HomeHome Metamath Proof Explorer
Theorem List (p. 249 of 315)
< 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
(1-21490)
  Hilbert Space Explorer  Hilbert Space Explorer
(21491-23013)
  Users' Mathboxes  Users' Mathboxes
(23014-31421)
 

Theorem List for Metamath Proof Explorer - 24801-24900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltrset 24801* A left translation is a set. (Contributed by FL, 28-Apr-2012.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  F  e.  _V )
 
Theoremltrran2 24802* The range of a left translation. The term  A is a constant. (Contributed by FL, 28-Apr-2012.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  ran  F  =  X )
 
Theoremltrooo 24803* A left translation is a bijection. The term  A is a constant. (Contributed by FL, 29-Apr-2012.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  F : X -1-1-onto-> X )
 
Theoremltrcmp 24804* Left translation expressed as a composite. (Contributed by FL, 3-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  F  =  ( G  o.  ( x  e.  X  |->  <. A ,  x >. ) ) )
 
Theoremltrinvlem 24805* The converse of a left translation. The term  A is a constant. (Contributed by FL, 30-Apr-2012.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   &    |-  N  =  ( inv `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X ) 
 ->  `' F  =  ( x  e.  X  |->  ( ( N `  A ) G x ) ) )
 
Theoremcmpltr2 24806* Composite of two left translations. The terms  A and 
B are constant. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   &    |-  H  =  ( x  e.  X  |->  ( B G x ) )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( F  o.  H )  =  ( x  e.  X  |->  ( ( A G B ) G x ) ) )
 
Theoremcmpltr 24807* Composite of two left translations. The terms  A and 
B are constant. Don't use. See cmpltr2 24806. (Contributed by FL, 2-Jul-2012.) (Revised by Mario Carneiro, 2-Jun-2014.)
 |-  F  =  ( x  e.  X  |->  ( A G x ) )   &    |-  X  =  ran  G   &    |-  H  =  ( x  e.  X  |->  ( B G x ) )   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( F  o.  H )  =  ( x  e.  X  |->  ( ( A G B ) G x ) ) )
 
Theoremcmperltr 24808* A right and left translation expressed as a composite. Note that  x and  y can't be the same. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( ( A G x ) G B ) )   &    |-  E  =  ( y  e.  X  |->  ( A G y ) )   &    |-  H  =  ( x  e.  X  |->  ( x G B ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  F  =  ( E  o.  H ) )
 
Theoremcmprltr 24809* Composite of two right and left translations. Note that  x and  y can't be the same. See cmprltr2 24810 for a more general version. (Contributed by FL, 2-Jul-2012.) (Proof shortened by Mario Carneiro, 26-Jul-2014.)
 |-  F  =  ( y  e.  X  |->  ( ( A G y ) G C ) )   &    |-  E  =  ( x  e.  X  |->  ( ( B G x ) G D ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( F  o.  E )  =  ( x  e.  X  |->  ( ( ( A G B ) G x ) G ( D G C ) ) ) )
 
Theoremcmprltr2 24810* Composite of two right and left translations. No restriction:  x and  z can be equal. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( z  e.  X  |->  ( ( A G z ) G C ) )   &    |-  E  =  ( x  e.  X  |->  ( ( B G x ) G D ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( F  o.  E )  =  ( x  e.  X  |->  ( ( ( A G B ) G x ) G ( D G C ) ) ) )
 
Theoremrltrdom 24811* The domain of a right and left translation. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( ( A G x ) G B ) )   &    |-  X  =  ran  G   =>    |-  dom 
 F  =  X
 
Theoremrltrset 24812* A right and left translation is a set. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( ( A G x ) G B ) )   &    |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  F  e.  _V )
 
Theoremrltrran 24813* The range of a right and left translation. Note that  A and  B are constant. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( ( A G x ) G B ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ran  F  =  X )
 
