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Theorem List for Metamath Proof Explorer - 27301-27400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremreldmdsmm 27301 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
 |-  Rel  dom  (+)m
 
Theoremdsmmval 27302* Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
 |-  B  =  { f  e.  ( Base `  ( S X_s R ) )  |  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin }   =>    |-  ( R  e.  V  ->  ( S  (+)m  R )  =  ( ( S
 X_s
 R )s  B ) )
 
Theoremdsmmbase 27303* Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
 |-  B  =  { f  e.  ( Base `  ( S X_s R ) )  |  { x  e.  dom  R  |  ( f `  x )  =/=  ( 0g `  ( R `  x ) ) }  e.  Fin }   =>    |-  ( R  e.  V  ->  B  =  ( Base `  ( S  (+)m  R ) ) )
 
Theoremdsmmval2 27304 Self-referential definition of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  B  =  ( Base `  ( S  (+)m  R ) )   =>    |-  ( S  (+)m  R )  =  ( ( S X_s R )s  B )
 
Theoremdsmmbas2 27305* Base set of the direct sum module using the fndmin 5648 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  P  =  ( S X_s R )   &    |-  B  =  {
 f  e.  ( Base `  P )  |  dom  ( f  \  ( 0g 
 o.  R ) )  e.  Fin }   =>    |-  ( ( R  Fn  I  /\  I  e.  V )  ->  B  =  ( Base `  ( S  (+)m  R ) ) )
 
Theoremdsmmfi 27306 For finite products, the direct sum is just the module product. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  (
 ( R  Fn  I  /\  I  e.  Fin )  ->  ( S  (+)m  R )  =  ( S X_s R ) )
 
Theoremdsmmelbas 27307* Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  P  =  ( S X_s R )   &    |-  C  =  ( S  (+)m  R )   &    |-  B  =  (
 Base `  P )   &    |-  H  =  ( Base `  C )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  Fn  I )   =>    |-  ( ph  ->  ( X  e.  H  <->  ( X  e.  B  /\  { a  e.  I  |  ( X `
  a )  =/=  ( 0g `  ( R `  a ) ) }  e.  Fin )
 ) )
 
Theoremdsmm0cl 27308 The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  P  =  ( S X_s R )   &    |-  H  =  (
 Base `  ( S  (+)m  R ) )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Mnd )   &    |-  .0.  =  ( 0g `  P )   =>    |-  ( ph  ->  .0.  e.  H )
 
Theoremdsmmacl 27309 The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  P  =  ( S X_s R )   &    |-  H  =  (
 Base `  ( S  (+)m  R ) )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Mnd )   &    |-  ( ph  ->  J  e.  H )   &    |-  ( ph  ->  K  e.  H )   &    |- 
 .+  =  ( +g  `  P )   =>    |-  ( ph  ->  ( J  .+  K )  e.  H )
 
Theoremprdsinvgd2 27310 Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  Y  =  ( S X_s R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Grp )   &    |-  B  =  (
 Base `  Y )   &    |-  N  =  ( inv g `  Y )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  J  e.  I
 )   =>    |-  ( ph  ->  (
 ( N `  X ) `  J )  =  ( ( inv g `  ( R `  J ) ) `  ( X `  J ) ) )
 
Theoremdsmmsubg 27311 The finite hull of a product of groups is additionally closed under negation and thus is a subgroup of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  P  =  ( S X_s R )   &    |-  H  =  (
 Base `  ( S  (+)m  R ) )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Grp )   =>    |-  ( ph  ->  H  e.  (SubGrp `  P )
 )
 
Theoremdsmmlss 27312* The finite hull of a product of modules is additionally closed under scalar multiplication and thus is a linear subspace of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  Ring )   &    |-  ( ph  ->  R : I --> LMod )   &    |-  (
 ( ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x ) )  =  S )   &    |-  P  =  ( S
 X_s
 R )   &    |-  U  =  (
 LSubSp `  P )   &    |-  H  =  ( Base `  ( S  (+)m  R ) )   =>    |-  ( ph  ->  H  e.  U )
 
