HomeHome Metamath Proof Explorer
Theorem List (p. 251 of 313)
< 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
(1-21423)
  Hilbert Space Explorer  Hilbert Space Explorer
(21424-22946)
  Users' Mathboxes  Users' Mathboxes
(22947-31284)
 

Theorem List for Metamath Proof Explorer - 25001-25100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnegveudr 25001* Existential uniqueness of vector negatives. (Contributed by FL, 29-May-2014.)
 |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  A  e.  ( RR  ^m  (
 1 ... N ) ) 
 /\  B  e.  ( RR  ^m  ( 1 ...
 N ) ) ) 
 ->  E! x  e.  ( RR  ^m  ( 1 ...
 N ) ) ( A + w x )  =  B )
 
Theoremissubcv 25002* Substraction of complex vectors in a space of dimension  n. (Contributed by FL, 15-Sep-2013.)
 |-  + w  =  (  + cv `  N )   &    |-  - w  =  (  - cv  `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ...
 N ) )  /\  V  e.  ( CC  ^m  ( 1 ... N ) ) )  ->  ( U - w V )  =  ( iota_ w  e.  ( CC  ^m  (
 1 ... N ) ) ( V + w w )  =  U ) )
 
Theoremsubaddv 25003 Relationship between subtraction and addition. (Contributed by FL, 30-May-2014.)
 |-  + w  =  (  + cv `  N )   &    |-  - w  =  (  - cv  `  N )   =>    |-  ( ( N  e.  NN  /\  ( A  e.  ( CC  ^m  ( 1
 ... N ) ) 
 /\  B  e.  ( CC  ^m  ( 1 ...
 N ) )  /\  C  e.  ( CC  ^m  ( 1 ... N ) ) ) ) 
 ->  ( ( A - w B )  =  C  <->  ( B + w C )  =  A )
 )
 
Theoremissubrv 25004* Addition of complex vectors. (Contributed by FL, 29-May-2014.)
 |-  - w  =  (  - cv  `  N )   =>    |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ...
 N ) )  /\  B  e.  ( CC  ^m  ( 1 ... N ) ) )  ->  ( A - w B )  =  ( x  e.  ( 1 ... N )  |->  ( ( A `
  x )  -  ( B `  x ) ) ) )
 
Theoremsubclcvd 25005 Closure law for vector substraction. (Contributed by FL, 15-Sep-2013.)
 |-  - w  =  (  - cv  `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ...
 N ) )  /\  V  e.  ( CC  ^m  ( 1 ... N ) ) )  ->  ( U - w V )  e.  ( CC  ^m  ( 1 ... N ) ) )
 
Theoremsubclrvd 25006 Closure law for vector substraction. (Contributed by FL, 29-May-2014.)
 |-  - w  =  (  - cv  `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ...
 N ) )  /\  V  e.  ( RR  ^m  ( 1 ... N ) ) )  ->  ( U - w V )  e.  ( RR  ^m  ( 1 ... N ) ) )
 
Syntaxcnegcv 25007 Extends class notation with the negative of a complex vector.
 class  - cv
 
Definitiondf-ucv 25008* Negative of a complex vector. Experimental. (Contributed by FL, 15-Sep-2013.)
 |-  - cv  =  ( n  e.  NN  |->  ( u  e.  ( CC  ^m  ( 1 ... n ) )  |->  ( ( 0 cv `  n ) (  - cv  `  n ) u ) ) )
 
Theoremisucv 25009 Negative of a complex vector. (Contributed by FL, 15-Sep-2013.)
 |-  ~ w  =  ( - cv `  N )   &    |-  0 w  =  ( 0 cv `  N )   &    |- 
 - w  =  (  - cv  `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ...
 N ) ) ) 
 ->  ( ~ w `  U )  =  (
 0 w - w U ) )
 
Theoremisucvr 25010 Negative of a complex vector. (Contributed by FL, 29-May-2014.)
 |-  ~ w  =  ( - cv `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ...
 N ) ) ) 
 ->  ( ~ w `  U )  e.  ( CC  ^m  ( 1 ...
 N ) ) )
 
Syntaxcsmcv 25011 Extends class notation with scalar multiplication of complex vectors.
 class  . cv
 
Definitiondf-mulcv 25012* Multiplication of complex vectors by a scalar in a space of dimension  n. Experimental. (Contributed by FL, 15-Sep-2013.)
 |-  . cv  =  ( n  e.  NN  |->  ( s  e. 
 CC ,  u  e.  ( CC  ^m  (
 1 ... n ) ) 
 |->  ( x  e.  (
 1 ... n )  |->  ( s  x.  ( u `
  x ) ) ) ) )
 
