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Theorem List for Metamath Proof Explorer - 24801-24900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-act 24801* Definition of an action law. The action is the function ( k ^m ( v ^m v ). Definitions equivalent through currying. (Contributed by FL, 24-Dec-2013.)
 |-  Action  =  {
 f  |  E. k E. v E. s ( ( k  =  (
 Base `  (Scalar `  f
 ) )  /\  v  =  ( Base `  f )  /\  s  =  ( .s `  f ) ) 
 /\  A. r  e.  k  A. w  e.  v  ( r s w )  e.  v ) }
 
16.12.22  Generic modules and vector spaces
 
Syntaxcvec 24802 Extend class notation with the class of all generic vector spaces and modules.
 class  Vec
 
Definitiondf-vec 24803* Definition of a vector space (
<. g ,  h >. is a field ), or of a module ( <. g ,  h >. is a ring ). (Contributed by FL, 12-Jul-2010.)
 |-  Vec  =  { z  |  E. g E. h E. a E. b ( z  = 
 <. <. g ,  h >. ,  <. a ,  b >.
 >.  /\  ( a  e.  AbelOp 
 /\  b : ( ran  g  X.  ran  a ) --> ran  a  /\  A. u  e.  ran  a ( ( (GId `  h ) b u )  =  u  /\  A. x  e.  ran  g
 ( A. v  e.  ran  a ( x b ( u a v ) )  =  ( ( x b u ) a ( x b v ) ) 
 /\  A. y  e.  ran  g ( ( ( x g y ) b u )  =  ( ( x b u ) a ( y b u ) )  /\  ( ( x h y ) b u )  =  ( x b ( y b u ) ) ) ) ) ) ) }
 
Theoremvecval1b 24804* The predicate "is a vector space" or "is a module". (Contributed by FL, 12-Jul-2010.)
 |-  X  =  ran  G   &    |-  W  =  ran  A   =>    |-  ( ( ( G  e.  M  /\  H  e.  N  /\  A  e.  O )  /\  B  e.  P )  ->  ( <. G ,  H >.  Vec  <. A ,  B >. 
 <->  ( A  e.  AbelOp  /\  B : ( X  X.  W ) --> W  /\  A. u  e.  W  ( ( (GId `  H ) B u )  =  u  /\  A. x  e.  X  ( A. v  e.  W  ( x B ( u A v ) )  =  ( ( x B u ) A ( x B v ) ) 
 /\  A. y  e.  X  ( ( ( x G y ) B u )  =  ( ( x B u ) A ( y B u ) ) 
 /\  ( ( x H y ) B u )  =  ( x B ( y B u ) ) ) ) ) ) ) )
 
Theoremvecval3b 24805* The "axioms" of a vector space or module. (Contributed by FL, 12-Jul-2010.)
 |-  X  =  ran  G   &    |-  G  =  ( 1st `  ( 1st `  R ) )   &    |-  H  =  ( 2nd `  ( 1st `  R ) )   &    |-  A  =  ( 1st `  ( 2nd `  R ) )   &    |-  B  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  A   =>    |-  ( R  e.  Vec  ->  ( A  e.  AbelOp  /\  B : ( X  X.  W ) --> W  /\  A. u  e.  W  ( ( (GId `  H ) B u )  =  u  /\  A. x  e.  X  ( A. v  e.  W  ( x B ( u A v ) )  =  ( ( x B u ) A ( x B v ) ) 
 /\  A. y  e.  X  ( ( ( x G y ) B u )  =  ( ( x B u ) A ( y B u ) ) 
 /\  ( ( x H y ) B u )  =  ( x B ( y B u ) ) ) ) ) ) )
 
Theoremvecax1 24806 1st "axiom" of a vector space or module. The vector addition is an abelian group. (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by FL, 14-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   =>    |-  ( R  e.  Vec  ->  + w  e.  AbelOp )
 
Theoremvecax2 24807 2nd "axiom" of a vector space or module. Domain, codomain and functionality of the multiplication of a vector by a scalar. (Contributed by FL, 14-Sep-2010.)
 |-  X  =  ran  ( 1st `  ( 1st `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   =>    |-  ( R  e.  Vec 
 ->  . w : ( X  X.  W ) --> W )
 
Theoremvecax3 24808* 3rd "axiom" of a vector space or module. Multiplication by 1. (Contributed by FL, 13-Sep-2010.)
 |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   &    |-  1 t  =  (GId `  ( 2nd `  ( 1st `  R ) ) )   =>    |-  ( R  e.  Vec 
 ->  A. u  e.  W  ( 1 t . w u )  =  u )
 
Theoremvecax4 24809* 4th "axiom" of a vector space or module. Multiplication by a scalar distributes over vector addition. (Contributed by FL, 13-Sep-2010.)
 |-  X  =  ran  ( 1st `  ( 1st `  R ) )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   =>    |-  ( R  e.  Vec  ->  A. u  e.  W  A. x  e.  X  A. v  e.  W  ( x . w ( u + w v ) )  =  ( ( x . w u ) + w ( x . w v ) ) )
 