Theoremdsmmlmod 27313* The direct sum of a family of modules is a module. (Contributed by Stefan O'Rear, 11-Jan-2015.)
 |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  Ring )   &    |-  ( ph  ->  R : I --> LMod )   &    |-  (
 ( ph  /\  x  e.  I )  ->  (Scalar `  ( R `  x ) )  =  S )   &    |-  C  =  ( S 
 (+)m  R )   =>    |-  ( ph  ->  C  e.  LMod )
 
18.17.44  Free modules
 
Syntaxcfrlm 27314 Class of free module generator.
 class freeLMod
 
Syntaxcuvc 27315 Class of basic unit vectors for an explicit free module.
 class unitVec
 
Definitiondf-frlm 27316* The  i-dimensional free module over a ring  r is the product of  i-many copies of the ring with componentwise addition and multiplication. If  i is infinite, the allowed vectors are restricted to those with finitely many nonzero coordinates; this ensures that the resulting module is actually spanned by its unit vectors. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- freeLMod  =  ( r  e.  _V ,  i  e.  _V  |->  ( r 
 (+)m  ( i  X.  {
 (ringLMod `  r ) }
 ) ) )
 
Definitiondf-uvc 27317*  ( ( R unitVec  I ) `  i
) is the unit vector in 
( R freeLMod  I ) along the  i axis. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |- unitVec  =  ( r  e.  _V ,  i  e.  _V  |->  ( j  e.  i  |->  ( k  e.  i  |->  if (
 k  =  j ,  ( 1r `  r
 ) ,  ( 0g
 `  r ) ) ) ) )
 
Theoremfrlmval 27318 Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   =>    |-  (
 ( R  e.  V  /\  I  e.  W )  ->  F  =  ( R  (+)m  ( I  X.  {
 (ringLMod `  R ) }
 ) ) )
 
Theoremfrlmlmod 27319 The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   =>    |-  (
 ( R  e.  Ring  /\  I  e.  W ) 
 ->  F  e.  LMod )
 
Theoremfrlmpws 27320 The free module as a restriction of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  F )   =>    |-  (
 ( R  e.  V  /\  I  e.  W )  ->  F  =  ( ( (ringLMod `  R )  ^s  I )s  B ) )
 
Theoremfrlmlss 27321 The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  F )   &    |-  U  =  ( LSubSp `  ( (ringLMod `  R )  ^s  I )
 )   =>    |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  B  e.  U )
 
Theoremfrlmpwsfi 27322 The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   =>    |-  (
 ( R  e.  V  /\  I  e.  Fin )  ->  F  =  ( (ringLMod `  R )  ^s  I ) )
 
Theoremfrlmsca 27323 The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   =>    |-  (
 ( R  e.  V  /\  I  e.  W )  ->  R  =  (Scalar `  F ) )
 
Theoremfrlm0 27324 Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 27321). (Contributed by Stefan O'Rear, 4-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  I  e.  W )  ->  ( I  X.  {  .0.  } )  =  ( 0g `  F ) )
 
Theoremfrlmbas 27325* Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  N  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  { k  e.  ( N  ^m  I
 )  |  ( `' k " ( _V  \  {  .0.  } )
 )  e.  Fin }   =>    |-  (
 ( R  e.  V  /\  I  e.  W )  ->  B  =  (
 Base `  F ) )
 
Theoremfrlmelbas 27326 Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  N  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  F )   =>    |-  ( ( R  e.  V  /\  I  e.  W )  ->  ( X  e.  B 
 <->  ( X  e.  ( N  ^m  I )  /\  ( `' X " ( _V  \  {  .0.  } )
 )  e.  Fin )
 ) )
 
Theoremfrlmrcl 27327 If a free module is inhabited, this is sufficient to conclude that the ring expression defines a set. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  F )   =>    |-  ( X  e.  B  ->  R  e.  _V )
 
Theoremfrlmbassup 27328 Elements of the free module are finitely supported. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  F )   =>    |-  ( ( I  e.  W  /\  X  e.  B )  ->  ( `' X " ( _V  \  {  .0.  } )
 )  e.  Fin )
 
Theoremfrlmbasmap 27329 Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  N  =  ( Base `  R )   &    |-  B  =  ( Base `  F )   =>    |-  (
 ( I  e.  W  /\  X  e.  B ) 
 ->  X  e.  ( N 
 ^m  I ) )
 