Theoremismulcv 25013* Multiplication of complex vectors by a scalar in a space of dimension  n. (Contributed by FL, 15-Sep-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  ( 1
 ... N ) ) )  ->  ( S . t U )  =  ( x  e.  (
 1 ... N )  |->  ( S  x.  ( U `
  x ) ) ) )
 
Theoremclsmulcv 25014 Closure of scalar multiplication. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  ( 1
 ... N ) ) )  ->  ( S . t U )  e.  ( CC  ^m  (
 1 ... N ) ) )
 
Theoremclsmulrv 25015 Closure of scalar multiplication. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( ( N  e.  NN  /\  S  e.  RR  /\  U  e.  ( RR  ^m  ( 1
 ... N ) ) )  ->  ( S . t U )  e.  ( RR  ^m  (
 1 ... N ) ) )
 
Theoremfnmulcv 25016 Functionality of scalar multiplication. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( N  e.  NN  ->  . t :
 ( CC  X.  ( CC  ^m  ( 1 ...
 N ) ) ) --> ( CC  ^m  (
 1 ... N ) ) )
 
Theoremmulone 25017 Multiplication of a vector by 1. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Jan-2015.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  (
 1 ... N ) ) )  ->  ( 1 . t U )  =  U )
 
Theoremvecscmonto 25018 Vector addition is onto. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( N  e.  NN  ->  . t :
 ( CC  X.  ( CC  ^m  ( 1 ...
 N ) ) )
 -onto-> ( CC  ^m  (
 1 ... N ) ) )
 
Theoremmulmulvec 25019 Connection between multiplication of complex numbers and scalar multiplication. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   =>    |-  ( ( N  e.  NN  /\  S  e.  CC  /\  ( T  e.  CC  /\  U  e.  ( CC  ^m  (
 1 ... N ) ) ) )  ->  (
 ( S  x.  T ) . t U )  =  ( S . t ( T . t U ) ) )
 
Theoremdistmlva 25020 Distribution of scalar multiplication over vector addition. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   &    |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  S  e.  CC  /\  ( U  e.  ( CC  ^m  ( 1 ... N ) )  /\  V  e.  ( CC  ^m  ( 1
 ... N ) ) ) )  ->  ( S . t ( U + w V ) )  =  ( ( S . t U ) + w ( S . t V ) ) )
 
Theoremdistsava 25021 "Distribution" of scalar addition. (Contributed by FL, 29-May-2014.)
 |-  . t  =  ( . cv `  N )   &    |-  + w  =  (  + cv `  N )   =>    |-  ( ( N  e.  NN  /\  S  e.  CC  /\  ( T  e.  CC  /\  U  e.  ( CC  ^m  (
 1 ... N ) ) ) )  ->  (
 ( S  +  T ) . t U )  =  ( ( S . t U ) + w ( T . t U ) ) )
 
Theoremtcnvec 25022 Nuples of complex numbers has a structure of vector space. (Contributed by FL, 29-May-2014.)
 |-  + w  =  (  + cv `  N )   &    |-  . t  =  ( . cv `  N )   =>    |-  ( N  e.  NN  ->  <. + w ,  . t >.  e.  CVec OLD )
 
Syntaxcdivcv 25023 Extends class notation with scalar division of complex vectors.
 class  / cv
 
Definitiondf-divcv 25024* Division of a complex vector by a scalar in a space of dimension  n. Experimental. (Contributed by FL, 29-May-2014.)
 |-  / cv  =  ( n  e.  NN  |->  ( u  e.  ( CC  ^m  ( 1 ... n ) ) ,  s  e.  ( CC  \  { 0 } )  |->  ( ( 1  /  s ) ( .
 cv `  n ) u ) ) )
 
Theoremisdivcv2 25025 Division of complex vectors by a scalar in a space of dimension  N. (Contributed by FL, 29-May-2014.)
 |-  / t  =  ( / cv `  N )   &    |- 
 . t  =  ( . cv `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ...
 N ) )  /\  S  e.  ( CC  \  { 0 } )
 )  ->  ( U / t S )  =  ( ( 1  /  S ) . t U ) )
 
