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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtcnvec 25701 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 25702 Extends class notation with scalar division of complex vectors.
 class  / cv
 
Definitiondf-divcv 25703* 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 25704 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 25705 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 25706 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 ) ) )
 
18.13.44  Calculus
 
Syntaxcintvl 25707 Extend class notation to include intervals.
 class  Intvl
 
Definitiondf-intvl 25708 The intervals of  RR. (Contributed by FL, 29-May-2014.)
 |-  Intvl  =  ( ( ran  (,)  u.  ( ran  (,]  u.  ( ran  [,)  u.  ran  [,] ) ) )  i^i 
 ~P RR )
 
Theoremintvlset 25709 The set of intervals is a set. (Contributed by FL, 29-May-2014.)
 |-  Intvl  e.  _V
 
Theoremintrr 25710 An interval is a part of  RR. (Contributed by FL, 29-May-2014.)
 |-  ( I  e.  Intvl  ->  I  C_ 
 RR )
 
Theoremicccon2 25711 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 25712 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 25713 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 25714 All the intervals of  RR are connected. (Contributed by FL, 29-May-2014.)
 |-  ( I  e.  Intvl  ->  (
 ( topGen `  ran  (,) )t  I
 )  e.  Con )
 
Syntaxcder 25715 Extend class notation to include the derivative of a function.
 class  der
 
Definitiondf-der 25716* 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 25595). We have  ( v  i^i  ( dom  f  \  { p } ) )  =  ( ( v  i^i  dom  f
)  \  { p } ) (indif2 3414) 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 17584. 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 25590 and this definition is used by df-der 25716.

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 25717 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 25718* 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 >. ) )
 
18.13.45  Directed multi graphs
 
Syntaxcmgra 25719 Extend class notation with the class of directed multi graphs.
 class  Dgra
 
Definitiondf-mgra 25720* 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 25721 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 ) ) )
 
18.13.46  Category and deductive system underlying "structure"
 
Syntaxcalg 25722 Extend class notation with the class of structures used by  Cat OLD and  Ded.
 class  Alg
 
Syntaxcdom_ 25723 Extend class notation with the function returning the function domain of a category.
 class  dom_
 
Syntaxccod_ 25724 Extend class notation with the function returning the function codomain of a category.
 class  cod_
 
Syntaxcid_ 25725 Extend class notation with the function returning the function identity of a category.
 class  id_
 
Syntaxco_ 25726 Extend class notation with the function returning the composition of morphisms of a category.
 class  o_
 
Definitiondf-alg 25727*  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_ 25728 Definition of  dom_. (Contributed by FL, 24-Oct-2007.)
 |-  dom_  =  ( 1st  o.  1st )
 
Definitiondf-cod_ 25729 Definition of  cod_. (Contributed by FL, 26-Oct-2007.)
 |-  cod_  =  ( 2nd  o.  1st )
 
Definitiondf-id_ 25730 Definition of  id_. (Contributed by FL, 26-Oct-2007.)
 |-  id_  =  ( 1st  o.  2nd )
 
Definitiondf-cmpa 25731 Definition of  o_. (Contributed by FL, 26-Oct-2007.)
 |-  o_  =  ( 2nd  o.  2nd )
 
Theoremisalg 25732 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 25733 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 25734 Value of the domain function expressed with the  1st function. (Contributed by FL, 24-Oct-2007.)
 |-  D  =  ( dom_ `  T )   =>    |-  D  =  ( 1st `  ( 1st `  T ) )
 
Theoremcodval 25735 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 25736 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 25737 Value of the identity function expressed with the  2nd functions. (Contributed by FL, 26-Oct-2007.)
 |-  G  =  ( o_ `  T )   =>    |-  G  =  ( 2nd `  ( 2nd `  T ) )
 
Theoremalgi 25738 "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 25739  ( 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 25740  ( 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 25741  ( 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 25742 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 25743  ( 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 25744  ( 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 )
 
18.13.47  Deductive systems
 
Syntaxcded 25745 Extend class notation with the class of deductive systems.
 class  Ded
 
Definitiondf-ded 25746* 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 25747* 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 25748* 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 25749 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 25750 Structure of a deductive system. (Contributed by FL, 28-Oct-2007.)
 |-  Ded  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
 
Theoremrelded 25751 A deductive system is a relation. (Contributed by FL, 28-Oct-2007.)
 |-  Rel  Ded
 
Theoremreldded 25752 The domain of a deductive system is a relation. (Contributed by FL, 28-Oct-2007.)
 |-  Rel  dom 
 Ded
 