Theoremfrlmbasf 27330 Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  N  =  ( Base `  R )   &    |-  B  =  ( Base `  F )   =>    |-  (
 ( I  e.  W  /\  X  e.  B ) 
 ->  X : I --> N )
 
Theoremfrlmplusgval 27331 Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |- 
 .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  Y )   =>    |-  ( ph  ->  ( F  .+b  G )  =  ( F  o F  .+  G ) )
 
Theoremfrlmvscafval 27332 Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   &    |-  K  =  ( Base `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  B )   &    |-  .xb  =  ( .s `  Y )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( ph  ->  ( A  .xb  X )  =  ( ( I  X.  { A } )  o F  .x.  X )
 )
 
Theoremfrlmvscaval 27333 Scalar multiplication in a free module at a coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   &    |-  K  =  ( Base `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  J  e.  I
 )   &    |-  .xb  =  ( .s `  Y )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ph  ->  ( ( A  .xb  X ) `
  J )  =  ( A  .x.  ( X `  J ) ) )
 
Theoremfrlmgsum 27334* Finite commutative sums in a free module are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 5-Jul-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  J  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ( ph  /\  y  e.  J )  ->  ( x  e.  I  |->  U )  e.  B )   &    |-  ( ph  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) ) "
 ( _V  \  {  .0.  } ) )  e. 
 Fin )   =>    |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
 
Theoremuvcfval 27335* Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  V  /\  I  e.  W )  ->  U  =  ( j  e.  I  |->  ( k  e.  I  |->  if ( k  =  j ,  .1.  ,  .0.  ) ) ) )
 
Theoremuvcval 27336* Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I ) 
 ->  ( U `  J )  =  ( k  e.  I  |->  if (
 k  =  J ,  .1.  ,  .0.  ) ) )
 
Theoremuvcvval 27337 Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  /\  K  e.  I )  ->  ( ( U `  J ) `
  K )  =  if ( K  =  J ,  .1.  ,  .0.  ) )
 
Theoremuvcvvcl 27338 A coodinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  .1.  =  ( 1r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( ( R  e.  V  /\  I  e.  W  /\  J  e.  I )  /\  K  e.  I )  ->  ( ( U `  J ) `
  K )  e. 
 {  .0.  ,  .1.  } )
 
Theoremuvcvvcl2 27339 A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  J  e.  I )   &    |-  ( ph  ->  K  e.  I
 )   =>    |-  ( ph  ->  (
 ( U `  J ) `  K )  e.  B )
 
Theoremuvcvv1 27340 The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  J  e.  I )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ph  ->  (
 ( U `  J ) `  J )  =  .1.  )
 
Theoremuvcvv0 27341 The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  J  e.  I )   &    |-  ( ph  ->  K  e.  I
 )   &    |-  ( ph  ->  J  =/=  K )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ph  ->  ( ( U `  J ) `  K )  =  .0.  )
 
Theoremuvcff 27342 Domain and range of the unit vector generator; ring condition required to be sure 1 and 0 are actually in the ring. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   =>    |-  (
 ( R  e.  Ring  /\  I  e.  W ) 
 ->  U : I --> B )
 
Theoremuvcf1 27343 In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   =>    |-  (
 ( R  e. NzRing  /\  I  e.  W )  ->  U : I -1-1-> B )
 
Theoremuvcresum 27344 Any element of a free module can be expressed as a finite linear combination of unit vectors. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Jul-2015.)
 |-  U  =  ( R unitVec  I )   &    |-  Y  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  Y )   &    |-  .x.  =  ( .s `  Y )   =>    |-  ( ( R  e.  Ring  /\  I  e.  W  /\  X  e.  B ) 
 ->  X  =  ( Y 
 gsumg  ( X  o F  .x.  U ) ) )
 
Theoremfrlmsplit2 27345* Restriction is homomoprhic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.)
 |-  Y  =  ( R freeLMod  U )   &    |-  Z  =  ( R freeLMod  V )   &    |-  B  =  ( Base `  Y )   &    |-  C  =  ( Base `  Z )   &    |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )   =>    |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y LMHom  Z ) )
 
