HomeHome Metamath Proof Explorer
Theorem List (p. 406 of 409)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26619)
  Hilbert Space Explorer  Hilbert Space Explorer
(26620-28142)
  Users' Mathboxes  Users' Mathboxes
(28143-40812)
 

Theorem List for Metamath Proof Explorer - 40501-40600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlcoss 40501 A set of vectors of a module is a subset of the set of all linear combinations of the set. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  (
 ( M  e.  LMod  /\  V  e.  ~P ( Base `  M ) ) 
 ->  V  C_  ( M LinCo  V ) )
 
Theoremlspsslco 40502 Lemma for lspeqlco 40504. (Contributed by AV, 17-Apr-2019.)
 |-  B  =  ( Base `  M )   =>    |-  (
 ( M  e.  LMod  /\  V  e.  ~P B )  ->  ( ( LSpan `  M ) `  V )  C_  ( M LinCo  V ) )
 
Theoremlcosslsp 40503 Lemma for lspeqlco 40504. (Contributed by AV, 20-Apr-2019.)
 |-  B  =  ( Base `  M )   =>    |-  (
 ( M  e.  LMod  /\  V  e.  ~P B )  ->  ( M LinCo  V )  C_  ( ( LSpan `  M ) `  V ) )
 
Theoremlspeqlco 40504 Equivalence of a span of a set of vectors of a left module defined as the intersection of all linear subspaces which each contain every vector in that set ( see df-lsp 18243) and as the set of all linear combinations of the vectors of the set with finite support. (Contributed by AV, 20-Apr-2019.)
 |-  B  =  ( Base `  M )   =>    |-  (
 ( M  e.  LMod  /\  V  e.  ~P B )  ->  ( M LinCo  V )  =  ( ( LSpan `  M ) `  V ) )
 
21.33.15.3  Linear independency

According to the definition in [Lang] p. 129: "A subset S of a module M is said to be linearly independent (over [the ring] A) if whenever we have a linear combination ∑x ∈S axx which is equal to 0, then ax=0 for all x∈S.". This definition does not care for the finiteness of the set S (because the definition of a linear combination in [Lang] p.129 does already assure that only a finite number of coefficients can be 0 in the sum). Our definition df-lininds 40507 does also neither claim that the subset must be finite, nor that almost all coefficients within the linear combination are 0. If this is required, it must be explicitly stated as precondition in the corresponding theorems.

Usually, the linear independency is defined for vector spaces, see Wikipedia ("Linear independence", 15-Apr-2019, https://en.wikipedia.org/wiki/Linear_independence): "In the theory of vector spaces, a set of vectors is said to be linearly dependent if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent.". Furthermore, "In order to allow the number of linearly independent vectors in a vector space to be countably infinite, it is useful to define linear dependence as follows. More generally, let V be a vector space over a field K, and let {vi | i∈I} be a family of elements of V. The family is linearly dependent over K if there exists a finite family {aj | j∈J} of elements of K, all non-zero, such that ∑j∈J ajvj=0. A set X of elements of V is linearly independent if the corresponding family{x}x∈X is linearly independent".
Remark 1: There are already definitions of (linearly) independent families (df-lindf 19412) and (linearly) independent sets (df-linds 19413). These definitions are based on the principle "of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements" or (see lbsind2 18352) "every element is not in the span of the remainder of the [set]". The equivalence of the definitions df-linds 19413 and df-lininds 40507 for (linear) independency for (left) modules is shown in lindslininds 40529.
Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly indepent (see df-lindeps 40509) or, according to Wikipedia, "if at least one of the vectors in the set can be defined as a linear combination of the others", see islindeps2 40548. The reversed implication is not valid for arbitrary modules (but for arbitrary vector spaces), because it requires a division by a coefficient. Therefore, the definition of Wikipedia is equivalent with our definition for (left) vector spaces (see isldepslvec2 40550) and not for (left) modules in general.

 
Syntaxclininds 40505 Extend class notation with the relation between a module and its linearly independent subsets.
 class linIndS
 
Syntaxclindeps 40506 Extend class notation with the relation between a module and its linearly dependent subsets.
 class linDepS
 
Definitiondf-lininds 40507* Define the relation between a module and its linearly independent subsets. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
 |- linIndS  =  { <. s ,  m >.  |  ( s  e.  ~P ( Base `  m )  /\  A. f  e.  (
 ( Base `  (Scalar `  m ) )  ^m  s ) ( ( f finSupp  ( 0g `  (Scalar `  m ) )  /\  ( f ( linC  `  m )
 s )  =  ( 0g `  m ) )  ->  A. x  e.  s  ( f `  x )  =  ( 0g `  (Scalar `  m ) ) ) ) }
 
Theoremrellininds 40508 The class defining the relation between a module and its linearly independent subsets is a relation. (Contributed by AV, 13-Apr-2019.)
 |-  Rel linIndS
 
Definitiondf-lindeps 40509* Define the relation between a module and its linearly dependent subsets. (Contributed by AV, 26-Apr-2019.)
 |- linDepS  =  { <. s ,  m >.  |  -.  s linIndS  m }
 
Theoremlinindsv 40510 The classes of the module and its linearly independent subsets are sets. (Contributed by AV, 13-Apr-2019.)
 |-  ( S linIndS  M  ->  ( S  e.  _V  /\  M  e.  _V ) )
 
Theoremislininds 40511* The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  ( 0g `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( S  e.  V  /\  M  e.  W )  ->  ( S linIndS  M  <->  ( S  e.  ~P B  /\  A. f  e.  ( E  ^m  S ) ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  )
 ) ) )
 
Theoremlinindsi 40512* The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  ( 0g `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( S linIndS  M  ->  ( S  e.  ~P B  /\  A. f  e.  ( E  ^m  S ) ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  )
 ) )
 
Theoremlinindslinci 40513* The implications of being a linearly independent subset and a linear combination of this subset being 0. (Contributed by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  ( 0g `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( S linIndS  M  /\  ( F  e.  ( E  ^m  S )  /\  F finSupp  .0.  /\  ( F ( linC  `  M ) S )  =  Z ) )  ->  A. x  e.  S  ( F `  x )  =  .0.  )
 