Theoremvecax5 24810* 5th "axiom" of a vector space or module. Multiplication by a sum of scalars. (Contributed by FL, 13-Sep-2010.)
 |-  X  =  ran  ( 1st `  ( 1st `  R ) )   &    |-  + t  =  ( 1st `  ( 1st `  R ) )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   =>    |-  ( R  e.  Vec 
 ->  A. u  e.  W  A. x  e.  X  A. y  e.  X  (
 ( x + t
 y ) . w u )  =  (
 ( x . w u ) + w
 ( y . w u ) ) )
 
Theoremvecax6 24811* 6th "axiom" of a vector space or module. Relation between scalar multiplication and vector multiplication. (Contributed by FL, 13-Sep-2010.)
 |-  X  =  ran  ( 1st `  ( 1st `  R ) )   &    |-  . t  =  ( 2nd `  ( 1st `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   =>    |-  ( R  e.  Vec  ->  A. u  e.  W  A. x  e.  X  A. y  e.  X  (
 ( x . t
 y ) . w u )  =  ( x . w ( y . w u ) ) )
 
Theoremvecax5b 24812 Multiplication by a sum of scalars. (Contributed by FL, 13-Sep-2010.)
 |-  X  =  ran  ( 1st `  ( 1st `  R ) )   &    |-  + t  =  ( 1st `  ( 1st `  R ) )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   =>    |-  ( ( R  e.  Vec  /\  ( U  e.  W  /\  A  e.  X  /\  B  e.  X ) )  ->  ( ( A + t B ) . w U )  =  (
 ( A . w U ) + w
 ( B . w U ) ) )
 
Theoremcladdinvvec 24813 Closure of the additive inverse of a vector. (Contributed by FL, 13-Sep-2010.)
 |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   &    |-  ~ w  =  ( inv `  ( 1st `  ( 2nd `  R ) ) )   =>    |-  ( ( R  e.  Vec  /\  U  e.  W ) 
 ->  ( ~ w `  U )  e.  W )
 
Theoremvec2inv 24814 Double inverse law for vector additive inverse. (Contributed by FL, 13-Sep-2010.)
 |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   &    |-  ~ w  =  ( inv `  ( 1st `  ( 2nd `  R ) ) )   =>    |-  ( ( R  e.  Vec  /\  U  e.  W ) 
 ->  ( ~ w `  ( ~ w `  U ) )  =  U )
 
Theoremsum2vv 24815 The sum of two vectors is a vector. (Contributed by FL, 13-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   =>    |-  ( ( R  e.  Vec  /\  V1  e.  W  /\  V 2  e.  W ) 
 ->  ( V1 + w V 2 )  e.  W )
 
Theoremaddnull1 24816 Addition of the null vector. (Contributed by FL, 13-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   &    |-  0 w  =  (GId `  + w )   =>    |-  (
 ( R  e.  Vec  /\  U  e.  W ) 
 ->  ( U + w
 0 w )  =  U )
 
Theoremaddnull2 24817 Addition of the null vector. (Contributed by FL, 13-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   &    |-  0 w  =  (GId `  + w )   =>    |-  (
 ( R  e.  Vec  /\  U  e.  W ) 
 ->  ( 0 w + w U )  =  U )
 
Theoremaddvecass 24818 The addition of vectors is associative. (Contributed by FL, 13-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   =>    |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W ) )  ->  ( V1 + w ( V 2 + w V 3 )
 )  =  ( (
 V1 + w V 2 ) + w V 3 ) )
 
Theoremaddvecom 24819 The addition of vectors is associative. (Contributed by FL, 13-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   =>    |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W ) )  ->  ( V1 + w V 2 )  =  ( V 2 + w V1 ) )
 
Theoreminvaddvec 24820 Additive inverse of a sum of vectors. (Contributed by FL, 13-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   &    |-  ~ w  =  ( inv `  + w )   =>    |-  (
 ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W ) )  ->  ( ~ w `  ( V1 + w V 2 )
 )  =  ( ( ~ w `  V1 ) + w ( ~ w `  V 2 ) ) )
 
Theoremprodvs 24821 The product of a vector by a scalar is a vector. (Contributed by FL, 12-Sep-2010.)
 |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   &    |-  X  =  ran  + t   &    |-  + t  =  ( 1st `  ( 1st `  R ) )   =>    |-  ( ( R  e.  Vec  /\  A  e.  X  /\  U  e.  W )  ->  ( A . w U )  e.  W )
 
Theoremvecsrcan 24822 Right cancellation law for vector subtraction. (Contributed by FL, 12-Sep-2010.)
 |-  - w  =  (  /g  `  + w )   &    |-  W  =  ran  + w   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   =>    |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W ) )  ->  ( ( V1 - w V 3 )  =  ( V 2 - w V 3 )  <->  V1  =  V 2
 ) )
 
Theoremvecslcan 24823 Left cancellation law for vector subtraction. (Contributed by FL, 12-Sep-2010.)
 |-  - w  =  (  /g  `  + w )   &    |-  W  =  ran  + w   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   =>    |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W ) )  ->  ( ( V 3 - w V1 )  =  ( V 3 - w V 2 )  <->  V1  =  V 2
 ) )
 
Theoremvwit 24824 A vector minus itself equals zero. (Contributed by FL, 12-Sep-2010.)
 |-  0 w  =  (GId `  + w )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  - w  =  (  /g  `  + w )   &    |-  W  =  ran  + w   =>    |-  ( ( R  e.  Vec  /\  U  e.  W ) 
 ->  ( U - w U )  =  0 w )
 