Theoremfrlmsslss 27346* A subset of a free module obtained by restricting the support set is a submodule.  J is the set of forbidden unit vectors. (Contributed by Stefan O'Rear, 4-Feb-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  U  =  ( LSubSp `  Y )   &    |-  B  =  ( Base `  Y )   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  { x  e.  B  |  ( x  |`  J )  =  ( J  X.  {  .0.  } ) }   =>    |-  ( ( R  e.  Ring  /\  I  e.  V  /\  J  C_  I )  ->  C  e.  U )
 
Theoremfrlmsslss2 27347* A subset of a free module obtained by restricting the support set is a submodule.  J is the set of permitted unit vectors. (Contributed by Stefan O'Rear, 5-Feb-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  U  =  ( LSubSp `  Y )   &    |-  B  =  ( Base `  Y )   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  { x  e.  B  |  ( `' x " ( _V  \  {  .0.  } )
 )  C_  J }   =>    |-  (
 ( R  e.  Ring  /\  I  e.  V  /\  J  C_  I )  ->  C  e.  U )
 
Theoremfrlmssuvc1 27348* A scalar multiple of a unit vector included in a support-restriction subspace is included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  B  =  ( Base `  F )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .s `  F )   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  { x  e.  B  |  ( `' x " ( _V  \  {  .0.  } )
 )  C_  J }   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  J 
 C_  I )   &    |-  ( ph  ->  L  e.  J )   &    |-  ( ph  ->  X  e.  K )   =>    |-  ( ph  ->  ( X  .x.  ( U `  L ) )  e.  C )
 
Theoremfrlmssuvc2 27349* A nonzero scalar multiple of a unit vector not included in a support-restriction subspace is not included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  B  =  ( Base `  F )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .s `  F )   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  { x  e.  B  |  ( `' x " ( _V  \  {  .0.  } )
 )  C_  J }   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  J 
 C_  I )   &    |-  ( ph  ->  L  e.  ( I  \  J ) )   &    |-  ( ph  ->  X  e.  ( K  \  {  .0.  } ) )   =>    |-  ( ph  ->  -.  ( X  .x.  ( U `  L ) )  e.  C )
 
Theoremfrlmsslsp 27350* A subset of a free module obtained by restricting the support set is spanned by the relevant unit vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.)
 |-  Y  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  K  =  ( LSpan `  Y )   &    |-  B  =  ( Base `  Y )   &    |-  .0.  =  ( 0g `  R )   &    |-  C  =  { x  e.  B  |  ( `' x " ( _V  \  {  .0.  } )
 )  C_  J }   =>    |-  (
 ( R  e.  Ring  /\  I  e.  V  /\  J  C_  I )  ->  ( K `  ( U
 " J ) )  =  C )
 
Theoremfrlmlbs 27351 The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  J  =  (LBasis `  F )   =>    |-  (
 ( R  e.  Ring  /\  I  e.  V ) 
 ->  ran  U  e.  J )
 
Theoremfrlmup1 27352* Any assignment of unit vectors to target vectors can be extended (uniquely) to a homomorphism from a free module to an arbitrary other module on the same base ring. (Contributed by Stefan O'Rear, 7-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  F )   &    |-  C  =  ( Base `  T )   &    |-  .x.  =  ( .s `  T )   &    |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  o F  .x.  A ) ) )   &    |-  ( ph  ->  T  e.  LMod )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  R  =  (Scalar `  T ) )   &    |-  ( ph  ->  A : I --> C )   =>    |-  ( ph  ->  E  e.  ( F LMHom  T ) )
 
Theoremfrlmup2 27353* The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  F )   &    |-  C  =  ( Base `  T )   &    |-  .x.  =  ( .s `  T )   &    |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  o F  .x.  A ) ) )   &    |-  ( ph  ->  T  e.  LMod )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  R  =  (Scalar `  T ) )   &    |-  ( ph  ->  A : I --> C )   &    |-  ( ph  ->  Y  e.  I )   &    |-  U  =  ( R unitVec  I )   =>    |-  ( ph  ->  ( E `  ( U `  Y ) )  =  ( A `  Y ) )
 