Theoremdivclcvd 25026 Closure law for vector division by a scalar. (Contributed by FL, 29-May-2014.)
 |-  / t  =  ( / cv `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ...
 N ) )  /\  S  e.  ( CC  \  { 0 } )
 )  ->  ( U / t S )  e.  ( CC  ^m  (
 1 ... N ) ) )
 
Theoremdivclrvd 25027 Closure law for vector division by a scalar. (Contributed by FL, 29-May-2014.)
 |-  / t  =  ( / cv `  N )   =>    |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ...
 N ) )  /\  S  e.  ( RR  \  { 0 } )
 )  ->  ( U / t S )  e.  ( RR  ^m  (
 1 ... N ) ) )
 
16.12.44  Calculus
 
Syntaxcintvl 25028 Extend class notation to include intervals.
 class  Intvl
 
Definitiondf-intvl 25029 The intervals of  RR. (Contributed by FL, 29-May-2014.)
 |-  Intvl  =  ( ( ran  (,)  u.  ( ran  (,]  u.  ( ran  [,)  u.  ran  [,] ) ) )  i^i 
 ~P RR )
 
Theoremintvlset 25030 The set of intervals is a set. (Contributed by FL, 29-May-2014.)
 |-  Intvl  e.  _V
 
Theoremintrr 25031 An interval is a part of  RR. (Contributed by FL, 29-May-2014.)
 |-  ( I  e.  Intvl  ->  I  C_ 
 RR )
 
Theoremicccon2 25032 A closed-below, open-above interval is connected. (Contributed by FL, 30-May-2014.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR* )  ->  ( ( topGen `  ran  (,) )t  ( A [,) B ) )  e.  Con )
 
Theoremicccon3 25033 An open-below, closed-above interval is connected. (Contributed by FL, 30-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR )  ->  ( ( topGen `  ran  (,) )t  ( A (,] B ) )  e.  Con )
 
Theoremicccon4 25034 An open interval is connected. (Contributed by FL, 30-May-2014.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( topGen `  ran  (,) )t  ( A (,) B ) )  e.  Con )
 
Theoremintvconlem1 25035 All the intervals of  RR are connected. (Contributed by FL, 29-May-2014.)
 |-  ( I  e.  Intvl  ->  (
 ( topGen `  ran  (,) )t  I
 )  e.  Con )
 
Syntaxcder 25036 Extend class notation to include the derivative of a function.
 class  der
 
Definitiondf-der 25037* Derivative of a function  f at  p. Meaningful when the domain of  f is an interval of  RR,  p belongs to the domain of  f, the domain of  f is not  { p } and the values of  f are in  ( RR  ^m  ( 1 ... n
) ).

Bourbaki doesn't explain why he requires the domain of  f be an interval. Here are some hints. The domain of  f is an interval,  p belongs to the domain of  f and  dom  f  =  { p } guarantee  p is not an isolated point in  dom  f (df-islpt 24916). We have  ( v  i^i  ( dom  f  \  { p } ) )  =  ( ( v  i^i  dom  f
)  \  { p } ) (indif2 3354) but  ( v  i^i  dom  f )  =/=  {
p } since  p is not an isolated point in  dom  f and  ( v  i^i  ( dom  f  \  { p } ) )  =/=  (/) what is the condition required by trfil2 17509. And in this case the class  { u  |  E. v  e.  ( ( nei `  ( topGen `
 ran  (,) )
) `  { p } ) u  =  ( v  i^i  ( dom  f  \  { p } ) ) } is a filter. This latter condition is required by df-flimfrs 24911 and this definition is used by df-der 25037.

This sort of derivative might be extended easily to work with functions  f whose domain is a field  A and whose values are in a topological vector space whose scalars are in  A. The topologies would be changed accordingly. The domain of  f would be a neighborhood of  p. Experimental. (Contributed by FL, 29-May-2014.)

 |-  der  =  ( n  e.  NN ,  i  e.  Intvl  |->  ( f  e.  ( ( RR 
 ^m  ( 1 ... n ) )  ^m  i ) ,  p  e.  i  |->  ( ( ( (  topX  `  ( x  e.  ( 1 ... n )  |->  ( topGen `  ran  (,) ) ) ) 
 fLimfrs  ( topGen `  ran  (,) )
 ) `  ( i  \  { p } )
 ) `  <. p ,  ( x  e.  (
 i  \  { p } )  |->  ( ( ( f `  x ) (  - cv  `  n ) ( f `  p ) ) ( / cv `  n ) ( x  -  p ) ) )
 >. ) ) )
 