Theoremsub2vec 24825 Definition of the subtraction of two vectors. (Contributed by FL, 12-Sep-2010.)
 |-  0 w  =  (GId `  + w )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  - w  =  (  /g  `  + w )   &    |-  W  =  ran  + w   &    |-  ~ w  =  ( inv `  + w )   =>    |-  (
 ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W ) )  ->  ( V1 - w V 2 )  =  ( V1 + w
 ( ~ w `  V 2 ) ) )
 
Theoremmvecrtol 24826 Moving a vector from the right member of an equation into the left member. (Contributed by FL, 12-Sep-2010.)
 |-  0 w  =  (GId `  + w )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  - w  =  (  /g  `  + w )   &    |-  W  =  ran  + w   =>    |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W ) )  ->  ( V1  =  V 2  <->  ( V1 - w V 2 )  =  0 w ) )
 
Theoremdblsubvec 24827 Double subtraction of vectors. (Contributed by FL, 12-Sep-2010.)
 |-  0 w  =  (GId `  + w )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  - w  =  (  /g  `  + w )   &    |-  W  =  ran  + w   =>    |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W ) )  ->  ( (
 V1 - w V 2
 ) - w V 3 )  =  ( V1
 - w ( V 2 + w V 3 ) ) )
 
Theoremvecrcan 24828 Right cancellation law for vector addition. (Contributed by FL, 12-Sep-2010.)
 |-  0 w  =  (GId `  + w )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  - w  =  (  /g  `  + w )   &    |-  W  =  ran  + w   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   =>    |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W ) )  ->  ( ( V1 + w V 3 )  =  ( V 2 + w V 3 )  <->  V1  =  V 2
 ) )
 
Theoremveclcan 24829 Left cancellation law for vector addition. (Contributed by FL, 12-Sep-2010.)
 |-  0 w  =  (GId `  + w )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  - w  =  (  /g  `  + w )   &    |-  W  =  ran  + w   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   =>    |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W ) )  ->  ( ( V 3 + w V1 )  =  ( V 3 + w V 2 )  <->  V1  =  V 2
 ) )
 
Theoremmvecrtol2 24830 Moving a vector from the right member of an equation into the left member. (Contributed by FL, 12-Sep-2010.)
 |-  0 w  =  (GId `  + w )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  - w  =  (  /g  `  + w )   &    |-  W  =  ran  + w   =>    |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W  /\  V 3  e.  W ) )  ->  ( V1  =  ( V 2 + w V 3 )  <->  ( V1 - w V 2 )  =  V 3 ) )
 
Theoremprvs 24831 The product of a vector by a scalar is a vector. (Contributed by FL, 12-Sep-2010.)
 |-  X  =  ran  ( 1st `  ( 1st `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   =>    |-  ( ( R  e.  Vec  /\  A  e.  X  /\  U  e.  W )  ->  ( A . w U )  e.  W )
 
Theoremmulveczer 24832 Multiplication of a vector by zero. (Contributed by FL, 12-Sep-2010.)
 |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   &    |-  0 t  =  (GId `  + t )   &    |-  + t  =  ( 1st `  ( 1st `  R ) )   &    |-  . t  =  ( 2nd `  ( 1st `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  0 w  =  (GId `  ( 1st `  ( 2nd `  R ) ) )   =>    |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  U  e.  W )  ->  ( 0 t . w U )  =  0 w )
 
Theoremmulinvsca 24833 Multiplication by the inverse of a scalar. (Contributed by FL, 12-Sep-2010.)
 |-  X  =  ran  + t   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  ~ t  =  ( inv `  + t )   &    |-  ~ w  =  ( inv `  ( 1st `  ( 2nd `  R ) ) )   &    |-  + t  =  ( 1st `  ( 1st `  R ) )   &    |-  . t  =  ( 2nd `  ( 1st `  R ) )   =>    |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  ( A  e.  X  /\  U  e.  W )
 )  ->  ( ( ~ t `  A ) . w U )  =  ( ~ w `  ( A . w U ) ) )
 
Theoremmuldisc 24834* Multiplication by a difference of scalars. (Contributed by FL, 12-Sep-2010.)
 |-  X  =  ran  ( 1st `  ( 1st `  R ) )   &    |-  + t  =  ( 1st `  ( 1st `  R ) )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  ( 1st `  ( 2nd `  R ) )   &    |-  - t  =  (  /g  `  + t )   &    |-  - w  =  (  /g  `  + w )   &    |-  . t  =  ( 2nd `  ( 1st `  R ) )   =>    |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps )  ->  A. u  e.  W  A. x  e.  X  A. y  e.  X  (
 ( x - t
 y ) . w u )  =  (
 ( x . w u ) - w
 ( y . w u ) ) )
 
Theoremglmrngo 24835 Generating a left module from a ring. (Contributed by FL, 29-May-2014.)
 |-  + t  =  ( 1st `  R )   &    |-  . t  =  ( 2nd `  R )   =>    |-  ( R  e.  RingOps  ->  <. <. + t ,  . t >. ,  <. + t ,  . t >.
 >.  e.  Vec  )
 