Theoremfrlmup3 27354* The range of such an evaluation map is the finite linear combinations of the target vectors and also the span of the target vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.)
 |-  F  =  ( R freeLMod  I )   &    |-  B  =  ( Base `  F )   &    |-  C  =  ( Base `  T )   &    |-  .x.  =  ( .s `  T )   &    |-  E  =  ( x  e.  B  |->  ( T  gsumg  ( x  o F  .x.  A ) ) )   &    |-  ( ph  ->  T  e.  LMod )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  R  =  (Scalar `  T ) )   &    |-  ( ph  ->  A : I --> C )   &    |-  K  =  ( LSpan `  T )   =>    |-  ( ph  ->  ran  E  =  ( K `  ran  A ) )
 
Theoremfrlmup4 27355* Universal propery of the free module by existential uniquenes. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  R  =  (Scalar `  T )   &    |-  F  =  ( R freeLMod  I )   &    |-  U  =  ( R unitVec  I )   &    |-  C  =  ( Base `  T )   =>    |-  (
 ( T  e.  LMod  /\  I  e.  X  /\  A : I --> C ) 
 ->  E! m  e.  ( F LMHom  T ) ( m  o.  U )  =  A )
 
Theoremellspd 27356* The elements of the span of an indexed collection of basic vectors are those vectors which can be written as finite linear combinations of basic vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.)
 |-  N  =  ( LSpan `  M )   &    |-  B  =  ( Base `  M )   &    |-  K  =  ( Base `  S )   &    |-  S  =  (Scalar `  M )   &    |-  .0.  =  ( 0g `  S )   &    |- 
 .x.  =  ( .s `  M )   &    |-  ( ph  ->  F : I --> B )   &    |-  ( ph  ->  M  e.  LMod
 )   &    |-  ( ph  ->  I  e.  _V )   =>    |-  ( ph  ->  ( X  e.  ( N `  ( F " I
 ) )  <->  E. f  e.  ( K  ^m  I ) ( ( `' f "
 ( _V  \  {  .0.  } ) )  e. 
 Fin  /\  X  =  ( M  gsumg  ( f  o F  .x.  F ) ) ) ) )
 
Theoremelfilspd 27357* Simplified version of ellspd 27356 when the spanning set is finite: all linear combinations are then acceptable. (Contributed by Stefan O'Rear, 7-Feb-2015.)
 |-  N  =  ( LSpan `  M )   &    |-  B  =  ( Base `  M )   &    |-  K  =  ( Base `  S )   &    |-  S  =  (Scalar `  M )   &    |-  .0.  =  ( 0g `  S )   &    |- 
 .x.  =  ( .s `  M )   &    |-  ( ph  ->  F : I --> B )   &    |-  ( ph  ->  M  e.  LMod
 )   &    |-  ( ph  ->  I  e.  Fin )   =>    |-  ( ph  ->  ( X  e.  ( N `  ( F " I
 ) )  <->  E. f  e.  ( K  ^m  I ) X  =  ( M  gsumg  ( f  o F  .x.  F ) ) ) )
 
18.17.45  Every set admits a group structure iff choice
 
Theoremunxpwdom3 27358* Weaker version of unxpwdom 7319 where a function is required only to be cancellative, not an injection.  D and  B are to be thought of as "large" "horizonal" sets, the others as "small". Because the operator is row-wise injective, but the whole row cannot inject into  A, each row must hit an element of 
B; by column injectivity, each row can be identified in at least one way by the  B element that it hits and the column in which it is hit. (Contributed by Stefan O'Rear, 8-Jul-2015.) MOVABLE
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  D  e.  X )   &    |-  (
 ( ph  /\  a  e.  C  /\  b  e.  D )  ->  (
 a  .+  b )  e.  ( A  u.  B ) )   &    |-  ( ( (
 ph  /\  a  e.  C )  /\  ( b  e.  D  /\  c  e.  D ) )  ->  ( ( a  .+  b )  =  (
 a  .+  c )  <->  b  =  c ) )   &    |-  ( ( ( ph  /\  d  e.  D ) 
 /\  ( a  e.  C  /\  c  e.  C ) )  ->  ( ( c  .+  d )  =  (
 a  .+  d )  <->  c  =  a ) )   &    |-  ( ph  ->  -.  D  ~<_  A )   =>    |-  ( ph  ->  C  ~<_*  ( D  X.  B ) )
 