Theoremhdrmp 25038 Hard to describe. A picture can help. (Contributed by FL, 29-May-2014.)
 |-  (
 ( ( A  i^i  B )  =/=  (/)  /\  ( C  =/=  (/)  /\  D  =/=  (/) )  /\  ( A  u.  B )  =  ( C  u.  D ) )  ->  ( ( ( A  i^i  C )  =/=  (/)  /\  ( A  i^i  D )  =/=  (/) )  \/  ( ( B  i^i  C )  =/=  (/)  /\  ( B  i^i  D )  =/=  (/) ) ) )
 
Theoremisder 25039* The derivative of  F at point  P is the limit of the slope  F ( x )  -  F ( P )  /  x  -  P when  x tends to  P. Definition 1 of [BourbakiFVR] p. I.11. (Contributed by FL, 29-May-2014.)
 |-  J  =  (  topX  `  ( x  e.  ( 1 ... N )  |->  ( topGen `  ran  (,) ) ) )   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  S  =  ( x  e.  ( I 
 \  { P }
 )  |->  ( ( ( F `  x ) (  - cv  `  N ) ( F `  P ) ) ( / cv `  N ) ( x  -  P ) ) )   &    |-  D  =  ( N der I )   =>    |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  ( I  =/=  { P }  /\  F : I --> ( RR 
 ^m  ( 1 ...
 N ) ) ) )  ->  ( F D P )  =  ( ( ( J  fLimfrs  K ) `  ( I 
 \  { P }
 ) ) `  <. P ,  S >. ) )
 
16.12.45  Directed multi graphs
 
Syntaxcmgra 25040 Extend class notation with the class of directed multi graphs.
 class  Dgra
 
Definitiondf-mgra 25041* Definition of a directed multi graph. Loops are allowed and there may be more than one edge between the same pair of vertices. Isolated points are allowed. (Contributed by FL, 10-Jan-2008.)
 |-  Dgra  =  { <. <. d ,  c >. ,  u >.  |  ( d : dom  d --> u  /\  c : dom  d
 --> u ) }
 
Theoremismgra 25042 The predicate "is a directed multi graph". (Contributed by FL, 10-Jan-2008.)
 |-  (
 ( D  e.  A  /\  C  e.  B  /\  U  e.  F )  ->  ( <. <. D ,  C >. ,  U >.  e.  Dgra  <->  ( D : dom  D --> U  /\  C : dom  D --> U ) ) )
 
16.12.46  Category and deductive system underlying "structure"
 
Syntaxcalg 25043 Extend class notation with the class of structures used by  Cat OLD and  Ded.
 class  Alg
 
Syntaxcdom_ 25044 Extend class notation with the function returning the function domain of a category.
 class  dom_
 
Syntaxccod_ 25045 Extend class notation with the function returning the function codomain of a category.
 class  cod_
 
Syntaxcid_ 25046 Extend class notation with the function returning the function identity of a category.
 class  id_
 
Syntaxco_ 25047 Extend class notation with the function returning the composition of morphisms of a category.
 class  o_
 
Definitiondf-alg 25048*  Ded and  Cat OLD structure. Metamath for internal reasons doesn't like too large definitions. Then  Cat OLD has been split giving birth to  Ded and  Alg. If  Ded has a real mathematical use,  Alg is only here to give relief to Metamath. (Contributed by FL, 24-Oct-2007.)
 |-  Alg  =  { x  |  E. d E. c E. j E. r ( x  = 
 <. <. d ,  c >. ,  <. j ,  r >.
 >.  /\  ( d : dom  d --> dom  j  /\  c : dom  d --> dom  j  /\  j : dom  j --> dom  d
 )  /\  ( Fun  r  /\  dom  r  C_  ( dom  d  X.  dom  d
 )  /\  ran  r  C_  dom  d ) ) }
 
Definitiondf-dom_ 25049 Definition of  dom_. (Contributed by FL, 24-Oct-2007.)
 |-  dom_  =  ( 1st  o.  1st )
 
Definitiondf-cod_ 25050 Definition of  cod_. (Contributed by FL, 26-Oct-2007.)
 |-  cod_  =  ( 2nd  o.  1st )
 
Definitiondf-id_ 25051 Definition of  id_. (Contributed by FL, 26-Oct-2007.)
 |-  id_  =  ( 1st  o.  2nd )
 
Definitiondf-cmpa 25052 Definition of  o_. (Contributed by FL, 26-Oct-2007.)
 |-  o_  =  ( 2nd  o.  2nd )
 