Theoremvecax5c 24836 Multiplication by a difference of scalars. (Contributed by FL, 12-Sep-2010.)
 |-  X  =  ran  + t   &    |-  + t  =  ( 1st `  ( 1st `  R ) )   &    |-  - t  =  (  /g  `  + t )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  - w  =  (  /g  `  + w )   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   &    |-  . t  =  ( 2nd `  ( 1st `  R ) )   =>    |-  ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps ) 
 ->  ( ( U  e.  W  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( ( A - t B ) . w U )  =  (
 ( A . w U ) - w
 ( B . w U ) ) ) )
 
Theoremsvli2 24837* If a finite sequence of vectors  U ( k ) are linearly independant, two combinations of those vectors are equal iff the scalars are equal. (Contributed by FL, 9-Nov-2010.)
 |-  X  =  ran  + t   &    |-  0 t  =  (GId `  + t )   &    |-  + t  =  ( 1st `  ( 1st `  R ) )   &    |-  . t  =  ( 2nd `  ( 1st `  R ) )   &    |-  0 w  =  (GId `  + w )   &    |-  + w  =  ( 1st `  ( 2nd `  R ) )   &    |-  W  =  ran  + w   &    |-  . w  =  ( 2nd `  ( 2nd `  R ) )   =>    |-  ( ( ( R  e.  Vec  /\  <. + t ,  . t >.  e.  RingOps  /\  N  e.  NN )  /\  ( A. k  e.  ( 1 ... N ) U  e.  W  /\  A. k  e.  ( 1 ... N ) S1  e.  X  /\  A. k  e.  ( 1
 ... N ) S 2  e.  X ) 
 /\  A. s  e.  ( X  ^m  ( 1 ...
 N ) ) (
 prod_ k  e.  (
 1 ... N ) + w ( ( s `
  k ) . w U )  =  0 w  ->  A. k  e.  ( 1 ... N ) ( s `  k )  =  0 t ) )  ->  ( prod_ k  e.  (
 1 ... N ) + w ( S1 . w U )  =  prod_ k  e.  ( 1 ...
 N ) + w
 ( S 2 . w U )  <->  A. k  e.  (
 1 ... N ) S1  =  S 2 ) )
 
Syntaxcsvec 24838 Extend class notation with the class of all generic subspace vector spaces and modules.
 class  SubVec
 
Definitiondf-svs 24839* A sub-vector space  v of a vector space  x is a vector space that has the same scalar set than 
x, whose addition and whose multiplication are restrictions of those of  x. (Contributed by FL, 30-Dec-2010.)
 |-  SubVec  =  ( x  e.  Vec  |->  { v  e.  Vec  |  ( ( 1st `  v )  =  ( 1st `  x )  /\  ( 1st `  ( 2nd `  v ) ) 
 C_  ( 1st `  ( 2nd `  x ) ) 
 /\  ( 2nd `  ( 2nd `  v ) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  ran  ( 1st `  ( 2nd `  v
 ) ) ) ) ) } )
 
Theoremsvs2 24840* A textbook definition. A sub-vector space of a vector space  x is a subset that is itself a vector space under the inherited operations. (Contributed by FL, 31-Dec-2010.)
 |-  SubVec  =  ( x  e.  Vec  |->  { v  e.  Vec  |  ( ( 1st `  v )  =  ( 1st `  x )  /\  E. w  e. 
 ~P  ran  ( 1st `  ( 2nd `  x ) ) ( ( 1st `  ( 2nd `  v ) )  =  ( ( 1st `  ( 2nd `  x ) )  |`  ( w  X.  w ) )  /\  ( 2nd `  ( 2nd `  v
 ) )  =  ( ( 2nd `  ( 2nd `  x ) )  |`  ( ran  ( 1st `  ( 1st `  x ) )  X.  w ) ) ) ) } )
 
Theoremsvs3 24841* A very concise definition of a subspace of a vector space. (Contributed by FL, 30-Dec-2010.)
 |-  SubVec  =  ( x  e.  Vec  |->  { v  e.  Vec  |  ( ( 1st `  v )  =  ( 1st `  x )  /\  ( 1st `  ( 2nd `  v ) ) 
 C_  ( 1st `  ( 2nd `  x ) ) 
 /\  ( 2nd `  ( 2nd `  v ) ) 
 C_  ( 2nd `  ( 2nd `  x ) ) ) } )
 
16.12.23  Real vector spaces
 
Syntaxcvr 24842 Extend class notation with the class of all real vector spaces.
 class  RVec
 
Definitiondf-vr 24843* Define the class of all real vector spaces. The definition of a  RVec and a  CVec OLD don't much differ. There may be a way to get both in only one definition. A  RVec seems mandatory if one wants to do classical cartesian geometry. We can't use a  CVec OLD instead. Changing the field changes important properties such as the dimension. (Contributed by FL, 16-Nov-2008.)
 |-  RVec  =  { <. g ,  s >.  |  ( g  e.  AbelOp 
 /\  s : ( RR  X.  ran  g
 ) --> ran  g  /\  A. x  e.  ran  g
 ( ( 1 s x )  =  x 
 /\  A. y  e.  RR  ( A. z  e.  ran  g ( y s ( x g z ) )  =  ( ( y s x ) g ( y s z ) ) 
 /\  A. z  e.  RR  ( ( ( y  +  z ) s x )  =  ( ( y s x ) g ( z s x ) ) 
 /\  ( ( y  x.  z ) s x )  =  ( y s ( z s x ) ) ) ) ) ) }
 