Theoremislinindfis 40514* The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  ( 0g `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( S  e.  Fin  /\  M  e.  W ) 
 ->  ( S linIndS  M  <->  ( S  e.  ~P B  /\  A. f  e.  ( E  ^m  S ) ( ( f ( linC  `  M ) S )  =  Z  ->  A. x  e.  S  ( f `  x )  =  .0.  )
 ) ) )
 
Theoremislinindfiss 40515* The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  ( 0g `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( M  e.  W  /\  S  e.  Fin  /\  S  e.  ~P B )  ->  ( S linIndS  M  <->  A. f  e.  ( E  ^m  S ) ( ( f ( linC  `  M ) S )  =  Z  ->  A. x  e.  S  ( f `  x )  =  .0.  )
 ) )
 
Theoremlinindscl 40516 A linearly independent set is a subset of (the base set of) a module. (Contributed by AV, 13-Apr-2019.)
 |-  ( S linIndS  M  ->  S  e.  ~P ( Base `  M )
 )
 
Theoremlindepsnlininds 40517 A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.)
 |-  (
 ( S  e.  V  /\  M  e.  W ) 
 ->  ( S linDepS  M  <->  -.  S linIndS  M )
 )
 
Theoremislindeps 40518* The property of being a linearly dependent subset. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  ( 0g `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( M  e.  W  /\  S  e.  ~P B )  ->  ( S linDepS  M 
 <-> 
 E. f  e.  ( E  ^m  S ) ( f finSupp  .0.  /\  ( f
 ( linC  `  M ) S )  =  Z  /\  E. x  e.  S  ( f `  x )  =/=  .0.  ) ) )
 
Theoremlincext1 40519* Property 1 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 29-Apr-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  N  =  ( invg `  R )   &    |-  F  =  ( z  e.  S  |->  if ( z  =  X ,  ( N `  Y ) ,  ( G `  z ) ) )   =>    |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B )  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X } ) ) ) )  ->  F  e.  ( E  ^m  S ) )
 
Theoremlincext2 40520* Property 2 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  N  =  ( invg `  R )   &    |-  F  =  ( z  e.  S  |->  if ( z  =  X ,  ( N `  Y ) ,  ( G `  z ) ) )   =>    |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B )  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X } ) ) ) 
 /\  G finSupp  .0.  )  ->  F finSupp  .0.  )
 
Theoremlincext3 40521* Property 3 of an extension of a linear combination. (Contributed by AV, 23-Apr-2019.) (Revised by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  N  =  ( invg `  R )   &    |-  F  =  ( z  e.  S  |->  if ( z  =  X ,  ( N `  Y ) ,  ( G `  z ) ) )   =>    |-  ( ( ( M  e.  LMod  /\  S  e.  ~P B )  /\  ( Y  e.  E  /\  X  e.  S  /\  G  e.  ( E  ^m  ( S  \  { X } ) ) ) 
 /\  ( G finSupp  .0.  /\  ( Y ( .s `  M ) X )  =  ( G ( linC  `  M ) ( S 
 \  { X }
 ) ) ) ) 
 ->  ( F ( linC  `  M ) S )  =  Z )
 
Theoremlindslinindsimp1 40522* Implication 1 for lindslininds 40529. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
 |-  R  =  (Scalar `  M )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   =>    |-  ( ( S  e.  V  /\  M  e.  LMod )  ->  (
 ( S  e.  ~P ( Base `  M )  /\  A. f  e.  ( B  ^m  S ) ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  )
 )  ->  ( S  C_  ( Base `  M )  /\  A. s  e.  S  A. y  e.  ( B 
 \  {  .0.  }
 )  -.  ( y
 ( .s `  M ) s )  e.  ( ( LSpan `  M ) `  ( S  \  { s } )
 ) ) ) )
 
Theoremlindslinindimp2lem1 40523* Lemma 1 for lindslinindsimp2 40528. (Contributed by AV, 25-Apr-2019.)
 |-  R  =  (Scalar `  M )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  Y  =  ( ( invg `  R ) `  (
 f `  x )
 )   &    |-  G  =  ( f  |`  ( S  \  { x } ) )   =>    |-  ( ( ( S  e.  V  /\  M  e.  LMod )  /\  ( S  C_  ( Base `  M )  /\  x  e.  S  /\  f  e.  ( B  ^m  S ) ) )  ->  Y  e.  B )
 
Theoremlindslinindimp2lem2 40524* Lemma 2 for lindslinindsimp2 40528. (Contributed by AV, 25-Apr-2019.)
 |-  R  =  (Scalar `  M )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  Y  =  ( ( invg `  R ) `  (
 f `  x )
 )   &    |-  G  =  ( f  |`  ( S  \  { x } ) )   =>    |-  ( ( ( S  e.  V  /\  M  e.  LMod )  /\  ( S  C_  ( Base `  M )  /\  x  e.  S  /\  f  e.  ( B  ^m  S ) ) )  ->  G  e.  ( B  ^m  ( S  \  { x } ) ) )
 
Theoremlindslinindimp2lem3 40525* Lemma 3 for lindslinindsimp2 40528. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
 |-  R  =  (Scalar `  M )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  Y  =  ( ( invg `  R ) `  (
 f `  x )
 )   &    |-  G  =  ( f  |`  ( S  \  { x } ) )   =>    |-  ( ( ( S  e.  V  /\  M  e.  LMod )  /\  ( S  C_  ( Base `  M )  /\  x  e.  S )  /\  (
 f  e.  ( B 
 ^m  S )  /\  f finSupp  .0.  ) )  ->  G finSupp  .0.  )
 