Theoremisalg 25053 The predicate "has the structure required by  Ded and  Cat OLD." (Contributed by FL, 24-Oct-2007.)
 |-  M  =  dom  D   &    |-  O  =  dom  J   =>    |-  ( ( ( D  e.  A  /\  C  e.  B  /\  J  e.  F )  /\  R  e.  G )  ->  ( <. <. D ,  C >. , 
 <. J ,  R >. >.  e.  Alg  <->  ( ( D : M --> O  /\  C : M --> O  /\  J : O --> M ) 
 /\  ( Fun  R  /\  dom  R  C_  ( M  X.  M )  /\  ran 
 R  C_  M )
 ) ) )
 
Theorem1alg 25054 CatOLDegory  1 has the structure required by  Ded and  Cat OLD. (Contributed by FL, 30-Oct-2007.)
 |-  A  e.  _V   =>    |- 
 <. <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. }
 >. ,  <. { <. A ,  <. A ,  A >. >. } ,  { <. <. <. A ,  A >. ,  <. A ,  A >. >. ,  <. A ,  A >. >. } >. >.  e.  Alg
 
Theoremdomval 25055 Value of the domain function expressed with the  1st function. (Contributed by FL, 24-Oct-2007.)
 |-  D  =  ( dom_ `  T )   =>    |-  D  =  ( 1st `  ( 1st `  T ) )
 
Theoremcodval 25056 Value of the function codomain expressed with the  1st and  2nd functions. (Contributed by FL, 26-Oct-2007.)
 |-  C  =  ( cod_ `  T )   =>    |-  C  =  ( 2nd `  ( 1st `  T ) )
 
Theoremidval 25057 Value of the identity function expressed with the  1st and  2nd functions. (Contributed by FL, 26-Oct-2007.)
 |-  J  =  ( id_ `  T )   =>    |-  J  =  ( 1st `  ( 2nd `  T ) )
 
Theoremcmpval 25058 Value of the identity function expressed with the  2nd functions. (Contributed by FL, 26-Oct-2007.)
 |-  G  =  ( o_ `  T )   =>    |-  G  =  ( 2nd `  ( 2nd `  T ) )
 
Theoremalgi 25059 "Axiomatic" properties of  Alg. (Contributed by FL, 24-Oct-2007.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_
 `  T )   &    |-  M  =  dom  D   &    |-  O  =  dom  J   =>    |-  ( T  e.  Alg  ->  ( ( D : M
 --> O  /\  C : M
 --> O  /\  J : O
 --> M )  /\  ( Fun  R  /\  dom  R  C_  ( M  X.  M )  /\  ran  R  C_  M ) ) )
 
Theoremdoma 25060  ( dom_ `  T ) is a mapping from the morphisms of  T to the objects of  T. (Contributed by FL, 24-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   =>    |-  ( T  e.  Alg  ->  D : M --> O )
 
Theoremcoda 25061  ( cod_ `  T ) is a mapping from the morphisms of  T to the objects of  T. (Contributed by FL, 26-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( T  e.  Alg  ->  C : M --> O )
 
Theoremida 25062  ( id_ `  A ) is a mapping from the objects of  T to the morphisms of  T. (Contributed by FL, 26-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   =>    |-  ( T  e.  Alg  ->  J : O --> M )
 
Theoremidmoa 25063 An identity is a morphism. (Contributed by FL, 10-Jan-2008.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   =>    |-  (
 ( T  e.  Alg  /\  A  e.  O ) 
 ->  ( J `  A )  e.  M )
 
Theoremcmppfa 25064  ( o_ `  T ) is a partial operation on the morphisms of  T. (Contributed by FL, 26-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( T  e.  Alg  ->  ( Fun  R  /\  dom  R 
 C_  ( M  X.  M )  /\  ran  R  C_  M ) )
 
Theoremdcsda 25065  ( dom_ `  T ) and  ( cod_ `  T ) have the same domain. (Contributed by FL, 10-Jan-2008.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( T  e.  Alg  ->  dom  D  =  dom  C )
 
16.12.47  Deductive systems
 
Syntaxcded 25066 Extend class notation with the class of deductive systems.
 class  Ded
 