Theoremvrrel 24844 The class of all real vector spaces is a relation. (Contributed by FL, 16-Nov-2008.)
 |-  Rel  RVec
 
Theoremvri 24845* The properties of a real vector space, which is an abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of real numbers. The variable  W was chosen because  _V is already used for the universal class. (Contributed by FL, 16-Nov-2008.)
 |-  G  =  ( 1st `  W )   &    |-  S  =  ( 2nd `  W )   &    |-  X  =  ran  G   =>    |-  ( W  e.  RVec  ->  ( G  e.  AbelOp  /\  S : ( RR  X.  X ) --> X  /\  A. x  e.  X  ( ( 1 S x )  =  x  /\  A. y  e.  RR  ( A. z  e.  X  ( y S ( x G z ) )  =  ( ( y S x ) G ( y S z ) )  /\  A. z  e.  RR  (
 ( ( y  +  z ) S x )  =  ( ( y S x ) G ( z S x ) )  /\  ( ( y  x.  z ) S x )  =  ( y S ( z S x ) ) ) ) ) ) )
 
16.12.24  Matrices
 
Syntaxcmmat 24846 Addition of matrices.
 class  + m
 
Syntaxcsmat 24847 Scalar multiplication of matrices.
 class  x m
 
Syntaxcxmat 24848 Multiplication of matrices.
 class  x m
 
Definitiondf-amat 24849* Matrix addition. Meaningful if  g is a  Magma (a binary internal operation) at least. Experimental. (Contributed by FL, 29-Aug-2010.)
 |-  + m  =  ( g  e.  _V  |->  { <. <. m ,  n >. ,  o >.  |  E. j  e.  NN  E. k  e.  NN  ( m :
 ( ( 1 ... j )  X.  (
 1 ... k ) ) --> dom  dom  g  /\  n : ( ( 1
 ... j )  X.  ( 1 ... k
 ) ) --> dom  dom  g  /\  o  =  ( x  e.  ( 1
 ... j ) ,  y  e.  ( 1
 ... k )  |->  ( ( m `  <. x ,  y >. ) g ( n `  <. x ,  y >. ) ) ) ) } )
 
Definitiondf-smat 24850* Matrix left scalar multiplication. Meaningful if  g is a binary external operation. Experimental. (Contributed by FL, 29-Aug-2010.)
 |-  x m  =  ( g  e.  _V  |->  { <. <. m ,  a >. ,  o >.  |  E. j  e.  NN  E. k  e.  NN  ( m :
 ( ( 1 ... j )  X.  (
 1 ... k ) ) --> ran  dom  g  /\  a  e.  dom  dom  g  /\  o  =  ( x  e.  ( 1 ... j ) ,  y  e.  ( 1 ... k
 )  |->  ( a g ( m `  <. x ,  y >. ) ) ) ) } )
 
Definitiondf-mmat 24851* Matrix multiplication. Meaningful if  <. g ,  h >. is a ring at least.  prod_ here should be 
sum_ (to be traditional). But in set.mm  sum_ is  CC oriented and has a limit definition embedded and thus doesn't fit the needs of this generic definition. Experimental. (Contributed by FL, 29-Aug-2010.)
 |-  x m  =  ( g  e.  _V ,  h  e. 
 _V  |->  { <. <. m ,  n >. ,  o >.  |  E. j  e.  NN  E. k  e.  NN  E. l  e. 
 NN  ( m :
 ( ( 1 ... j )  X.  (
 1 ... k ) ) --> dom  dom  g  /\  n : ( ( 1
 ... k )  X.  ( 1 ... l
 ) ) --> dom  dom  g  /\  o  =  ( x  e.  ( 1
 ... j ) ,  y  e.  ( 1
 ... l )  |->  prod_
 z  e.  ( 1
 ... k ) g ( ( m `  <. x ,  z >. ) h ( n `  <. z ,  y >. ) ) ) ) }
 )
 
16.12.25  Affine spaces
 
Syntaxcraffsp 24852 Extend class notation with the class of all R affine spaces.
 class  RAffSp
 
Definitiondf-raffsp 24853* Define a  RR affine space id est a  RR vector space  x ( called the free vectors class ) together with a function  y.  y associates to each vector a bijection from a set  t (called the space) to itself (here  t is retrieved from the operation.) Technically speaking,  y is a faithful (i.e. injective) and transitive group action (id est a group homomorphism whose range is the underlying set of a symmetry group ). Informally speaking the aim of all of that is to associate to each point of  t a unique point of  t through the "action" of a vector of  x and thus to formalize the idea of translation. When we have embedded the idea of translation it is easy to define a repere and thus all the cartesian geometry is available. (Contributed by FL, 29-Aug-2010.)
 |-  RAffSp  =  { <. x ,  y >.  |  ( x  e.  RVec  /\ 
 E. t ( y  e.  ( ( 1st `  x ) GrpOpHom  ( SymGrp `  t ) )  /\  y : ran  ( 1st `  x ) -1-1-> t  /\  A. a  e.  t  A. b  e.  t  E. v  e.  ran  ( 1st `  x ) ( ( y `  v ) `
  a )  =  b ) ) }
 