Theoremlindslinindimp2lem4 40526* Lemma 4 for lindslinindsimp2 40528. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
 |-  R  =  (Scalar `  M )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  Y  =  ( ( invg `  R ) `  (
 f `  x )
 )   &    |-  G  =  ( f  |`  ( S  \  { x } ) )   =>    |-  ( ( ( S  e.  V  /\  M  e.  LMod )  /\  ( S  C_  ( Base `  M )  /\  x  e.  S )  /\  (
 f  e.  ( B 
 ^m  S )  /\  f finSupp  .0.  /\  ( f
 ( linC  `  M ) S )  =  Z ) )  ->  ( M  gsumg  (
 y  e.  ( S 
 \  { x }
 )  |->  ( ( f `
  y ) ( .s `  M ) y ) ) )  =  ( Y ( .s `  M ) x ) )
 
Theoremlindslinindsimp2lem5 40527* Lemma 5 for lindslinindsimp2 40528. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
 |-  R  =  (Scalar `  M )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   =>    |-  ( ( ( S  e.  V  /\  M  e.  LMod )  /\  ( S  C_  ( Base `  M )  /\  x  e.  S ) )  ->  ( ( f  e.  ( B  ^m  S )  /\  ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z ) )  ->  ( A. y  e.  ( B  \  {  .0.  } ) A. g  e.  ( B  ^m  ( S  \  { x } ) ) ( -.  g finSupp  .0.  \/ 
 -.  ( y ( .s `  M ) x )  =  ( g ( linC  `  M ) ( S  \  { x } ) ) )  ->  ( f `  x )  =  .0.  ) ) )
 
Theoremlindslinindsimp2 40528* Implication 2 for lindslininds 40529. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
 |-  R  =  (Scalar `  M )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   =>    |-  ( ( S  e.  V  /\  M  e.  LMod )  ->  (
 ( S  C_  ( Base `  M )  /\  A. s  e.  S  A. y  e.  ( B  \  {  .0.  } )  -.  ( y ( .s
 `  M ) s )  e.  ( (
 LSpan `  M ) `  ( S  \  { s } ) ) ) 
 ->  ( S  e.  ~P ( Base `  M )  /\  A. f  e.  ( B  ^m  S ) ( ( f finSupp  .0.  /\  ( f ( linC  `  M ) S )  =  Z )  ->  A. x  e.  S  ( f `  x )  =  .0.  )
 ) ) )
 
Theoremlindslininds 40529 Equivalence of definitions df-linds 19413 and df-lininds 40507 for (linear) independency for (left) modules. (Contributed by AV, 26-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  (
 ( S  e.  V  /\  M  e.  LMod )  ->  ( S linIndS  M  <->  S  e.  (LIndS `  M ) ) )
 
Theoremlinds0 40530 The empty set is always a linearly independet subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  ( M  e.  V  ->  (/) linIndS  M )
 
Theoremel0ldep 40531 A set containing the zero element of a module is always linearly dependent, if the underlying ring has at least two elements. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  (
 ( ( M  e.  LMod  /\  1  <  ( # `  ( Base `  (Scalar `  M ) ) ) ) 
 /\  S  e.  ~P ( Base `  M )  /\  ( 0g `  M )  e.  S )  ->  S linDepS  M )
 
Theoremel0ldepsnzr 40532 A set containing the zero element of a module over a nonzero ring is always linearly dependent. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
 |-  (
 ( ( M  e.  LMod  /\  (Scalar `  M )  e. NzRing )  /\  S  e.  ~P ( Base `  M )  /\  ( 0g `  M )  e.  S )  ->  S linDepS  M )
 
Theoremlindsrng01 40533 Any subset of a module is always linearly independent if the underlying ring has at most one element. Since the underlying ring cannot be the empty set (see lmodsn0 18152), this means that the underlying ring has only one element, so it is a zero ring. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   =>    |-  (
 ( M  e.  LMod  /\  ( ( # `  E )  =  0  \/  ( # `  E )  =  1 )  /\  S  e.  ~P B )  ->  S linIndS  M )
 
Theoremlindszr 40534 Any subset of a module over a zero ring is always linearly independent. (Contributed by AV, 27-Apr-2019.)
 |-  (
 ( M  e.  LMod  /\ 
 -.  (Scalar `  M )  e. NzRing  /\  S  e.  ~P ( Base `  M )
 )  ->  S linIndS  M )
 
Theoremsnlindsntorlem 40535* Lemma for snlindsntor 40536. (Contributed by AV, 15-Apr-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  S  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  .x.  =  ( .s `  M )   =>    |-  ( ( M  e.  LMod  /\  X  e.  B ) 
 ->  ( A. f  e.  ( S  ^m  { X } ) ( ( f ( linC  `  M ) { X } )  =  Z  ->  ( f `
  X )  =  .0.  )  ->  A. s  e.  S  ( ( s 
 .x.  X )  =  Z  ->  s  =  .0.  )
 ) )
 
Theoremsnlindsntor 40536* A singleton is linearly independent iff it does not contain a torsion element. According to Wikipedia ("Torsion (algebra)", 15-Apr-2019, https://en.wikipedia.org/wiki/Torsion_(algebra)): "An element m of a module M over a ring R is called a torsion element of the module if there exists a regular element r of the ring (an element that is neither a left nor a right zero divisor) that annihilates m, i.e.,  ( r  .x.  m )  =  0. In an integral domain (a commutative ring without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero element of the integral domain." Analogously, the definition in [Lang] p. 147 states that "An element x of [a module] E [over a ring R] is called a torsion element if there exists  a  e.  R,  a  =/=  0, such that  a  .x.  x  =  0. This definition includes the zero element of the module. Some authors, however, exclude the zero element from the definition of torsion elements. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  S  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  .x.  =  ( .s `  M )   =>    |-  ( ( M  e.  LMod  /\  X  e.  B ) 
 ->  ( A. s  e.  ( S  \  {  .0.  } ) ( s 
 .x.  X )  =/=  Z  <->  { X } linIndS  M )
 )
 