Theoremrelrded 25753 The range of a deductive system is a relation. (Contributed by FL, 28-Oct-2007.)
 |-  Rel  ran 
 Ded
 
Theoremdedalg 25754 A deductive system is an "algebra". (Contributed by FL, 28-Oct-2007.)
 |-  ( T  e.  Ded  ->  T  e.  Alg  )
 
Theoremidosd 25755 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 25756  ( 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 25757 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 25758 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 25759 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 25760 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 25761 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 25762 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 ) )
 
18.13.48  Categories
 
SyntaxccatOLD 25763 Extend class notation with the class of categories.
 class  Cat OLD
 
Definitiondf-catOLD 25764* 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 25765* 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 25766* 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 25767 Lemma for 0ded 25768. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Alg
 
Theorem0ded 25768 A deductive system with no object and no morphism. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Ded
 
Theorem0catOLD 25769 Category  0 has no object and no morphism. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Cat OLD
 
Theorem1cat 25770 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 25771 Structure of a category. (Contributed by FL, 26-Oct-2007.)
 |-  Cat OLD  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
 
Theoremrelcat 25772 A category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  Cat
 OLD
 
Theoremreldcat 25773 The domain of a category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  dom 
 Cat OLD
 
Theoremrelrcat 25774 The range of a category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  ran 
 Cat OLD
 
Theoremcatded 25775 A category is a deductive system. (Contributed by FL, 26-Oct-2007.)
 |-  ( T  e.  Cat OLD  ->  T  e.  Ded )
 
Theoremdomc 25776 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 25777 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 25778 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 25779 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 ) )
 
Theoremidosc 25780 The 5th "axiom" of a category: identities are morphisms whose domains and codomains are equal. (Contributed by FL, 5-Dec-2007.)
 |-  O  =  dom  J   &    |-  D  =  (
 dom_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O )  ->  ( ( D `
  ( J `  A ) )  =  A  /\  ( C `
  ( J `  A ) )  =  A ) )
 
Theoremcmppfcd 25781 The 6th "axiom" of a category:  ( G ( o_ `  T ) F ) is only defined when the domain of  F equals the codomain of 
G. (Contributed by FL, 10-Mar-2008.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  F  e.  M  /\  G  e.  M ) 
 ->  ( <. G ,  F >.  e.  dom  R  <->  ( D `  G )  =  ( C `  F ) ) )
 
Theoremdomcmpc 25782 The 7th "axiom" of a category: when  ( G ( o_ `  T ) F ) is defined its domain is the domain of 
F. (Contributed by FL, 10-Mar-2008.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  F  e.  M  /\  G  e.  M ) 
 ->  ( ( D `  G )  =  ( C `  F )  ->  ( D `  ( G R F ) )  =  ( D `  F ) ) )
 
Theoremcodcmpc 25783 The 8th "axiom" of a category: when  ( G ( o_ `  T ) F ) is defined its codomain is the codomain of  G. (Contributed by FL, 10-Jan-2008.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  F  e.  M  /\  G  e.  M ) 
 ->  ( ( D `  G )  =  ( C `  F )  ->  ( C `  ( G R F ) )  =  ( C `  G ) ) )
 
Theoremcmpasso 25784 The 9th "axiom" of a category:  ( o_ `  T ) is associative. (Contributed by FL, 29-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  ( F  e.  M  /\  G  e.  M  /\  H  e.  M ) )  ->  ( (
 ( D `  H )  =  ( C `  G )  /\  ( D `  G )  =  ( C `  F ) )  ->  ( H R ( G R F ) )  =  ( ( H R G ) R F ) ) )
 
Theoremcmpida 25785 The 10th "axiom" of a category:  ( J `  A ) is a left neutral element. (Contributed by FL, 29-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O  /\  F  e.  M ) 
 ->  ( ( C `  F )  =  A  ->  ( ( J `  A ) R F )  =  F )
 )
 
Theoremcmpidb 25786 The 11th "axiom" of a category:  ( J `  A ) is a right neutral element. (Contributed by FL, 24-Oct-2007.)
 |-  M  =  dom  D   &    |-  D  =  (
 dom_ `  T )   &    |-  O  =  dom  J   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O  /\  F  e.  M ) 
 ->  ( ( D `  F )  =  A  ->  ( F R ( J `  A ) )  =  F ) )
 