16.12.26  Intervals of reals and extended reals
 
Theorembsi 24854* Membership to the set of open intervals implied the existence of two bounds in the set of the extended reals. (Contributed by FL, 31-Oct-2007.) (Revised by Mario Carneiro, 3-Nov-2013.)
 |-  ( A  e.  ran  (,)  <->  E. x  e.  RR*  E. y  e.  RR*  A  =  ( x (,) y ) )
 
Theoremelioo1t3 24855 If an open interval has an element, then  A  <  B. (Contributed by FL, 14-Aug-2007.)
 |-  ( C  e.  ( A (,) B )  ->  A  <  B )
 
Theoremoisbmi 24856 An open interval with its upper bound equal to  -oo is empty. (Contributed by FL, 12-Sep-2007.)
 |-  ( A (,)  -oo )  =  (/)
 
Theoremoisbmj 24857 An open interval with its lower bound equal to  +oo is empty. (Contributed by FL, 12-Sep-2007.)
 |-  (  +oo (,) A )  =  (/)
 
Theoremtruni1 24858 Closure of translation in a half-infinite interval. (Contributed by FL, 11-Sep-2007.)
 |-  (
 ( A  e.  RR*  /\  D  e.  RR  /\  0  <  D )  ->  ( C  e.  ( A (,)  +oo )  ->  ( C  +  D )  e.  ( A (,)  +oo ) ) )
 
Theoremtruni2 24859 Closure of translation in a half-infinite interval. (Contributed by FL, 26-Jan-2009.)
 |-  (
 ( D  e.  RR  /\  0  <  D ) 
 ->  ( C  e.  ( A (,)  +oo )  ->  ( C  +  D )  e.  ( A (,)  +oo ) ) )
 
Theoremtruni3 24860 Closure of translation in a half-infinite interval. (Contributed by FL, 26-Jan-2009.)
 |-  (
 ( D  e.  RR  /\  0  <  D ) 
 ->  ( C  e.  (  -oo (,) A )  ->  ( C  -  D )  e.  (  -oo (,)
 A ) ) )
 
Theoremcbci 24861 The center belongs to a centered interval. (Contributed by FL, 5-Jan-2009.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR+ )  ->  A  e.  ( ( A  -  B ) (,) ( A  +  B ) ) )
 
Theoremoibbi1 24862 An open interval is included in a bound below interval. (Contributed by FL, 26-Jan-2009.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  ( A (,) B )  C_  ( A (,)  +oo )
 
Theoremoibbi2 24863 An open interval is included in a bound above interval. (Contributed by FL, 26-Jan-2009.)
 |-  ( A (,) B )  C_  (  -oo (,) B )
 
Theoremnelioo5 24864 Membership in an open interval of extended reals. (Contributed by FL, 7-Dec-2010.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  ( -.  C  e.  ( A (,) B )  <->  ( C  <_  A  \/  B  <_  C ) ) )
 
16.12.27  Topology
 
Theoremtopnem 24865 A topology is not empty. (Contributed by FL, 1-Jun-2008.)
 |-  ( J  e.  Top  ->  J  =/= 
 (/) )
 
Theoremclsint 24866 The closure of an intersection is included in the intersection of the closures. (Contributed by FL, 23-Feb-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  X )  ->  ( ( cls `  J ) `  ( S  i^i  T ) )  C_  (
 ( ( cls `  J ) `  S )  i^i  ( ( cls `  J ) `  S ) ) )
 
Theoremislp3 24867* The predicate " P is a limit point of  S " in terms of open sets. see islp2 16825, elcls 16758, islp 16820. (Contributed by FL, 31-Jul-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  S  C_  X  /\  P  e.  X )  ->  ( P  e.  (
 ( limPt `  J ) `  S )  <->  A. x  e.  J  ( P  e.  x  ->  ( x  i^i  ( S  \  { P }
 ) )  =/=  (/) ) ) )
 
Theoreminttop2 24868* The intersection of a family of topologies is a topology. (Contributed by FL, 19-Sep-2011.)
 |-  (
 ( I  =/=  (/)  /\  A. x  e.  I  J  e.  Top )  ->  |^|_ x  e.  I  J  e.  Top )
 
Theoreminttop3 24869 The intersection of a family of topologies is a topology. (Contributed by FL, 19-Sep-2011.)
 |-  (
 ( J  =/=  (/)  /\  J  C_ 
 Top )  ->  |^| J  e.  Top )
 
Theoreminttop4 24870 The intersection of two topologies is a topology. (Contributed by FL, 19-Sep-2011.)
 |-  (
 ( J  e.  Top  /\  K  e.  Top )  ->  ( J  i^i  K )  e.  Top )
 
Theoremunint2t 24871 The intersection of two topologies over the same underlying set  U. J is a topology over  U. J. compare uniin 3807. (Contributed by FL, 27-Nov-2011.)
 |-  (
 ( J  e.  Top  /\  K  e.  Top  /\  U. J  =  U. K )  ->  U. ( J  i^i  K )  =  U. J )
 
Theoremintfmu2 24872* The intersection of a family of topologies over the same underying set  U. J is a topology over  U. J. (Contributed by FL, 27-Nov-2011.)
 |-  (
 ( T  C_  Top  /\  J  e.  T  /\  A. x  e.  T  U. x  =  U. J ) 
 ->  U. |^| T  =  U. J )
 