Theoremldepsprlem 40537 Lemma for ldepspr 40538. (Contributed by AV, 16-Apr-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  S  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  .x.  =  ( .s `  M )   &    |-  .1.  =  ( 1r `  R )   &    |-  N  =  ( invg `  R )   =>    |-  ( ( M  e.  LMod  /\  ( X  e.  B  /\  Y  e.  B  /\  A  e.  S )
 )  ->  ( X  =  ( A  .x.  Y )  ->  ( (  .1.  .x.  X ) ( +g  `  M ) ( ( N `  A ) 
 .x.  Y ) )  =  Z ) )
 
Theoremldepspr 40538 If a vector is a scalar multiple of another vector, the (unordered pair containing the) two vectors are linearly dependent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  S  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  .x.  =  ( .s `  M )   =>    |-  ( ( M  e.  LMod  /\  ( X  e.  B  /\  Y  e.  B  /\  X  =/=  Y ) ) 
 ->  ( ( A  e.  S  /\  X  =  ( A  .x.  Y )
 )  ->  { X ,  Y } linDepS  M )
 )
 
Theoremlincresunit3lem3 40539 Lemma 3 for lincresunit3 40546. (Contributed by AV, 18-May-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  N  =  ( invg `  R )   &    |-  .x.  =  ( .s `  M )   =>    |-  ( ( ( M  e.  LMod  /\  X  e.  B  /\  Y  e.  B )  /\  A  e.  U )  ->  ( ( ( N `  A )  .x.  X )  =  ( ( N `  A )  .x.  Y )  <->  X  =  Y )
 )
 
Theoremlincresunitlem1 40540 Lemma 1 for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  N  =  ( invg `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  G  =  ( s  e.  ( S  \  { X } )  |->  ( ( I `  ( N `
  ( F `  X ) ) ) 
 .x.  ( F `  s ) ) )   =>    |-  ( ( ( S  e.  ~P B  /\  M  e.  LMod  /\  X  e.  S )  /\  ( F  e.  ( E  ^m  S )  /\  ( F `  X )  e.  U ) )  ->  ( I `  ( N `
  ( F `  X ) ) )  e.  E )
 
Theoremlincresunitlem2 40541 Lemma for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  N  =  ( invg `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  G  =  ( s  e.  ( S  \  { X } )  |->  ( ( I `  ( N `
  ( F `  X ) ) ) 
 .x.  ( F `  s ) ) )   =>    |-  ( ( ( ( S  e.  ~P B  /\  M  e.  LMod  /\  X  e.  S )  /\  ( F  e.  ( E  ^m  S )  /\  ( F `  X )  e.  U ) )  /\  Y  e.  S )  ->  ( ( I `  ( N `  ( F `
  X ) ) )  .x.  ( F `  Y ) )  e.  E )
 
Theoremlincresunit1 40542* Property 1 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  N  =  ( invg `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  G  =  ( s  e.  ( S  \  { X } )  |->  ( ( I `  ( N `
  ( F `  X ) ) ) 
 .x.  ( F `  s ) ) )   =>    |-  ( ( ( S  e.  ~P B  /\  M  e.  LMod  /\  X  e.  S )  /\  ( F  e.  ( E  ^m  S )  /\  ( F `  X )  e.  U ) )  ->  G  e.  ( E  ^m  ( S  \  { X } ) ) )
 
Theoremlincresunit2 40543* Property 2 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  N  =  ( invg `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  G  =  ( s  e.  ( S  \  { X } )  |->  ( ( I `  ( N `
  ( F `  X ) ) ) 
 .x.  ( F `  s ) ) )   =>    |-  ( ( ( S  e.  ~P B  /\  M  e.  LMod  /\  X  e.  S )  /\  ( F  e.  ( E  ^m  S )  /\  ( F `  X )  e.  U  /\  F finSupp  .0.  )
 )  ->  G finSupp  .0.  )
 
Theoremlincresunit3lem1 40544* Lemma 1 for lincresunit3 40546. (Contributed by AV, 17-May-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  N  =  ( invg `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  G  =  ( s  e.  ( S  \  { X } )  |->  ( ( I `  ( N `
  ( F `  X ) ) ) 
 .x.  ( F `  s ) ) )   =>    |-  ( ( ( S  e.  ~P B  /\  M  e.  LMod  /\  X  e.  S )  /\  ( F  e.  ( E  ^m  S )  /\  ( F `  X )  e.  U  /\  z  e.  ( S  \  { X } ) ) ) 
 ->  ( ( N `  ( F `  X ) ) ( .s `  M ) ( ( G `  z ) ( .s `  M ) z ) )  =  ( ( F `
  z ) ( .s `  M ) z ) )
 
Theoremlincresunit3lem2 40545* Lemma 2 for lincresunit3 40546. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  N  =  ( invg `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  G  =  ( s  e.  ( S  \  { X } )  |->  ( ( I `  ( N `
  ( F `  X ) ) ) 
 .x.  ( F `  s ) ) )   =>    |-  ( ( ( S  e.  ~P B  /\  M  e.  LMod  /\  X  e.  S )  /\  ( F  e.  ( E  ^m  S )  /\  ( F `  X )  e.  U  /\  F finSupp  .0.  )
 )  ->  ( ( N `  ( F `  X ) ) ( .s `  M ) ( M  gsumg  ( z  e.  ( S  \  { X }
 )  |->  ( ( G `
  z ) ( .s `  M ) z ) ) ) )  =  ( ( F  |`  ( S  \  { X } )
 ) ( linC  `  M ) ( S  \  { X } ) ) )
 
Theoremlincresunit3 40546* Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  N  =  ( invg `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  G  =  ( s  e.  ( S  \  { X } )  |->  ( ( I `  ( N `
  ( F `  X ) ) ) 
 .x.  ( F `  s ) ) )   =>    |-  ( ( ( S  e.  ~P B  /\  M  e.  LMod  /\  X  e.  S )  /\  ( F  e.  ( E  ^m  S )  /\  ( F `  X )  e.  U  /\  F finSupp  .0.  )  /\  ( F ( linC  `  M ) S )  =  Z )  ->  ( G ( linC  `  M ) ( S 
 \  { X }
 ) )  =  X )
 