Theoremenfixsn 27359* Given two equipollent sets, a bijection can always be chosen which fixes a single point. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  (
 ( A  e.  X  /\  B  e.  Y  /\  X  ~~  Y )  ->  E. f ( f : X -1-1-onto-> Y  /\  ( f `
  A )  =  B ) )
 
Theoremmapfien2 27360* Equinumerousity relation for sets of finitely supported functions. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  S  =  { x  e.  ( B  ^m  A )  |  ( `' x "
 ( _V  \  {  .0.  } ) )  e. 
 Fin }   &    |-  T  =  { x  e.  ( D  ^m  C )  |  ( `' x " ( _V  \  { W } )
 )  e.  Fin }   &    |-  ( ph  ->  A  ~~  C )   &    |-  ( ph  ->  B  ~~  D )   &    |-  ( ph  ->  .0. 
 e.  B )   &    |-  ( ph  ->  W  e.  D )   =>    |-  ( ph  ->  S  ~~  T )
 
Theoremfsuppeq 27361 Two ways of writing the support of a function with known codomain. MOVABLE SHORTEN nn0supp (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  ( F : I --> S  ->  ( `' F " ( _V  \  { X } )
 )  =  ( `' F " ( S 
 \  { X }
 ) ) )
 
Theorempwfi2f1o 27362* The pw2f1o 6983 bijection relates finitely supported indicator functions on a two-element set to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
 |-  S  =  { y  e.  ( 2o  ^m  A )  |  ( `' y "
 ( _V  \  { (/)
 } ) )  e. 
 Fin }   &    |-  F  =  ( x  e.  S  |->  ( `' x " { 1o } ) )   =>    |-  ( A  e.  V  ->  F : S -1-1-onto-> ( ~P A  i^i  Fin ) )
 
Theorempwfi2en 27363* Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
 |-  S  =  { y  e.  ( 2o  ^m  A )  |  ( `' y "
 ( _V  \  { (/)
 } ) )  e. 
 Fin }   =>    |-  ( A  e.  V  ->  S  ~~  ( ~P A  i^i  Fin )
 )
 
Theoremfrlmpwfi 27364 Formal linear combinations over Z/2Z are equivalent to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.)
 |-  R  =  (ℤ/n `  2 )   &    |-  Y  =  ( R freeLMod  I )   &    |-  B  =  (
 Base `  Y )   =>    |-  ( I  e.  V  ->  B  ~~  ( ~P I  i^i  Fin )
 )
 
Theoremgicabl 27365 Being Abelian is a group invariant. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |-  ( G  ~=ph𝑔 
 H  ->  ( G  e.  Abel 
 <->  H  e.  Abel )
 )
 
Theoremimasgim 27366 A relabeling of the elements of a group induces an isomorphism to the relabeled group. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.) (Revised by Mario Carneiro, 11-Aug-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R ) )   &    |-  ( ph  ->  V  =  (
 Base `  R ) )   &    |-  ( ph  ->  F : V
 -1-1-onto-> B )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  F  e.  ( R GrpIso  U ) )
 
Theorembasfn 27367 Functionality of the base set extractor. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |-  Base  Fn 
 _V
 
Theoremisnumbasgrplem1 27368 A set which is equipollent to the base set of a definable Abelian group is the base set of some (relabeled) Abelian group. (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |-  B  =  ( Base `  R )   =>    |-  (
 ( R  e.  Abel  /\  C  ~~  B ) 
 ->  C  e.  ( Base "
 Abel ) )
 
Theoremharn0 27369 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  ( S  e.  V  ->  (har `  S )  =/=  (/) )
 
Theoremnuminfctb 27370 A numerable infinite set contains a countable subset. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  (
 ( S  e.  dom  card  /\  -.  S  e.  Fin )  ->  om  ~<_  S )
 
Theoremisnumbasgrplem2 27371 If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  (
 ( S  u.  (har `  S ) )  e.  ( Base " Grp )  ->  S  e.  dom  card )
 
Theoremisnumbasgrplem3 27372 Every nonempty numerable set can be given the structure of an Abelian group, either a finite cyclic group or a vector space over Z/2Z. (Contributed by Stefan O'Rear, 10-Jul-2015.)
 |-  (
 ( S  e.  dom  card  /\  S  =/=  (/) )  ->  S  e.  ( Base "
 Abel ) )
 