Theoremapnei 24873* Any point has a neighborhood. (Contributed by FL, 15-Oct-2012.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  e.  X ) 
 ->  E. v  v  e.  ( ( nei `  J ) `  { A }
 ) )
 
Theoremnpmp 24874 A neighborhood of a point can't be empty. (Contributed by FL, 15-Oct-2012.)
 |-  (
 ( J  e.  Top  /\  A  e.  X  /\  N  e.  ( ( nei `  J ) `  { A } ) ) 
 ->  N  =/=  (/) )
 
Theorembasexre 24875 A basis for the standard topology over the extended reals. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
 |-  ( ran  (,)  u.  { RR* } )  e.  TopBases
 
Theoremstovr 24876 The standard topology over  RR*. (Contributed by FL, 15-Sep-2013.)
 |-  ( topGen `
  ( ran  (,)  u. 
 { RR* } ) )  e.  Top
 
Theoremcldifemp 24877 The closure of a class  S is empty iff  S is empty. (Contributed by FL, 15-Sep-2013.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  S  C_  X )  ->  ( ( ( cls `  J ) `  S )  =  (/)  <->  S  =  (/) ) )
 
16.12.28  Continuous functions
 
Theoremcnrsfin 24878 A mapping remains continuous when the topology associated to its domain is replaced by a finer one. (Contributed by FL, 22-May-2008.)
 |-  (
 ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  ->  ( ( F  e.  ( J  Cn  L ) 
 /\  U. J  =  U. K  /\  J  C_  K )  ->  F  e.  ( K  Cn  L ) ) )
 
Theoremcnrscoa 24879 A mapping remains continuous when the topology associated to its range is replaced by a coarser one. (Contributed by FL, 1-Jun-2008.)
 |-  (
 ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  ->  ( ( F  e.  ( J  Cn  L ) 
 /\  U. L  =  U. K  /\  K  C_  L )  ->  F  e.  ( J  Cn  K ) ) )
 
Theoremmapdiscn 24880 Any mapping whose domain is associated to the discrete topology is continuous. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Mario Carneiro, 7-Apr-2015.)
 |-  B  =  U. J   =>    |-  ( ( J  e.  Top  /\  F : A --> B  /\  A  e.  D )  ->  F  e.  ( ~P A  Cn  J ) )
 
Theoremmapudiscn 24881 Any mapping whose range is associated to the undiscrete topology is continuous. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  A  =  U. J   &    |-  B  e.  _V   =>    |-  (
 ( J  e.  Top  /\  F : A --> B ) 
 ->  F  e.  ( J  Cn  { (/) ,  B } ) )
 
Theoremsallnei 24882* Two ways to state the set of all the neighborhoods. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  U. ran  ( nei `  J )  =  { v  |  ( v  C_  X  /\  E. g  e.  J  g  C_  v ) }
 )
 
Theoremnsn 24883* The neighborhoods of the singletons are neighborhoods. (Contributed by FL, 2-Aug-2009.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  U_ x  e.  X  ( ( nei `  J ) `  { x }
 )  C_  U. ran  ( nei `  J ) )
 
Theoremosneisi 24884* The non empty open sets are neighborhoods of the singletons. (Contributed by FL, 16-Jul-2009.)
 |-  X  =  U. J   =>    |-  ( ( J  e.  Top  /\  A  =/=  (/) )  ->  ( A  e.  J  ->  A  e.  U_ x  e.  X  ( ( nei `  J ) `  { x } ) ) )
 
Theoremelsubops 24885 The elements of a subbase are open sets. (Contributed by FL, 16-Apr-2012.) (Revised by Mario Carneiro, 14-Dec-2013.)
 |-  K  =  ( topGen `  ( fi `  S ) )   =>    |-  ( S  e.  A  ->  S  C_  K )
 
16.12.29  Homeomorphisms
 
Theoremdmhmph 24886  ~= is a relation whose domain is included in  Top. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)
 |-  dom  ~=  C_  Top
 
Theoremrnhmph 24887  ~= is a relation whose range is included in  Top. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)
 |-  ran  ~=  C_  Top
 
Theoremhmeogrplem 24888* Lemma for hmeogrp 24890. (Contributed by FL, 30-May-2014.)
 |-  G  =  ( x  e.  ( J  Homeo  J ) ,  y  e.  ( J 
 Homeo  J )  |->  ( x  o.  y ) )   =>    |-  ( ( A  e.  ( J  Homeo  J ) 
 /\  B  e.  ( J  Homeo  J ) ) 
 ->  ( A G B )  =  ( A  o.  B ) )
 
Theoremhmeogrpi 24889* Lemma for hmeogrp 24890. (Contributed by FL, 31-May-2014.)
 |-  G  =  ( x  e.  ( J  Homeo  J ) ,  y  e.  ( J 
 Homeo  J )  |->  ( x  o.  y ) )   &    |-  J  e.  Top   =>    |-  G  e.  GrpOp
 
Theoremhmeogrp 24890* Homeomorphisms on a topology  J is a group for composition. This means from Felix Klein's point of view that a set equipped with a topology is a geometry, namely the so-called rubber sheet geometry. (Contributed by FL, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 31-May-2014.)
 |-  G  =  ( x  e.  ( J  Homeo  J ) ,  y  e.  ( J 
 Homeo  J )  |->  ( x  o.  y ) )   =>    |-  ( J  e.  Top  ->  G  e.  GrpOp )
 