Theoremlincreslvec3 40547* Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  Z  =  ( 0g
 `  M )   &    |-  N  =  ( invg `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  G  =  ( s  e.  ( S  \  { X } )  |->  ( ( I `  ( N `
  ( F `  X ) ) ) 
 .x.  ( F `  s ) ) )   =>    |-  ( ( ( S  e.  ~P B  /\  M  e.  LVec  /\  X  e.  S )  /\  ( F  e.  ( E  ^m  S )  /\  ( F `  X )  =/= 
 .0.  /\  F finSupp  .0.  )  /\  ( F ( linC  `  M ) S )  =  Z )  ->  ( G ( linC  `  M ) ( S 
 \  { X }
 ) )  =  X )
 
Theoremislindeps2 40548* Conditions for being a linearly dependent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  ( 0g `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( M  e.  LMod  /\  S  e.  ~P B  /\  R  e. NzRing )  ->  ( E. s  e.  S  E. f  e.  ( E  ^m  ( S  \  { s } )
 ) ( f finSupp  .0.  /\  ( f ( linC  `  M ) ( S  \  { s } )
 )  =  s ) 
 ->  S linDepS  M ) )
 
Theoremislininds2 40549* Implication of being a linearly independent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  ( 0g `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( M  e.  LMod  /\  S  e.  ~P B  /\  R  e. NzRing )  ->  ( S linIndS  M  ->  A. s  e.  S  A. f  e.  ( E  ^m  ( S  \  { s }
 ) ) ( -.  f finSupp  .0.  \/  ( f ( linC  `  M )
 ( S  \  {
 s } ) )  =/=  s ) ) )
 
Theoremisldepslvec2 40550* Alternative definition of being a linearly dependent subset of a (left) vector space. In this case, the reverse implication of islindeps2 40548 holds, so that both definitions are equivalent (see theorem 1.6 in [Roman] p. 46 and the note in [Roman] p. 112: if a nontrivial linear combination of elements (where not all of the coefficients are 0) in an R-vector space is 0, then and only then each of the elements is a linear combination of the others. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  ( 0g `  M )   &    |-  R  =  (Scalar `  M )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( M  e.  LVec  /\  S  e.  ~P B )  ->  ( E. s  e.  S  E. f  e.  ( E  ^m  ( S  \  { s }
 ) ) ( f finSupp  .0.  /\  ( f ( linC  `  M ) ( S 
 \  { s }
 ) )  =  s )  <->  S linDepS  M ) )
 
Theoremlindssnlvec 40551 A singleton not containing the zero element of a vector space is always linearly independent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 28-Apr-2019.)
 |-  (
 ( M  e.  LVec  /\  S  e.  ( Base `  M )  /\  S  =/=  ( 0g `  M ) )  ->  { S } linIndS  M )
 
21.33.15.4  Simple left modules and the ` ZZ `-module
 
Theoremlmod1lem1 40552* Lemma 1 for lmod1 40557. (Contributed by AV, 28-Apr-2019.)
 |-  M  =  ( { <. ( Base ` 
 ndx ) ,  { I } >. ,  <. ( +g  ` 
 ndx ) ,  { <.
 <. I ,  I >. ,  I >. } >. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R ) ,  y  e.  { I }  |->  y ) >. } )   =>    |-  ( ( I  e.  V  /\  R  e.  Ring  /\  r  e.  ( Base `  R ) ) 
 ->  ( r ( .s
 `  M ) I )  e.  { I } )
 
Theoremlmod1lem2 40553* Lemma 2 for lmod1 40557. (Contributed by AV, 28-Apr-2019.)
 |-  M  =  ( { <. ( Base ` 
 ndx ) ,  { I } >. ,  <. ( +g  ` 
 ndx ) ,  { <.
 <. I ,  I >. ,  I >. } >. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R ) ,  y  e.  { I }  |->  y ) >. } )   =>    |-  ( ( I  e.  V  /\  R  e.  Ring  /\  r  e.  ( Base `  R ) ) 
 ->  ( r ( .s
 `  M ) ( I ( +g  `  M ) I ) )  =  ( ( r ( .s `  M ) I ) ( +g  `  M ) ( r ( .s `  M ) I ) ) )
 
Theoremlmod1lem3 40554* Lemma 3 for lmod1 40557. (Contributed by AV, 29-Apr-2019.)
 |-  M  =  ( { <. ( Base ` 
 ndx ) ,  { I } >. ,  <. ( +g  ` 
 ndx ) ,  { <.
 <. I ,  I >. ,  I >. } >. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R ) ,  y  e.  { I }  |->  y ) >. } )   =>    |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  (
 q  e.  ( Base `  R )  /\  r  e.  ( Base `  R )
 ) )  ->  (
 ( q ( +g  `  (Scalar `  M )
 ) r ) ( .s `  M ) I )  =  ( ( q ( .s
 `  M ) I ) ( +g  `  M ) ( r ( .s `  M ) I ) ) )
 
Theoremlmod1lem4 40555* Lemma 4 for lmod1 40557. (Contributed by AV, 29-Apr-2019.)
 |-  M  =  ( { <. ( Base ` 
 ndx ) ,  { I } >. ,  <. ( +g  ` 
 ndx ) ,  { <.
 <. I ,  I >. ,  I >. } >. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R ) ,  y  e.  { I }  |->  y ) >. } )   =>    |-  ( ( ( I  e.  V  /\  R  e.  Ring )  /\  (
 q  e.  ( Base `  R )  /\  r  e.  ( Base `  R )
 ) )  ->  (
 ( q ( .r
 `  (Scalar `  M )
 ) r ) ( .s `  M ) I )  =  ( q ( .s `  M ) ( r ( .s `  M ) I ) ) )
 
Theoremlmod1lem5 40556* Lemma 5 for lmod1 40557. (Contributed by AV, 28-Apr-2019.)
 |-  M  =  ( { <. ( Base ` 
 ndx ) ,  { I } >. ,  <. ( +g  ` 
 ndx ) ,  { <.
 <. I ,  I >. ,  I >. } >. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R ) ,  y  e.  { I }  |->  y ) >. } )   =>    |-  ( ( I  e.  V  /\  R  e.  Ring
 )  ->  ( ( 1r `  (Scalar `  M ) ) ( .s
 `  M ) I )  =  I )
 
