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Theorem List for Metamath Proof Explorer - 26501-26600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremislssfg2 26501* Property of a finitely generated left (sub-)module, with a relaxed constraint on the spanning vectors. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  X  =  ( Ws  U )   &    |-  S  =  (
 LSubSp `  W )   &    |-  N  =  ( LSpan `  W )   &    |-  B  =  ( Base `  W )   =>    |-  (
 ( W  e.  LMod  /\  U  e.  S ) 
 ->  ( X  e. LFinGen  <->  E. b  e.  ( ~P B  i^i  Fin )
 ( N `  b
 )  =  U ) )
 
Theoremislssfgi 26502 Finitely spanned subspaces are finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  N  =  ( LSpan `  W )   &    |-  V  =  ( Base `  W )   &    |-  X  =  ( Ws  ( N `  B ) )   =>    |-  ( ( W  e.  LMod  /\  B  C_  V  /\  B  e.  Fin )  ->  X  e. LFinGen )
 
Theoremfglmod 26503 Finitely generated left modules are left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  ( M  e. LFinGen  ->  M  e.  LMod
 )
 
Theoremlsmfgcl 26504 The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  U  =  ( LSubSp `  W )   &    |-  .(+)  =  (
 LSSum `  W )   &    |-  D  =  ( Ws  A )   &    |-  E  =  ( Ws  B )   &    |-  F  =  ( Ws  ( A  .(+)  B ) )   &    |-  ( ph  ->  W  e.  LMod )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  B  e.  U )   &    |-  ( ph  ->  D  e. LFinGen )   &    |-  ( ph  ->  E  e. LFinGen )   =>    |-  ( ph  ->  F  e. LFinGen )
 
16.16.41  Noetherian left modules I
 
Syntaxclnm 26505 Extend class notation with the class of Noetherian left modules.
 class LNoeM
 
Definitiondf-lnm 26506* A left-module is Noetherian iff it is hereditarily finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |- LNoeM  =  { w  e.  LMod  |  A. i  e.  ( LSubSp `  w ) ( ws  i )  e. LFinGen }
 
Theoremislnm 26507* Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  S  =  ( LSubSp `  M )   =>    |-  ( M  e. LNoeM  <->  ( M  e.  LMod  /\  A. i  e.  S  ( Ms  i )  e. LFinGen )
 )
 
Theoremislnm2 26508* Property of being a Noetherian left module with finite generation expanded in terms of spans. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  B  =  ( Base `  M )   &    |-  S  =  ( LSubSp `  M )   &    |-  N  =  ( LSpan `  M )   =>    |-  ( M  e. LNoeM  <->  ( M  e.  LMod  /\  A. i  e.  S  E. g  e.  ( ~P B  i^i  Fin )
 i  =  ( N `
  g ) ) )
 
Theoremlnmlmod 26509 A Noetherian left module is a left module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  ( M  e. LNoeM  ->  M  e.  LMod
 )
 
Theoremlnmlssfg 26510 A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  S  =  ( LSubSp `  M )   &    |-  R  =  ( Ms  U )   =>    |-  ( ( M  e. LNoeM  /\  U  e.  S ) 
 ->  R  e. LFinGen )
 
Theoremlnmlsslnm 26511 All submodules of a Noetherian module are Noetherian. (Contributed by Stefan O'Rear, 1-Jan-2015.)
 |-  S  =  ( LSubSp `  M )   &    |-  R  =  ( Ms  U )   =>    |-  ( ( M  e. LNoeM  /\  U  e.  S ) 
 ->  R  e. LNoeM )
 
Theoremlnmfg 26512 A Noetherian left module is finitely generated. (Contributed by Stefan O'Rear, 12-Dec-2014.)
 |-  ( M  e. LNoeM  ->  M  e. LFinGen )
 
Theoremkercvrlsm 26513 The domain of a linear function is the subspace sum of the kernel and any subspace which covers the range. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  U  =  ( LSubSp `  S )   &    |-  .(+)  =  (
 LSSum `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  K  =  ( `' F " {  .0.  } )   &    |-  B  =  (
 Base `  S )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   &    |-  ( ph  ->  D  e.  U )   &    |-  ( ph  ->  ( F " D )  = 
 ran  F )   =>    |-  ( ph  ->  ( K  .(+)  D )  =  B )
 