16.12.30  Initial and final topologies
 
Theoremintopcoaconlem3b 24891* The underlying set of the initial topology is the domain of the mappings  F. (Contributed by FL, 24-Apr-2012.) (Revised by Mario Carneiro, 25-Nov-2013.)
 |-  J  =  ( topGen `  ( fi ` 
 { x  |  E. i  e.  I  E. o  e.  K  x  =  ( `' F "
 o ) } )
 )   =>    |-  ( ( ( I  e.  A  /\  X  e.  B )  /\  I  =/= 
 (/)  /\  A. i  e.  I  ( K  e.  Top  /\  F : X --> U. K ) )  ->  U. J  =  X )
 
Theoremintopcoaconlem3 24892* The underlying set of the initial topology is the domain of the mappings  F. (Contributed by FL, 21-Apr-2012.) (Revised by Mario Carneiro, 25-Nov-2013.)
 |-  J  =  ( topGen `  ( fi ` 
 { x  |  E. i  e.  I  E. o  e.  K  x  =  ( `' F "
 o ) } )
 )   &    |-  I  e.  A   &    |-  X  e.  B   &    |-  A. i  e.  I  ( K  e.  Top  /\  F : X --> U. K )   &    |-  I  =/=  (/)   =>    |- 
 U. J  =  X
 
Theoremintopcoaconb 24893* The initial topology is the coarsest one making the functions  F continuous . (Contributed by FL, 14-May-2012.) (Revised by Mario Carneiro, 26-Nov-2013.)
 |-  J  =  ( topGen `  ( fi ` 
 { x  |  E. i  e.  I  E. o  e.  K  x  =  ( `' F "
 o ) } )
 )   &    |-  I  e.  A   &    |-  X  e.  B   &    |-  A. i  e.  I  ( K  e.  Top  /\  F : X --> U. K )   &    |-  I  =/=  (/)   =>    |-  J  =  |^| { t  e.  Top  |  A. i  e.  I  F  e.  ( t  Cn  K ) }
 
Theoremintopcoaconc 24894* The initial topology makes the functions  F continuous. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 26-Nov-2013.)
 |-  J  =  ( topGen `  ( fi ` 
 { x  |  E. i  e.  I  E. o  e.  K  x  =  ( `' F "
 o ) } )
 )   &    |-  I  e.  A   &    |-  X  e.  B   &    |-  A. i  e.  I  ( K  e.  Top  /\  F : X --> U. K )   &    |-  I  =/=  (/)   =>    |- 
 A. i  e.  I  F  e.  ( J  Cn  K )
 
Theoremqusp 24895* A quotient space is a topology. (Contributed by FL, 4-Jun-2007.)
 |-  X  =  U. J   &    |-  R  Er  A   =>    |-  ( J  e.  Top  ->  { x  |  ( x  C_  ( X /. R )  /\  U. x  e.  J ) }  e.  Top )
 
Theoremintcont 24896 If  F is continous over two topologies  J and  K then it is continuous over  ( J  i^i  K
). (Contributed by FL, 27-Nov-2011.)
 |-  (
 ( ( J  e.  Top  /\  K  e.  Top  /\  L  e.  Top )  /\  U. J  =  U. K  /\  ( F  e.  ( J  Cn  L )  /\  F  e.  ( K  Cn  L ) ) ) 
 ->  F  e.  ( ( J  i^i  K )  Cn  L ) )
 
Syntaxctopx 24897 Extend class notation with a function whose value is a product topology.
 class  topX
 
Definitiondf-prtop 24898* The product topology of a family  f of topologies is the coarsest topology over the product of the underlying sets that makes the projections continuous. (Bourbaki TG I.14 ex. 3) Experimental. (Contributed by FL, 4-Dec-2011.)
 |-  topX  =  { <. f ,  y >.  |  ( f : dom  f --> Top  /\  y  =  |^| { t  e.  Top  |  ( U. t  =  X_ x  e. 
 dom  f U. (
 f `  x )  /\  A. i  e.  dom  f ( X_ x  e.  dom  f U. (
 f `  x )  pr  i )  e.  (
 t  Cn  ( f `  i ) ) ) } ) }
 
Theoremusptoplem 24899* Lemma for usptop 24903. (Contributed by FL, 5-Dec-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)
 |-  (
 ( F : I --> Top  /\  I  e.  A )  ->  ~P X_ x  e.  I  U. ( F `  x )  e.  { t  e.  Top  |  ( U. t  =  X_ x  e.  I  U. ( F `
  x )  /\  A. i  e.  I  (
 X_ x  e.  I  U. ( F `  x )  pr  i )  e.  ( t  Cn  ( F `  i ) ) ) } )
 
Theoremistopx 24900* Definition of the product topology of a family of topologies  F. (Contributed by FL, 4-Dec-2011.) (Revised by Mario Carneiro, 26-Jan-2015.)
 |-  (
 ( F : I --> Top  /\  I  e.  A )  ->  (  topX  `  F )  =  |^| { t  e.  Top  |  ( U. t  =  X_ x  e.  I  U. ( F `
  x )  /\  A. i  e.  I  (
 X_ x  e.  I  U. ( F `  x )  pr  i )  e.  ( t  Cn  ( F `  i ) ) ) } )
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