Theoremisnumbasabl 27373 A set is numerable iff it and its Hartogs number can be jointly given the structure of an Abelian group. (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  ( S  e.  dom  card  <->  ( S  u.  (har `  S ) )  e.  ( Base " Abel ) )
 
Theoremisnumbasgrp 27374 A set is numerable iff it and its Hartogs number can be jointly given the structure of a group. (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  ( S  e.  dom  card  <->  ( S  u.  (har `  S ) )  e.  ( Base " Grp ) )
 
Theoremdfacbasgrp 27375 A choice equivalent in abstract algebra: All nonempty sets admit a group structure. From http://mathoverflow.net/a/12988. (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  (CHOICE  <->  ( Base " Grp )  =  ( _V  \  { (/) } ) )
 
18.17.46  Independent sets and families
 
Syntaxclindf 27376 The class relationship of independent families in a module.
 class LIndF
 
Syntaxclinds 27377 The class generator of independent sets in a module.
 class LIndS
 
Definitiondf-lindf 27378* An independent family is a family of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements of the family. This is almost, but not quite, the same as a function into an independent set.

This is a defined concept because it matters in many cases whether independence is taken at a set or family level. For instance, a number is transcedental iff its nonzero powers are linearly independent. Is 1 transcedental? It has only one nonzero power.

We can almost define family independence as a family of unequal elements with independent range, as islindf3 27398, but taking that as primitive would lead to unpleasant corner case behavior with the zero ring.

This is equivalent to the common definition of having no nontrivial representations of zero (islindf4 27410) and only one representation for each element of the range (islindf5 27411). (Contributed by Stefan O'Rear, 24-Feb-2015.)

 |- LIndF  =  { <. f ,  w >.  |  ( f : dom  f
 --> ( Base `  w )  /\  [. (Scalar `  w )  /  s ]. A. x  e.  dom  f A. k  e.  ( ( Base `  s
 )  \  { ( 0g `  s ) }
 )  -.  ( k
 ( .s `  w ) ( f `  x ) )  e.  ( ( LSpan `  w ) `  ( f "
 ( dom  f  \  { x } ) ) ) ) }
 
Definitiondf-linds 27379* An independent set is a set which is independent as a family. See also islinds3 27406 and islinds4 27407. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- LIndS  =  ( w  e.  _V  |->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s ) LIndF 
 w } )
 
Theoremrellindf 27380 The independent-family predicate is a proper relation and can be used with brrelexi 4745. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  Rel LIndF
 
Theoremislinds 27381 Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   =>    |-  ( W  e.  V  ->  ( X  e.  (LIndS `  W )  <->  ( X  C_  B  /\  (  _I  |`  X ) LIndF  W ) ) )
 
Theoremlinds1 27382 An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   =>    |-  ( X  e.  (LIndS `  W )  ->  X  C_  B )
 
Theoremlinds2 27383 An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  ( X  e.  (LIndS `  W )  ->  (  _I  |`  X ) LIndF  W )
 
Theoremislindf 27384* Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( LSpan `  W )   &    |-  S  =  (Scalar `  W )   &    |-  N  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ( W  e.  Y  /\  F  e.  X )  ->  ( F LIndF  W  <->  ( F : dom  F --> B  /\  A. x  e. 
 dom  F A. k  e.  ( N  \  {  .0.  } )  -.  (
 k  .x.  ( F `  x ) )  e.  ( K `  ( F " ( dom  F  \  { x } )
 ) ) ) ) )
 
Theoremislinds2 27385* Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( LSpan `  W )   &    |-  S  =  (Scalar `  W )   &    |-  N  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( W  e.  Y  ->  ( F  e.  (LIndS `  W )  <->  ( F  C_  B  /\  A. x  e.  F  A. k  e.  ( N  \  {  .0.  } )  -.  (
 k  .x.  x )  e.  ( K `  ( F  \  { x }
 ) ) ) ) )
 