Theoremlmod1 40557* The (smallest) structure representing a zero module over an arbitrary ring. (Contributed by AV, 29-Apr-2019.)
 |-  M  =  ( { <. ( Base ` 
 ndx ) ,  { I } >. ,  <. ( +g  ` 
 ndx ) ,  { <.
 <. I ,  I >. ,  I >. } >. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R ) ,  y  e.  { I }  |->  y ) >. } )   =>    |-  ( ( I  e.  V  /\  R  e.  Ring
 )  ->  M  e.  LMod
 )
 
Theoremlmod1zr 40558 The (smallest) structure representing a zero module over a zero ring. (Contributed by AV, 29-Apr-2019.)
 |-  R  =  { <. ( Base `  ndx ) ,  { Z } >. ,  <. ( +g  ` 
 ndx ) ,  { <.
 <. Z ,  Z >. ,  Z >. } >. ,  <. ( .r `  ndx ) ,  { <. <. Z ,  Z >. ,  Z >. } >. }   &    |-  M  =  ( { <. ( Base `  ndx ) ,  { I } >. , 
 <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
 >. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  { <. <. Z ,  I >. ,  I >. }
 >. } )   =>    |-  ( ( I  e.  V  /\  Z  e.  W )  ->  M  e.  LMod
 )
 
Theoremlmod1zrnlvec 40559 There is a (left) module (a zero module) which is not a (left) vector space. (Contributed by AV, 29-Apr-2019.)
 |-  R  =  { <. ( Base `  ndx ) ,  { Z } >. ,  <. ( +g  ` 
 ndx ) ,  { <.
 <. Z ,  Z >. ,  Z >. } >. ,  <. ( .r `  ndx ) ,  { <. <. Z ,  Z >. ,  Z >. } >. }   &    |-  M  =  ( { <. ( Base `  ndx ) ,  { I } >. , 
 <. ( +g  `  ndx ) ,  { <. <. I ,  I >. ,  I >. }
 >. ,  <. (Scalar `  ndx ) ,  R >. }  u.  { <. ( .s `  ndx ) ,  { <. <. Z ,  I >. ,  I >. }
 >. } )   =>    |-  ( ( I  e.  V  /\  Z  e.  W )  ->  M  e/  LVec
 )
 
Theoremlmodn0 40560 Left modules exist. (Contributed by AV, 29-Apr-2019.)
 |-  LMod  =/=  (/)
 
Theoremzlmodzxzequa 40561 Example of an equation within the 
ZZ-module  ZZ  X.  ZZ (see example in [Roman] p. 112 for a linearly dependent set). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  .0.  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  .xb  =  ( .s `  Z )   &    |-  .-  =  ( -g `  Z )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6 >. }   &    |-  B  =  { <. 0 ,  2
 >. ,  <. 1 ,  4
 >. }   =>    |-  ( ( 2  .xb  A )  .-  ( 3  .xb 
 B ) )  =  .0.
 
Theoremzlmodzxznm 40562 Example of a linearly dependent set whose elements are not linear combinations of the others, see note in [Roman] p. 112). (Contributed by AV, 23-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  .0.  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  .xb  =  ( .s `  Z )   &    |-  .-  =  ( -g `  Z )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6 >. }   &    |-  B  =  { <. 0 ,  2
 >. ,  <. 1 ,  4
 >. }   =>    |- 
 A. i  e.  ZZ  ( ( i  .xb  A )  =/=  B  /\  ( i  .xb  B )  =/=  A )
 
Theoremzlmodzxzldeplem 40563 A and B are not equal. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   =>    |-  A  =/=  B
 
Theoremzlmodzxzequap 40564 Example of an equation within the 
ZZ-module  ZZ  X.  ZZ (see example in [Roman] p. 112 for a linearly dependent set), written as a sum. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   &    |-  .0.  =  { <. 0 ,  0 >. ,  <. 1 ,  0
 >. }   &    |-  .+  =  ( +g  `  Z )   &    |-  .xb  =  ( .s `  Z )   =>    |-  ( ( 2  .xb  A )  .+  ( -u 3  .xb  B ) )  =  .0.
 
Theoremzlmodzxzldeplem1 40565 Lemma 1 for zlmodzxzldep 40569. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   &    |-  F  =  { <. A ,  2 >. ,  <. B ,  -u 3 >. }   =>    |-  F  e.  ( ZZ 
 ^m  { A ,  B } )
 
Theoremzlmodzxzldeplem2 40566 Lemma 2 for zlmodzxzldep 40569. (Contributed by AV, 24-May-2019.) (Revised by AV, 30-Jul-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   &    |-  F  =  { <. A ,  2 >. ,  <. B ,  -u 3 >. }   =>    |-  F finSupp  0
 
Theoremzlmodzxzldeplem3 40567 Lemma 3 for zlmodzxzldep 40569. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   &    |-  F  =  { <. A ,  2 >. ,  <. B ,  -u 3 >. }   =>    |-  ( F ( linC  `  Z ) { A ,  B } )  =  ( 0g `  Z )
 
Theoremzlmodzxzldeplem4 40568* Lemma 4 for zlmodzxzldep 40569. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   &    |-  F  =  { <. A ,  2 >. ,  <. B ,  -u 3 >. }   =>    |- 
 E. y  e.  { A ,  B }  ( F `  y )  =/=  0
 
Theoremzlmodzxzldep 40569 { A , B } is a linearly dependent set within the  ZZ-module  ZZ  X.  ZZ (see example in [Roman] p. 112). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   =>    |- 
 { A ,  B } linDepS  Z
 
Theoremldepsnlinclem1 40570 Lemma 1 for ldepsnlinc 40573. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   =>    |-  ( F  e.  (
 ( Base ` ring )  ^m  { B } )  ->  ( F ( linC  `  Z ) { B } )  =/= 
 A )
 