Definitiondf-ded 25067* Definition of a deductive system. Lambeck and Scott. Introduction to higher order categorical logic. p. 47. 1986. Unformally we can say a deductive system is a directed multi graph where for each object a specific morphism called identity of the object exists and where for some pairs of morphisms the composite exists. Deductive system are named so because morphisms may be interpreted as logical deductions, objects as sets of formulas and compositions as inferences. (Contributed by FL, 24-Oct-2007.)
 |-  Ded  =  { x  |  E. d E. c E. j E. r ( x  = 
 <. <. d ,  c >. ,  <. j ,  r >.
 >.  /\  ( ( <. <.
 d ,  c >. , 
 <. j ,  r >. >.  e.  Alg  /\  A. a  e. 
 dom  j ( ( d `  ( j `
  a ) )  =  a  /\  (
 c `  ( j `  a ) )  =  a )  /\  A. f  e.  dom  d A. g  e.  dom  d (
 <. g ,  f >.  e. 
 dom  r  <->  ( d `  g )  =  (
 c `  f )
 ) )  /\  ( A. f  e.  dom  d A. g  e.  dom  d ( ( d `
  g )  =  ( c `  f
 )  ->  ( d `  ( g r f ) )  =  ( d `  f ) )  /\  A. f  e.  dom  d A. g  e.  dom  d ( ( d `  g )  =  ( c `  f )  ->  ( c `
  ( g r f ) )  =  ( c `  g
 ) ) ) ) ) }
 
Theoremisded 25068* The predicate "is a deductive system". (Contributed by FL, 24-Oct-2007.)
 |-  M  =  dom  D   &    |-  O  =  dom  J   =>    |-  ( ( ( D  e.  A  /\  C  e.  B  /\  J  e.  F )  /\  R  e.  G )  ->  ( <. <. D ,  C >. , 
 <. J ,  R >. >.  e.  Ded  <->  ( ( <. <. D ,  C >. , 
 <. J ,  R >. >.  e.  Alg  /\  A. a  e.  O  ( ( D `
  ( J `  a ) )  =  a  /\  ( C `
  ( J `  a ) )  =  a )  /\  A. f  e.  M  A. g  e.  M  ( <. g ,  f >.  e.  dom  R  <-> 
 ( D `  g
 )  =  ( C `
  f ) ) )  /\  ( A. f  e.  M  A. g  e.  M  ( ( D `
  g )  =  ( C `  f
 )  ->  ( D `  ( g R f ) )  =  ( D `  f ) )  /\  A. f  e.  M  A. g  e.  M  ( ( D `
  g )  =  ( C `  f
 )  ->  ( C `  ( g R f ) )  =  ( C `  g ) ) ) ) ) )
 
Theoremdedi 25069* Properties of a deductive system. (Contributed by FL, 24-Oct-2007.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_
 `  T )   &    |-  M  =  dom  D   &    |-  O  =  dom  J   =>    |-  ( T  e.  Ded  ->  ( ( <. <. D ,  C >. ,  <. J ,  R >. >.  e.  Alg  /\  A. a  e.  O  ( ( D `  ( J `  a ) )  =  a  /\  ( C `  ( J `  a ) )  =  a )  /\  A. f  e.  M  A. g  e.  M  ( <. g ,  f >.  e.  dom  R  <-> 
 ( D `  g
 )  =  ( C `
  f ) ) )  /\  ( A. f  e.  M  A. g  e.  M  ( ( D `
  g )  =  ( C `  f
 )  ->  ( D `  ( g R f ) )  =  ( D `  f ) )  /\  A. f  e.  M  A. g  e.  M  ( ( D `
  g )  =  ( C `  f
 )  ->  ( C `  ( g R f ) )  =  ( C `  g ) ) ) ) )
 
Theorem1ded 25070 Category  1 is a deductive system. We can think of the morphism of Category  1 as corresponding to  ph |-  ph. (Contributed by FL, 30-Oct-2007.)
 |-  A  e.  _V   =>    |- 
 <. <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. }
 >. ,  <. { <. A ,  <. A ,  A >. >. } ,  { <. <. <. A ,  A >. ,  <. A ,  A >. >. ,  <. A ,  A >. >. } >. >.  e.  Ded
 
Theoremstrded 25071 Structure of a deductive system. (Contributed by FL, 28-Oct-2007.)
 |-  Ded  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
 
Theoremrelded 25072 A deductive system is a relation. (Contributed by FL, 28-Oct-2007.)
 |-  Rel  Ded
 
Theoremreldded 25073 The domain of a deductive system is a relation. (Contributed by FL, 28-Oct-2007.)
 |-  Rel  dom 
 Ded
 