Theoremislindf2 27386* Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( LSpan `  W )   &    |-  S  =  (Scalar `  W )   &    |-  N  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ( W  e.  Y  /\  I  e.  X  /\  F : I --> B ) 
 ->  ( F LIndF  W  <->  A. x  e.  I  A. k  e.  ( N  \  {  .0.  }
 )  -.  ( k  .x.  ( F `  x ) )  e.  ( K `  ( F "
 ( I  \  { x } ) ) ) ) )
 
Theoremlindff 27387 Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   =>    |-  (
 ( F LIndF  W  /\  W  e.  Y )  ->  F : dom  F --> B )
 
Theoremlindfind 27388 A linearly independent family is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  L  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  L )   &    |-  K  =  ( Base `  L )   =>    |-  ( ( ( F LIndF  W  /\  E  e.  dom  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  -.  ( A  .x.  ( F `  E ) )  e.  ( N `  ( F " ( dom 
 F  \  { E } ) ) ) )
 
Theoremlindsind 27389 A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  .x.  =  ( .s `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  L  =  (Scalar `  W )   &    |-  .0.  =  ( 0g `  L )   &    |-  K  =  ( Base `  L )   =>    |-  ( ( ( F  e.  (LIndS `  W )  /\  E  e.  F )  /\  ( A  e.  K  /\  A  =/=  .0.  ) )  ->  -.  ( A  .x.  E )  e.  ( N `  ( F  \  { E }
 ) ) )
 
Theoremlindfind2 27390 In a linearly independent family in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  K  =  ( LSpan `  W )   &    |-  L  =  (Scalar `  W )   =>    |-  (
 ( ( W  e.  LMod  /\  L  e. NzRing )  /\  F LIndF  W 
 /\  E  e.  dom  F )  ->  -.  ( F `  E )  e.  ( K `  ( F " ( dom  F  \  { E } )
 ) ) )
 
Theoremlindsind2 27391 In a linearly independent set in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  K  =  ( LSpan `  W )   &    |-  L  =  (Scalar `  W )   =>    |-  (
 ( ( W  e.  LMod  /\  L  e. NzRing )  /\  F  e.  (LIndS `  W )  /\  E  e.  F ) 
 ->  -.  E  e.  ( K `  ( F  \  { E } ) ) )
 
Theoremlindff1 27392 A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  B  =  ( Base `  W )   &    |-  L  =  (Scalar `  W )   =>    |-  (
 ( W  e.  LMod  /\  L  e. NzRing  /\  F LIndF  W )  ->  F : dom  F
 -1-1-> B )
 
Theoremlindfrn 27393 The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  (
 ( W  e.  LMod  /\  F LIndF  W )  ->  ran  F  e.  (LIndS `  W ) )
 
Theoremf1lindf 27394 Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  (
 ( W  e.  LMod  /\  F LIndF  W  /\  G : K -1-1-> dom  F )  ->  ( F  o.  G ) LIndF  W )
 
Theoremlindfres 27395 Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  (
 ( W  e.  LMod  /\  F LIndF  W )  ->  ( F  |`  X ) LIndF  W )
 
Theoremlindsss 27396 Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  (
 ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  G  e.  (LIndS `  W ) )
 
Theoremf1linds 27397 A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  (
 ( W  e.  LMod  /\  S  e.  (LIndS `  W )  /\  F : D -1-1-> S )  ->  F LIndF  W )
 
Theoremislindf3 27398 In a nonzero ring, independent families can be equivalently characterized as renamings of independent sets. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  L  =  (Scalar `  W )   =>    |-  (
 ( W  e.  LMod  /\  L  e. NzRing )  ->  ( F LIndF  W  <->  ( F : dom  F -1-1-> _V  /\  ran  F  e.  (LIndS `  W )
 ) ) )
 
Theoremlindfmm 27399 Linear independence of a family is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  ( Base `  T )   =>    |-  (
 ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F : I
 --> B )  ->  ( F LIndF  S  <->  ( G  o.  F ) LIndF  T ) )
 
Theoremlindsmm 27400 Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.)
 |-  B  =  ( Base `  S )   &    |-  C  =  ( Base `  T )   =>    |-  (
 ( G  e.  ( S LMHom  T )  /\  G : B -1-1-> C  /\  F  C_  B )  ->  ( F  e.  (LIndS `  S ) 
 <->  ( G " F )  e.  (LIndS `  T ) ) )
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