Theoremldepsnlinclem2 40571 Lemma 2 for ldepsnlinc 40573. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.)
 |-  Z  =  (ring freeLMod  { 0 ,  1 } )   &    |-  A  =  { <. 0 ,  3 >. ,  <. 1 ,  6
 >. }   &    |-  B  =  { <. 0 ,  2 >. ,  <. 1 ,  4
 >. }   =>    |-  ( F  e.  (
 ( Base ` ring )  ^m  { A } )  ->  ( F ( linC  `  Z ) { A } )  =/= 
 B )
 
21.33.15.5  Differences between (left) modules and (left) vector spaces
 
Theoremlvecpsslmod 40572 The class of all (left) vector spaces is a proper subclass of the class of all (left) modules. Although it is obvious (and proven by lveclmod 18377) that every left vector space is a left module, there is (at least) one left module which is no left vector space, for example the zero module over the zero ring, see lmod1zrnlvec 40559. (Contributed by AV, 29-Apr-2019.)
 |-  LVec  C. 
 LMod
 
Theoremldepsnlinc 40573* The reverse implication of islindeps2 40548 does not hold for arbitrary (left) modules, see note in [Roman] p. 112: "... if a nontrivial linear combination of the elements ... in an R-module M is 0, ... where not all of the coefficients are 0, then we cannot conclude ... that one of the elements ... is a linear combination of the others." This means that there is at least one left module having a linearly dependent subset in which there is at least one element which is not a linear combinantion of the other elements of this subset. Such a left module can be constructed by using zlmodzxzequa 40561 and zlmodzxznm 40562. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.)
 |-  E. m  e.  LMod  E. s  e.  ~P  ( Base `  m )
 ( s linDepS  m  /\  A. v  e.  s  A. f  e.  ( ( Base `  (Scalar `  m ) )  ^m  ( s 
 \  { v }
 ) ) ( f finSupp  ( 0g `  (Scalar `  m ) )  ->  ( f ( linC  `  m )
 ( s  \  {
 v } ) )  =/=  v ) )
 
Theoremldepslinc 40574* For (left) vector spaces, isldepslvec2 40550 provides an alternative definition of being a linearly dependent subset, whereas ldepsnlinc 40573 indicates that there is not an analogous alternative definition for arbitrary (left) modules. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.)
 |-  ( A. m  e.  LVec  A. s  e.  ~P  ( Base `  m ) ( s linDepS  m  <->  E. v  e.  s  E. f  e.  ( ( Base `  (Scalar `  m ) )  ^m  ( s 
 \  { v }
 ) ) ( f finSupp  ( 0g `  (Scalar `  m ) )  /\  ( f ( linC  `  m )
 ( s  \  {
 v } ) )  =  v ) ) 
 /\  -.  A. m  e. 
 LMod  A. s  e.  ~P  ( Base `  m )
 ( s linDepS  m  <->  E. v  e.  s  E. f  e.  (
 ( Base `  (Scalar `  m ) )  ^m  ( s 
 \  { v }
 ) ) ( f finSupp  ( 0g `  (Scalar `  m ) )  /\  ( f ( linC  `  m )
 ( s  \  {
 v } ) )  =  v ) ) )
 
21.33.16  Complexity theory
 
21.33.16.1  Auxiliary theorems
 
Theoremoffval0 40575* Value of an operation applied to two functions. (Contributed by AV, 15-May-2020.)
 |-  (
 ( F  e.  V  /\  G  e.  W ) 
 ->  ( F  oF R G )  =  ( x  e.  ( dom 
 F  i^i  dom  G ) 
 |->  ( ( F `  x ) R ( G `  x ) ) ) )
 
Theoremsuppdm 40576 If the range of a function does not contain the zero, the support of the function equals its domain. (Contributed by AV, 20-May-2020.)
 |-  (
 ( ( Fun  F  /\  F  e.  V  /\  Z  e.  W )  /\  Z  e/  ran  F )  ->  ( F supp  Z )  =  dom  F )
 
Theoremeluz2gt1 40577 An integer greater than or equal to 2 is greater than 1. (Contributed by AV, 24-May-2020.)
 |-  ( N  e.  ( ZZ>= `  2 )  ->  1  <  N )
 
Theoremeluz2n0 40578 An integer greater than or equal to 2 is not 0. (Contributed by AV, 25-May-2020.)
 |-  ( N  e.  ( ZZ>= `  2 )  ->  N  =/=  0 )
 
Theoremeluz2cnn0n1 40579 An integer greater than 1 is a complex number not equal to 0 or 1. (Contributed by AV, 23-May-2020.)
 |-  ( B  e.  ( ZZ>= `  2 )  ->  B  e.  ( CC  \  { 0 ,  1 } )
 )
 
Theoremdivge1b 40580 The ratio of a real number to a positive real number is greater than or equal to 1 iff the divisor (the positive real number) is less than or equal to the dividend (the real number). (Contributed by AV, 26-May-2020.)
 |-  (
 ( A  e.  RR+  /\  B  e.  RR )  ->  ( A  <_  B  <->  1 
 <_  ( B  /  A ) ) )
 
Theoremdivgt1b 40581 The ratio of a real number to a positive real number is greater than 1 iff the divisor (the positive real number) is less than the dividend (the real number). (Contributed by AV, 30-May-2020.)
 |-  (
 ( A  e.  RR+  /\  B  e.  RR )  ->  ( A  <  B  <->  1  <  ( B  /  A ) ) )
 
Theoremdivlt1lt 40582 A real number divided by a positive real number is less than 1 iff the real number is less than the positive real number. (Contributed by AV, 25-May-2020.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( ( A  /  B )  <  1  <->  A  <  B ) )
 
Theoremltsubaddb 40583 Equivalence for the "less than" relation between differences and sums. (Contributed by AV, 6-Jun-2020.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  -  C )  <  ( B  -  D )  <->  ( A  +  D )  <  ( B  +  C ) ) )
 