Theoremdmo 25787 The domain of a morphism is an object. (Contributed by FL, 2-Jan-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  O  =  dom  ( id_ `  T )   &    |-  D  =  ( dom_ `  T )   =>    |-  (
 ( T  e.  Cat OLD  /\  F  e.  M ) 
 ->  ( D `  F )  e.  O )
 
Theoremcdmo 25788 The codomain of a morphism is an object. (Contributed by FL, 2-Jan-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  O  =  dom  ( id_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  (
 ( T  e.  Cat OLD  /\  F  e.  M ) 
 ->  ( C `  F )  e.  O )
 
Theoremjdmo 25789 An identity is a morphism. (Contributed by FL, 10-Jan-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  O  =  dom  ( id_ `  T )   &    |-  J  =  ( id_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O )  ->  ( J `  A )  e.  M )
 
Theoremcmpmorp 25790 Conditions for a composite to be a morphism. (Contributed by FL, 10-Mar-2008.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  (
 cod_ `  T )   &    |-  R  =  ( o_ `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  F  e.  M  /\  G  e.  M ) 
 ->  ( ( D `  G )  =  ( C `  F )  ->  ( G R F )  e.  M ) )
 
Theoremmorcat 25791 Two ways to define the set of the morphisms of a category. (Contributed by FL, 19-Sep-2009.)
 |-  ( T  e.  Cat OLD  ->  dom  ( dom_ `  T )  =  dom  ( cod_ `  T ) )
 
Theoremcmppfc1 25792 Composition is a function. (Contributed by FL, 5-Oct-2009.)
 |-  R  =  ( o_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  Fun  R )
 
Theoremdualalg 25793 The dual of a  Alg is a  Alg. (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_
 `  T )   =>    |-  ( T  e.  Alg 
 ->  <. <. C ,  D >. ,  <. J , tpos  R >.
 >.  e.  Alg  )
 
Theoremdualded 25794 The dual of a deductive system is a deductive system. (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_
 `  T )   =>    |-  ( T  e.  Ded 
 ->  <. <. C ,  D >. ,  <. J , tpos  R >.
 >.  e.  Ded )
 
Theoremdualcat2 25795 The dual of a category is a category. Joy of cats 3.5 (Contributed by FL, 4-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
 |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  R  =  ( o_
 `  T )   =>    |-  ( T  e.  Cat
 OLD  ->  <. <. C ,  D >. ,  <. J , tpos  R >.
 >.  e.  Cat OLD  )
 
18.13.49  Homsets
 
SyntaxchomOLD 25796 Extend class notation with the function returning all the morphisms between two objects.
 class  hom
 
Definitiondf-homOLD 25797*  ( hom `  x ) is a function which returns for each pair of objects  <. a ,  b >. the morphisms whose domain is  a and codomain  b. JFM CAT1 def. 6 (Contributed by FL, 6-May-2007.)
 |-  hom  =  ( x  e.  Cat OLD  |->  ( a  e.  dom  ( id_ `  x ) ,  b  e.  dom  ( id_ `  x )  |->  { f  e.  dom  ( dom_ `  x )  |  ( ( ( dom_ `  x ) `  f
 )  =  a  /\  ( ( cod_ `  x ) `  f )  =  b ) } )
 )
 
Theoremishoma 25798* Definition of  ( hom `  T
). (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
 |-  O  =  dom  ( id_ `  T )   &    |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( T  e.  Cat OLD  ->  ( hom `  T )  =  ( a  e.  O ,  b  e.  O  |->  { f  e.  M  |  ( ( D `  f )  =  a  /\  ( C `  f
 )  =  b ) } ) )
 
Theoremishomb 25799* The homset  ( ( hom `  T ) `  <. A ,  B >. ). (Contributed by FL, 18-May-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  T  e.  Cat OLD   =>    |-  ( ( A  e.  O  /\  B  e.  O )  ->  ( H `  <. A ,  B >. )  =  { f  e.  M  |  ( ( D `  f )  =  A  /\  ( C `  f )  =  B ) } )
 
Theoremishomc 25800 The predicate  F  e.  ( ( hom `  T
) `  <. A ,  B >. ) JFM vol. 1.2 p. 411 th. 18. (Contributed by FL, 30-Nov-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  M  =  dom  ( dom_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  T  e.  Cat OLD   =>    |-  ( ( A  e.  O  /\  B  e.  O )  ->  ( F  e.  ( H ` 
 <. A ,  B >. )  <-> 
 ( F  e.  M  /\  ( D `  F )  =  A  /\  ( C `  F )  =  B ) ) )
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