  ( x  .X.  ( F `  y ) ) ) ) )
 
Theoremislmhm3 15620* Property of a module homomorphism, similar to ismhm 14252. (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 15621 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 15622 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 15623 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 15624 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 15625 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 15626 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 15627 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 15628 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 15629 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 15630* A one-equation proof of linearity of a left module homomorphism, similar to df-lss 15525. (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 15631* 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 15632 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 15633 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 15634 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 15635 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 15636 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 15637 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 15638 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 15639 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 15640 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 15641 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 15642 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 15643 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 15644 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 15645 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 15646 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 15647 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 15648* A linear function is completely determined (or overdetermined) by its values on a spanning subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  B  =  ( Base `  S )   &    |-  K  =  (
 LSpan `  S )   =>    |-  ( ( X 
 C_  B  /\  ( K `  X )  =  B )  ->  E* g
 ( g  e.  ( S LMHom  T )  /\  (
 g  |`  X )  =  F ) )
 
Theorempwsdiaglmhm 15649* 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 15650 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 15651 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 15652 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 15653 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 15654 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 15655 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 15656 The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑚 
 S 
 <->  ( R LMIso  S )  =/=  (/) )
 
Theorembrlmici 15657 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( F  e.  ( R LMIso  S )  ->  R  ~=ph𝑚  S )
 
Theoremlmiclcl 15658 Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑚 
 S  ->  R  e.  LMod
 )
 
Theoremlmicrcl 15659 Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( R  ~=ph𝑚 
 S  ->  S  e.  LMod
 )
 
Theoremlmicsym 15660 Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  ( R  ~=ph𝑚 
 S  ->  S  ~=ph𝑚  R )
 
Theoremlmhmpropd 15661* 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 15662 Extend class notation with the set of bases for a vector space.
 class LBasis
 
Definitiondf-lbs 15663* 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 15664* 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 15665 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 15666 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 15667 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 15668 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 15669 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 15670 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 15671 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 15672* 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 15673* 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 15674 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 15675 Subspace sum of spans of subsets is the span of their union. (spanuni 21953 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 15676 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 15677 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 15678 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 15679 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 15680 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 15681* 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 15682* 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 15683 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 15684 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 15685 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 15686 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 15687* 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 15688 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 15689 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 15690 Extend class notation with class of all left vector spaces.
 class  LVec
 
Definitiondf-lvec 15691 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 15692 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 15693 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 15694 A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)
 |-  ( W  e.  LVec  ->  W  e.  LMod )
 
Theoremlsslvec 15695 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 15696 If a scalar product is zero, one of its factors must be zero. (hvmul0or 21434 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 15697 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 15698 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 15699 Cancellation law for scalar multiplication. (hvmulcan 21481 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 15700 Cancellation law for scalar multiplication. (hvmulcan2 21482 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 ) )
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