Theoremltsubsubb 40584 Equivalence for the "less than" relation between differences. (Contributed by AV, 6-Jun-2020.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  -  C )  <  ( B  -  D )  <->  ( A  -  B )  <  ( C  -  D ) ) )
 
Theoremltsubadd2b 40585 Equivalence for the "less than" relation between differences and sums. (Contributed by AV, 6-Jun-2020.)
 |-  (
 ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( D  -  C )  <  ( B  -  A )  <->  ( A  +  D )  <  ( B  +  C ) ) )
 
Theoremdivsub1dir 40586 Distribution of division over subtraction by 1. (Contributed by AV, 6-Jun-2020.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( ( A  /  B )  -  1
 )  =  ( ( A  -  B ) 
 /  B ) )
 
Theoremexpnegico01 40587 An integer greater than 1 to the power of a negative integer is in the closed-below, open-above interval between 0 and 1. (Contributed by AV, 24-May-2020.)
 |-  (
 ( B  e.  ( ZZ>=
 `  2 )  /\  N  e.  ZZ  /\  N  <  0 )  ->  ( B ^ N )  e.  ( 0 [,) 1
 ) )
 
Theoremelfzolborelfzop1 40588 An element of a half-open integer interval is either equal to the left bound of the interval or an element of a half-open integer interval with a left bound increased by 1. (Contributed by AV, 2-Jun-2020.)
 |-  ( K  e.  ( M..^ N )  ->  ( K  =  M  \/  K  e.  ( ( M  +  1 )..^ N ) ) )
 
Theorem3halfnz 40589 Three halfs is not an integer. (Contributed by AV, 2-Jun-2020.)
 |-  -.  ( 3  /  2
 )  e.  ZZ
 
Theorempw2m1lepw2m1 40590 2 to the power of a positive integer decreased by 1 is less than or equal to 2 to the power of the integer minus 1. (Contributed by AV, 30-May-2020.)
 |-  ( I  e.  NN  ->  ( 2 ^ ( I  -  1 ) ) 
 <_  ( ( 2 ^ I )  -  1
 ) )
 
Theoremzgtp1leeq 40591 If an integer is between another integer and its predecessor, the integer is equal to the other integer. (Contributed by AV, 7-Jun-2020.)
 |-  (
 ( I  e.  ZZ  /\  A  e.  ZZ )  ->  ( ( ( A  -  1 )  < 
 I  /\  I  <_  A )  ->  I  =  A ) )
 
Theoremflsubz 40592 An integer can be moved in and out of the floor of a difference. (Contributed by AV, 29-May-2020.)
 |-  (
 ( A  e.  RR  /\  N  e.  ZZ )  ->  ( |_ `  ( A  -  N ) )  =  ( ( |_ `  A )  -  N ) )
 
21.33.16.2  The modulo (remainder) operation (extension)
 
Theoremfldivmod 40593 Expressing the floor of a division by the modulo operator. (Contributed by AV, 6-Jun-2020.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  ( |_ `  ( A  /  B ) )  =  ( ( A  -  ( A  mod  B ) )  /  B ) )
 
Theoremmod0mul 40594* If an integer is 0 modulo a positive integer, this integer must be the product of another integer and the modulus. (Contributed by AV, 7-Jun-2020.)
 |-  (
 ( A  e.  ZZ  /\  N  e.  NN )  ->  ( ( A  mod  N )  =  0  ->  E. x  e.  ZZ  A  =  ( x  x.  N ) ) )
 
Theoremmodn0mul 40595* If an integer is not 0 modulo a positive integer, this integer must be the sum of the product of another integer and the modulus and a positive integer less than the modulus. (Contributed by AV, 7-Jun-2020.)
 |-  (
 ( A  e.  ZZ  /\  N  e.  NN )  ->  ( ( A  mod  N )  =/=  0  ->  E. x  e.  ZZ  E. y  e.  ( 1..^ N ) A  =  ( ( x  x.  N )  +  y
 ) ) )
 
Theoremm1modmmod 40596 An integer decreased by 1 modulo a positive integer minus the integer modulo the same modulus is either -1 or the modulus minus 1. (Contributed by AV, 7-Jun-2020.)
 |-  (
 ( A  e.  ZZ  /\  N  e.  NN )  ->  ( ( ( A  -  1 )  mod  N )  -  ( A 
 mod  N ) )  =  if ( ( A 
 mod  N )  =  0 ,  ( N  -  1 ) ,  -u 1
 ) )
 
Theoremdifmodm1lt 40597 The difference between an integer modulo a positive integer and the integer decreased by 1 modulo the same modulus is less than the modulus decreased by 1 (if the modulus is greater than 2). This theorem would not be valid for an odd  A and  N  =  2, since  ( ( A  mod  N )  -  ( ( A  - 
1 )  mod  N
) ) would be  ( 1  -  0 )  =  1 which is not less than  ( N  -  1 )  =  1. (Contributed by AV, 6-Jun-2012.)
 |-  (
 ( A  e.  ZZ  /\  N  e.  NN  /\  2  <  N )  ->  ( ( A  mod  N )  -  ( ( A  -  1 ) 
 mod  N ) )  < 
 ( N  -  1
 ) )
 
21.33.16.3  Even and odd integers
 
Theoremnn0enne 40598 A positive integer is an even nonnegative integer iff it is an even positive integer. (Contributed by AV, 30-May-2020.)
 |-  ( N  e.  NN  ->  ( ( N  /  2
 )  e.  NN0  <->  ( N  / 
 2 )  e.  NN ) )
 
Theoremnn0o1gt2 40599 An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.)
 |-  (
 ( N  e.  NN0  /\  ( ( N  +  1 )  /  2
 )  e.  NN0 )  ->  ( N  =  1  \/  2  <  N ) )
 
Theoremnno 40600 An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.)
 |-  (
 ( N  e.  ( ZZ>=
 `  2 )  /\  ( ( N  +  1 )  /  2
 )  e.  NN0 )  ->  ( ( N  -  1 )  /  2
 )  e.  NN )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40812
  Copyright terms: Public domain < Previous  Next >