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Theorem List for Metamath Proof Explorer - 26601-26700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempwssplit3 26601* Splitting for structure powers, part 3: restriction is a module homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  U )   &    |-  Z  =  ( W  ^s  V )   &    |-  B  =  ( Base `  Y )   &    |-  C  =  ( Base `  Z )   &    |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )   =>    |-  ( ( W  e.  LMod  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y LMHom  Z ) )
 
Theorempwssplit4 26602* Splitting for structure powers 4: maps isomorphically onto the other half. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  E  =  ( R  ^s  ( A  u.  B ) )   &    |-  G  =  ( Base `  E )   &    |-  .0.  =  ( 0g `  R )   &    |-  K  =  { y  e.  G  |  ( y  |`  A )  =  ( A  X.  {  .0.  } ) }   &    |-  F  =  ( x  e.  K  |->  ( x  |`  B )
 )   &    |-  C  =  ( R 
 ^s 
 A )   &    |-  D  =  ( R  ^s  B )   &    |-  L  =  ( Es  K )   =>    |-  ( ( R  e.  LMod  /\  ( A  u.  B )  e.  V  /\  ( A  i^i  B )  =  (/) )  ->  F  e.  ( L LMIso  D ) )
 
Theoremfilnm 26603 Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  B  =  ( Base `  W )   =>    |-  (
 ( W  e.  LMod  /\  B  e.  Fin )  ->  W  e. LNoeM )
 
Theorempwslnmlem0 26604 Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  (/) )   =>    |-  ( W  e.  LMod 
 ->  Y  e. LNoeM )
 
Theorempwslnmlem1 26605* First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  { i } )   =>    |-  ( W  e. LNoeM  ->  Y  e. LNoeM )
 
Theorempwslnmlem2 26606 A sum of powers is Noetherian. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  X  =  ( W  ^s  A )   &    |-  Y  =  ( W  ^s  B )   &    |-  Z  =  ( W  ^s  ( A  u.  B ) )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  X  e. LNoeM )   &    |-  ( ph  ->  Y  e. LNoeM )   =>    |-  ( ph  ->  Z  e. LNoeM )
 
Theorempwslnm 26607 Finite powers of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  I )   =>    |-  (
 ( W  e. LNoeM  /\  I  e.  Fin )  ->  Y  e. LNoeM )
 
18.17.43  Direct sum of left modules
 
Syntaxcdsmm 26608 Class of module direct sum generator.
 class  (+)m
 
Definitiondf-dsmm 26609* The direct sum of a family of Abelian groups or left modules is the induced group structure on finite linear combinations of elements, here represented as functions with finite support. (Contributed by Stefan O'Rear, 7-Jan-2015.)
 |-  (+)m  =  ( s  e.  _V ,  r  e.  _V  |->  ( ( s X_s r
 )s  { f  e.  X_ x  e.  dom  r ( Base `  ( r `  x ) )  |  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x ) ) }  e.  Fin
 } ) )
 
Theoremreldmdsmm 26610 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
 |-  Rel  dom  (+)m
 
Theoremdsmmval 26611* 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 26612* 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 26613 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 26614* Base set of the direct sum module using the fndmin 5593 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 26615 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 26616* 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 26617 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 26618 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 26619 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 26620 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 26621* 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 26622* 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 26623 Class of free module generator.
 class freeLMod
 
Syntaxcuvc 26624 Class of basic unit vectors for an explicit free module.
 class unitVec
 
Definitiondf-frlm 26625* 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 26626*  ( ( 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 26627 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 26628 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 26629 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 26630 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 26631 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 26632 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 26633 Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 26630). (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 26634* 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 26635 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 26636 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 26637 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 26638 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 26639 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 26640 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 26641 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 26642 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 26643* 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 26644* 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 26645* 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 26646 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 26647 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 26648 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 26649 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 26650 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 26651 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 26652 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 26653 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 26654* 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 26655* 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 26656* 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 26657* 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 26658* 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 26659* 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 26660 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 26661* 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 26662* 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 26663* 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 26664* 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 26665* 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 26666* Simplified version of ellspd 26665 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 26667* Weaker version of unxpwdom 7298 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 26668* 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 26669* 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 26670 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 26671* The pw2f1o 6962 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 26672* 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 26673 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 26674 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 26675 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 26676 Functionality of the base set extractor. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |-  Base  Fn 
 _V
 
Theoremisnumbasgrplem1 26677 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 26678 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  ( S  e.  V  ->  (har `  S )  =/=  (/) )
 
Theoremnuminfctb 26679 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 26680 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 26681 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 26682 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 26683 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 26684 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 26685 The class relationship of independent families in a module.
 class LIndF
 
Syntaxclinds 26686 The class generator of independent sets in a module.
 class LIndS
 
Definitiondf-lindf 26687* 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 26707, 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 26719) and only one representation for each element of the range (islindf5 26720). (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 26688* An independent set is a set which is independent as a family. See also islinds3 26715 and islinds4 26716. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |- LIndS  =  ( w  e.  _V  |->  { s  e.  ~P ( Base `  w )  |  (  _I  |`  s ) LIndF 
 w } )
 
Theoremrellindf 26689 The independent-family predicate is a proper relation and can be used with brrelexi 4727. (Contributed by Stefan O'Rear, 24-Feb-2015.)
 |-  Rel LIndF
 
Theoremislinds 26690 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 26691 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 26692 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 26693* 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 26694* 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 26695* 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 26696 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 26697 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 26698 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 26699 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 26700 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 } ) ) )
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