Theoremrltrooo 24814* A right and left translation is a bijection. (Contributed by FL, 2-Jul-2012.)
 |-  F  =  ( x  e.  X  |->  ( ( A G x ) G B ) )   &    |-  X  =  ran  G   =>    |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  F : X -1-1-onto-> X )
 
18.12.19  Fields and Rings
 
Theoremcom2i 24815* Property of a commutative structure with two operation. (Contributed by FL, 14-Feb-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  Com2  ->  A. a  e.  X  A. b  e.  X  (
 a H b )  =  ( b H a ) )
 
Theoremrngmgmbs3 24816* The domain of the first variable of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.)
 |-  (
 ( G : ( X  X.  X ) --> X  /\  E. u  e.  X  A. x  e.  X  ( ( x G u )  =  x  /\  ( u G x )  =  x ) )  ->  dom  dom  G  =  X )
 
Theoremrngodmdmrn 24817 In a unital ring the range of the multiplication equals the domain of the first variable. (Contributed by FL, 24-Jan-2010.)
 |-  H  =  ( 2nd `  R )   =>    |-  ( R  e.  RingOps  ->  dom  dom  H  =  ran  H )
 
Theoremrngodmeqrn 24818 In a unital ring the domain of the first operand of the addition equals the domain of the second operand of the addition. (Contributed by FL, 11-Feb-2010.)
 |-  G  =  ( 1st `  R )   =>    |-  ( R  e.  RingOps  ->  dom  dom  G  =  ran  dom 
 G )
 
Theoremununr 24819* The unit of a unital ring is unique. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 23-Dec-2013.)
 |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  ( 1st `  R )   =>    |-  ( R  e.  RingOps  ->  E! x  e.  X  A. y  e.  X  ( ( x H y )  =  y  /\  ( y H x )  =  y ) )
 
Theoremrngoinvcl 24820 The additive inverse of a unital ring element pertains to the unital ring. (Contributed by FL, 18-Apr-2010.)
 |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  (
 ( <. G ,  H >.  e.  RingOps  /\  A  e.  X )  ->  ( N `
  A )  e.  X )
 
Theoremmultinv 24821 Multiplication by an additive inverse. (Contributed by FL, 2-Sep-2009.)
 |-  X  =  ran  G   &    |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( inv `  G ) `  A ) H B )  =  ( ( inv `  G ) `  ( A H B ) ) )
 
Theoremmultinvb 24822 Multiplication by an additive inverse. (Contributed by FL, 6-Sep-2009.)
 |-  X  =  ran  G   &    |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H ( ( inv `  G ) `  B ) )  =  ( ( inv `  G ) `  ( A H B ) ) )
 
Theoremmult2inv 24823 Multiplication of two additive inverses. (Contributed by FL, 6-Sep-2009.)
 |-  X  =  ran  G   &    |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( inv `  G ) `  A ) H ( ( inv `  G ) `  B ) )  =  ( A H B ) )
 
Theoremrngounval2 24824* The value of the unit of a ring. (Contributed by FL, 12-Feb-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  U  =  (GId `  H )   =>    |-  ( R  e.  RingOps  ->  U  =  ( iota_ u  e.  X A. x  e.  X  ( ( u H x )  =  x  /\  ( x H u )  =  x ) ) )
 
TheoremisfldOLD 24825* The predicate "is a field". (Contributed by FL, 6-Sep-2009.)
 |-  (
 ( G  e.  A  /\  H  e.  B ) 
 ->  ( <. G ,  H >.  e.  Fld  <->  ( <. G ,  H >.  e.  DivRingOps  /\  A. x  e.  ran  G A. y  e.  ran  G ( x H y )  =  ( y H x ) ) ) )
 
Theoremfldi 24826* The "axioms" of a field. (Contributed by FL, 15-Sep-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  Fld  ->  ( ( G  e.  AbelOp  /\  H : ( X  X.  X ) --> X ) 
 /\  ( A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( ( x H y ) H z )  =  ( x H ( y H z ) )  /\  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) )  /\  ( ( x G y ) H z )  =  ( ( x H z ) G ( y H z ) ) )  /\  E. x  e.  X  A. y  e.  X  ( ( x H y )  =  y  /\  ( y H x )  =  y ) )  /\  ( ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
 ) ) )  e. 
 GrpOp  /\  A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) ) ) )
 