Theoremlmhmfgima 26514 A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( Ts  ( F " A ) )   &    |-  X  =  ( Ss  A )   &    |-  U  =  (
 LSubSp `  S )   &    |-  ( ph  ->  X  e. LFinGen )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   =>    |-  ( ph  ->  Y  e. LFinGen )
 
Theoremlnmepi 26515 Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  B  =  ( Base `  T )   =>    |-  (
 ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  T  e. LNoeM )
 
Theoremlmhmfgsplit 26516 If the kernel and range of a homomorphism of left modules are finitely generated, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
 |-  .0.  =  ( 0g `  T )   &    |-  K  =  ( `' F " {  .0.  } )   &    |-  U  =  ( Ss  K )   &    |-  V  =  ( Ts 
 ran  F )   =>    |-  ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen ) 
 ->  S  e. LFinGen )
 
Theoremlmhmlnmsplit 26517 If the kernel and range of a homomorphism of left modules are Noetherian, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 12-Jun-2015.)
 |-  .0.  =  ( 0g `  T )   &    |-  K  =  ( `' F " {  .0.  } )   &    |-  U  =  ( Ss  K )   &    |-  V  =  ( Ts 
 ran  F )   =>    |-  ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM ) 
 ->  S  e. LNoeM )
 
Theoremlnmlmic 26518 Noetherian is an invariant property of modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑚 
 S  ->  ( R  e. LNoeM  <->  S  e. LNoeM ) )
 
16.16.42  Addenda for structure powers
 
Theorempwssplit0 26519* Splitting for structure powers, part 0: restriction is a function. (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.  T  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
 
Theorempwssplit1 26520* Splitting for structure powers, part 1: restriction is an onto function. The only actual monoid law we need here is that the base set is nonempty. (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.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B -onto-> C )
 
Theorempwssplit2 26521* Splitting for structure powers, part 2: restriction is a group 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.  Grp  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y  GrpHom  Z ) )
 
Theorempwssplit3 26522* 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 26523* 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 26524 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 26525 Zeroeth powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  (/) )   =>    |-  ( W  e.  LMod 
 ->  Y  e. LNoeM )
 
Theorempwslnmlem1 26526* First powers are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( W  ^s  { i } )   =>    |-  ( W  e. LNoeM  ->  Y  e. LNoeM )
 
Theorempwslnmlem2 26527 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 26528 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 )
 
16.16.43  Direct sum of left modules
 
Syntaxcdsmm 26529 Class of module direct sum generator.
 class  (+)m
 
Definitiondf-dsmm 26530* 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 26531 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
 |-  Rel  dom  (+)m
 
Theoremdsmmval 26532* 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 26533* 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 26534 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 26535* Base set of the direct sum module using the fndmin 5531 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 26536 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 26537* 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 26538 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 26539 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 26540 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 26541 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 26542* 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 26543* 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 )
 
16.16.44  Free modules
 
Syntaxcfrlm 26544 Class of free module generator.
 class freeLMod
 
Syntaxcuvc 26545 Class of basic unit vectors for an explicit free module.
 class unitVec
 
Definitiondf-frlm 26546* 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 26547*  ( ( 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 26548 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 26549 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 26550 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 26551 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 26552 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 26553 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 26554 Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 26551). (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 26555* 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 26556 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 26557 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 26558 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 26559 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 26560 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 26561 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 26562 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 26563 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 26564* 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 26565* 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 26566* 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 26567 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 26568 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 26569 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 26570 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 26571 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 26572 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 26573 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 26574 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 26575* 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 26576* 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 26577* 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 26578* 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 26579* 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 26580* 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 26581 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 26582* 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 26583* 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 26584* 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 26585* 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 26586* 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 26587* Simplified version of ellspd 26586 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 ) ) ) )
 
16.16.45  Every set admits a group structure iff choice
 
Theoremunxpwdom3 26588* Weaker version of unxpwdom 7236 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 26589* 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 26590* 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 26591 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 26592* The pw2f1o 6900 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 26593* 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 26594 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 26595 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 26596 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 26597 Functionality of the base set extractor. MOVABLE (Contributed by Stefan O'Rear, 8-Jul-2015.)
 |-  Base  Fn 
 _V
 
Theoremisnumbasgrplem1 26598 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 26599 The Hartogs number of a set is never zero. MOVABLE (Contributed by Stefan O'Rear, 9-Jul-2015.)
 |-  ( S  e.  V  ->  (har `  S )  =/=  (/) )
 
Theoremnuminfctb 26600 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 )
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