Theoremrelrded 25074 The range of a deductive system is a relation. (Contributed by FL, 28-Oct-2007.)
 |-  Rel  ran 
 Ded
 
Theoremdedalg 25075 A deductive system is an "algebra". (Contributed by FL, 28-Oct-2007.)
 |-  ( T  e.  Ded  ->  T  e.  Alg  )
 
Theoremidosd 25076 The identity is a morphism which has the same object as its domain and its codomain. (Contributed by FL, 28-Oct-2007.)
 |-  O  =  dom  J   &    |-  D  =  (
 dom_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( ( T  e.  Ded  /\  A  e.  O ) 
 ->  ( ( D `  ( J `  A ) )  =  A  /\  ( C `  ( J `
  A ) )  =  A ) )
 
Theoremcmppfd 25077  ( G
( o_ `  T
) F ) is only defined when the domain of  G is the codomain of  F. (Contributed by FL, 29-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Ded  /\  F  e.  M  /\  G  e.  M )  ->  ( <. G ,  F >.  e.  dom  R  <->  ( D `  G )  =  ( C `  F ) ) )
 
Theoremdomcmpd 25078 When  ( G
( o_ `  T
) F ) is defined its domain is the domain of  F. (Contributed by FL, 29-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Ded  /\  F  e.  M  /\  G  e.  M )  ->  ( ( D `  G )  =  ( C `  F )  ->  ( D `  ( G R F ) )  =  ( D `  F ) ) )
 
Theoremcodcmpd 25079 When  ( G
( o_ `  T
) F ) is defined its codomain is the codomain of  G. (Contributed by FL, 29-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Ded  /\  F  e.  M  /\  G  e.  M )  ->  ( ( D `  G )  =  ( C `  F )  ->  ( C `  ( G R F ) )  =  ( C `  G ) ) )
 
Theoremrdmob 25080 The range of  ( dom_ `  T
) is the class of the objects. (Contributed by FL, 10-Jan-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  D  =  ( dom_ `  T )   =>    |-  ( T  e.  Ded  ->  ran  D  =  O )
 
Theoremrcmob 25081 The range of  ( cod_ `  T
) is the class of the objects. (Contributed by FL, 10-Jan-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( T  e.  Ded  ->  ran  C  =  O )
 
Theoremaidm2 25082 The underlying directed multi graph of a deductive system. (Contributed by FL, 20-Sep-2009.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( T  e.  Ded  ->  <. <. D ,  C >. ,  ran  D >.  e.  Dgra )
 
Theoremdmrngcmp 25083 Domain and range of the domain of the composition. (Contributed by FL, 5-Oct-2009.)
 |-  R  =  ( o_ `  T )   &    |-  M  =  dom  ( dom_ `  T )   =>    |-  ( T  e.  Ded 
 ->  ( dom  dom  R  =  M  /\  ran  dom  R  =  M ) )
 
16.12.48  Categories
 
SyntaxccatOLD 25084 Extend class notation with the class of categories.
 class  Cat OLD
 
Definitiondf-catOLD 25085* A category is a deductive system where composition is associative, and identities are neutral elements. A category is an abstraction of a structure (a group, a topology, an order...) Category theory consists in finding new formulation of the concepts associated to those structures (product, substructure...) using morphisms instead of the belonging relation. That trick has the interesting property that heterogeneous structures like topologies or groups for instance become comparable. (Note: in category theory morphisms are also called arrows.) (Contributed by FL, 24-Oct-2007.)
 |-  Cat OLD 
 =  { x  |  E. d E. c E. j E. r ( x  =  <. <. d ,  c >. ,  <. j ,  r >.
 >.  /\  ( ( <. <.
 d ,  c >. , 
 <. j ,  r >. >.  e.  Ded  /\  A. f  e. 
 dom  d A. g  e.  dom  d A. h  e.  dom  d ( ( ( d `  h )  =  ( c `  g )  /\  (
 d `  g )  =  ( c `  f
 ) )  ->  ( h r ( g r f ) )  =  ( ( h r g ) r f ) ) ) 
 /\  ( A. a  e.  dom  j A. f  e.  dom  d ( ( c `  f )  =  a  ->  (
 ( j `  a
 ) r f )  =  f )  /\  A. a  e.  dom  j A. f  e.  dom  d ( ( d `
  f )  =  a  ->  ( f
 r ( j `  a ) )  =  f ) ) ) ) }
 