Theoremfldax1 24827 1st "axiom" of a field. The addition is an abelian group. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   =>    |-  ( R  e.  Fld  ->  G  e.  AbelOp )
 
Theoremfldax2 24828 2nd "axiom" of a field. The multiplication is an internal operation. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  Fld  ->  H : ( X  X.  X ) --> X )
 
Theoremfldax3 24829* 3rd "axiom" of a field. The multiplication is associative. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  Fld  ->  A. x  e.  X  A. y  e.  X  A. z  e.  X  (
 ( x H y ) H z )  =  ( x H ( y H z ) ) )
 
Theoremfldax4 24830* 4th "axiom" of a field. The multiplication is distributive. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  Fld  ->  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) ) )
 
Theoremfldax5 24831* 5th "axiom" of a field. Existence of a neutral element. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  Fld  ->  E. x  e.  X  A. y  e.  X  ( y H x )  =  y )
 
Theoremfldax6 24832 6th "axiom" of a field. The multiplication is a group on the underlying set deprived from zero. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  Fld  ->  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
 ) ) )  e. 
 GrpOp )
 
Theoremfldax7 24833* 7th "axiom" of a field. The multiplication is commutative. (Contributed by FL, 11-Jul-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e.  Fld  ->  A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) )
 
Theoremzrfld 24834 The zero ring is not a field. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  A  e.  _V   =>    |- 
 -.  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. }
 >.  e.  Fld
 
Theoremzerdivemp1 24835* In a unitary ring a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  ( X 
 \  Z )  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) )
 
Theoremrngoridfz 24836* In a unitary ring a left invertible element is different from zero iff  1  =/=  0. (Contributed by FL, 18-Apr-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( A  =/=  Z  <->  U  =/=  Z ) )
 
Theoremzintdom 24837  ZZ is a commutative ring. (Contributed by FL, 18-Apr-2010.)
 |-  <.  +  ,  x.  >.  e.  ( RingOps  i^i  Com2 )
 
Syntaxctofld 24838 Extend class notation with the class of all totally ordered fields.
 class  Tofld
 
Definitiondf-tofld 24839* Definition of a totally ordered field. Experimental. (Contributed by FL, 27-Jun-2011.)
 |-  Tofld  =  { <.
 <. g ,  h >. ,  r >.  |  ( <. g ,  h >.  e. 
 Fld  /\  ( r  e.  TosetRel  /\  U. U. r  = 
 ran  g )  /\  A. x  e.  U. U. r A. y  e.  U. U. r A. z  e. 
 U. U. r ( ( x r y  ->  ( x g z ) r ( y g z ) )  /\  ( (GId `  g )
 r z  ->  ( x h z ) r ( y h z ) ) ) ) }
 
Syntaxczerodiv 24840 Extend class notation with the class of all the zero divisors.
 class  zeroDiv
 
Definitiondf-zd 24841* Definition of the zero divisors of a ring. Experimental. (Contributed by FL, 27-Jun-2011.)
 |-  zeroDiv  =  { <. r ,  y >.  |  y  =  { a  e.  ran  ( 1st `  r
 )  |  ( a  =/=  (GId `  ( 1st `  r ) ) 
 /\  E. b  e.  ran  ( 1st `  r )
 ( b  =/=  (GId `  ( 1st `  r
 ) )  /\  (
 a ( 2nd `  r
 ) b )  =  (GId `  ( 1st `  r ) ) ) ) } }
 