TheoremiscatOLD 25086* The predicate "is a category". (Contributed by FL, 24-Oct-2007.)
 |-  M  =  dom  D   &    |-  O  =  dom  J   =>    |-  ( ( ( D  e.  A  /\  C  e.  B  /\  J  e.  F )  /\  R  e.  G )  ->  ( <. <. D ,  C >. , 
 <. J ,  R >. >.  e.  Cat OLD  <->  ( ( <. <. D ,  C >. , 
 <. J ,  R >. >.  e.  Ded  /\  A. f  e.  M  A. g  e.  M  A. h  e.  M  ( ( ( D `  h )  =  ( C `  g )  /\  ( D `
  g )  =  ( C `  f
 ) )  ->  ( h R ( g R f ) )  =  ( ( h R g ) R f ) ) )  /\  ( A. a  e.  O  A. f  e.  M  ( ( C `  f
 )  =  a  ->  ( ( J `  a ) R f )  =  f ) 
 /\  A. a  e.  O  A. f  e.  M  ( ( D `  f
 )  =  a  ->  ( f R ( J `  a ) )  =  f ) ) ) ) )
 
Theoremcati 25087* Definitional properties of a category. (Contributed by FL, 24-Oct-2007.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_
 `  T )   &    |-  M  =  dom  D   &    |-  O  =  dom  J   =>    |-  ( T  e.  Cat OLD 
 ->  ( ( <. <. D ,  C >. ,  <. J ,  R >. >.  e.  Ded  /\  A. f  e.  M  A. g  e.  M  A. h  e.  M  ( ( ( D `  h )  =  ( C `  g )  /\  ( D `
  g )  =  ( C `  f
 ) )  ->  ( h R ( g R f ) )  =  ( ( h R g ) R f ) ) )  /\  ( A. a  e.  O  A. f  e.  M  ( ( C `  f
 )  =  a  ->  ( ( J `  a ) R f )  =  f ) 
 /\  A. a  e.  O  A. f  e.  M  ( ( D `  f
 )  =  a  ->  ( f R ( J `  a ) )  =  f ) ) ) )
 
Theorem0alg 25088 Lemma for 0ded 25089. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Alg
 
Theorem0ded 25089 A deductive system with no object and no morphism. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Ded
 
Theorem0catOLD 25090 Category  0 has no object and no morphism. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Cat OLD
 
Theorem1cat 25091 Category  1 has one object and one morphism. (Contributed by FL, 30-Oct-2007.)
 |-  A  e.  _V   =>    |- 
 <. <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. }
 >. ,  <. { <. A ,  <. A ,  A >. >. } ,  { <. <. <. A ,  A >. ,  <. A ,  A >. >. ,  <. A ,  A >. >. } >. >.  e.  Cat OLD
 
Theoremstrcat 25092 Structure of a category. (Contributed by FL, 26-Oct-2007.)
 |-  Cat OLD  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
 
Theoremrelcat 25093 A category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  Cat
 OLD
 
Theoremreldcat 25094 The domain of a category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  dom 
 Cat OLD
 
Theoremrelrcat 25095 The range of a category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  ran 
 Cat OLD
 
Theoremcatded 25096 A category is a deductive system. (Contributed by FL, 26-Oct-2007.)
 |-  ( T  e.  Cat OLD  ->  T  e.  Ded )
 
Theoremdomc 25097 The 1st "axiom" of a category:  ( dom_ `  T ) is a mapping from the morphisms of  T to the objects of  T. (Contributed by FL, 2-Jan-2008.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   =>    |-  ( T  e.  Cat OLD  ->  D : M --> O )
 
Theoremcodc 25098 The 2nd "axiom" of a category  ( cod_ `  T ) is a mapping from the morphisms of  T to the objects of  T. (Contributed by FL, 2-Jan-2008.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( T  e.  Cat OLD  ->  C : M --> O )
 
Theoremidc 25099 The 3rd "axiom" of a category  ( id_ `  T ) is a mapping from the objects of  T to the morphisms of  T. (Contributed by FL, 5-Dec-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   =>    |-  ( T  e.  Cat OLD  ->  J : O --> M )
 
Theoremcmppfc 25100 The 4th "axiom" of a category:  ( o_ `  T ) is a partial operation from the morphisms of  T to the morphisms of  T. (Contributed by FL, 10-Mar-2008.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  ( Fun  R  /\  dom 
 R  C_  ( M  X.  M )  /\  ran  R 
 C_  M ) )
    < 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 31201-31284
  Copyright terms: Public domain < Previous  Next >