18.12.20  Ideals
 
Syntaxcidln 24842 Extend class notation with the class of ideals.
 class IdlNEW
 
Definitiondf-idlNEW 24843* Define the class of (two-sided) ideals of a ring  R. A subset of  R is an ideal if it contains  0, is closed under addition, and is closed under multiplication on either side by any element of  R. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
 |- IdlNEW  =  ( r  e.  Ring  |->  { i  e.  ~P ( Base `  r
 )  |  ( ( 0g `  r )  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x ( +g  `  r
 ) y )  e.  i  /\  A. z  e.  ( Base `  r )
 ( ( z ( .r `  r ) x )  e.  i  /\  ( x ( .r
 `  r ) z )  e.  i ) ) ) } )
 
TheoremidlvalNEW 24844* The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
 |-  P  =  ( +g  `  R )   &    |-  T  =  ( .r
 `  R )   &    |-  B  =  ( Base `  R )   &    |-  Z  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  (IdlNEW `  R )  =  { i  e.  ~P B  |  ( Z  e.  i  /\  A. x  e.  i  ( A. y  e.  i  ( x P y )  e.  i  /\  A. z  e.  B  ( ( z T x )  e.  i  /\  ( x T z )  e.  i ) ) ) } )
 
TheoremisidlNEW 24845* The predicate "is an ideal of the ring  R." (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by FL, 29-Oct-2014.)
 |-  P  =  ( +g  `  R )   &    |-  T  =  ( .r
 `  R )   &    |-  B  =  ( Base `  R )   &    |-  Z  =  ( 0g `  R )   =>    |-  ( R  e.  Ring  ->  ( I  e.  (IdlNEW `  R )  <->  ( I  C_  B  /\  Z  e.  I  /\  A. x  e.  I  ( A. y  e.  I  ( x P y )  e.  I  /\  A. z  e.  B  (
 ( z T x )  e.  I  /\  ( x T z )  e.  I ) ) ) ) )
 
18.12.21  Generic modules and vector spaces (New Structure builder)
 
Syntaxcact 24846 Extend class notation to include actions.
 class  Action
 
Definitiondf-act 24847* 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 ) }
 
18.12.22  Generic modules and vector spaces
 
Syntaxcvec 24848 Extend class notation with the class of all generic vector spaces and modules.
 class  Vec
 
Definitiondf-vec 24849* 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 24850* 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 24851* 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 24852 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 24853 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 24854* 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 24855* 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 24856* 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 24857* 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 24858 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 24859 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 24860 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 24861 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 24862 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 24863 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 24864 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 24865 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 24866 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 24867 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 24868 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 24869 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 24870 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 24871 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 24872 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 24873 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 24874 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 24875 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 24876 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 24877 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 24878 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 24879 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 24880* 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 24881 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 24882 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 24883* 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 24884 Extend class notation with the class of all generic subspace vector spaces and modules.
 class  SubVec
 
Definitiondf-svs 24885* 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 24886* 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 24887* 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 ) ) ) } )
 
18.12.23  Real vector spaces
 
Syntaxcvr 24888 Extend class notation with the class of all real vector spaces.
 class  RVec
 
Definitiondf-vr 24889* 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 24890 The class of all real vector spaces is a relation. (Contributed by FL, 16-Nov-2008.)
 |-  Rel  RVec
 
Theoremvri 24891* 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 ) ) ) ) ) ) )
 
18.12.24  Matrices
 
Syntaxcmmat 24892 Addition of matrices.
 class  + m
 
Syntaxcsmat 24893 Scalar multiplication of matrices.
 class  x m
 
Syntaxcxmat 24894 Multiplication of matrices.
 class  x m
 
Definitiondf-amat 24895* 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 24896* 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 24897* 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 >. ) ) ) ) }
 )
 
18.12.25  Affine spaces
 
Syntaxcraffsp 24898 Extend class notation with the class of all R affine spaces.
 class  RAffSp
 
Definitiondf-raffsp 24899* 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 ) ) }
 
18.12.26  Intervals of reals and extended reals
 
Theorembsi 24900* 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 ) )
    < 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 31401-31421
  Copyright terms: Public domain < Previous  Next >