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Theorem List for Metamath Proof Explorer - 25801-25900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremintvlset 25801 The set of intervals is a set. (Contributed by FL, 29-May-2014.)
 |-  Intvl  e.  _V
 
Theoremintrr 25802 An interval is a part of  RR. (Contributed by FL, 29-May-2014.)
 |-  ( I  e.  Intvl  ->  I  C_ 
 RR )
 
Theoremicccon2 25803 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 25804 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 25805 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 25806 All the intervals of  RR are connected. (Contributed by FL, 29-May-2014.)
 |-  ( I  e.  Intvl  ->  (
 ( topGen `  ran  (,) )t  I
 )  e.  Con )
 
Syntaxcder 25807 Extend class notation to include the derivative of a function.
 class  der
 
Definitiondf-der 25808* 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 25687). We have  ( v  i^i  ( dom  f  \  { p } ) )  =  ( ( v  i^i  dom  f
)  \  { p } ) (indif2 3425) 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 17598. 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 25682 and this definition is used by df-der 25808.

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 25809 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 25810* 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 25811 Extend class notation with the class of directed multi graphs.
 class  Dgra
 
Definitiondf-mgra 25812* 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 25813 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 25814 Extend class notation with the class of structures used by  Cat OLD and  Ded.
 class  Alg
 
Syntaxcdom_ 25815 Extend class notation with the function returning the function domain of a category.
 class  dom_
 
Syntaxccod_ 25816 Extend class notation with the function returning the function codomain of a category.
 class  cod_
 
Syntaxcid_ 25817 Extend class notation with the function returning the function identity of a category.
 class  id_
 
Syntaxco_ 25818 Extend class notation with the function returning the composition of morphisms of a category.
 class  o_
 
Definitiondf-alg 25819*  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_ 25820 Definition of  dom_. (Contributed by FL, 24-Oct-2007.)
 |-  dom_  =  ( 1st  o.  1st )
 
Definitiondf-cod_ 25821 Definition of  cod_. (Contributed by FL, 26-Oct-2007.)
 |-  cod_  =  ( 2nd  o.  1st )
 
Definitiondf-id_ 25822 Definition of  id_. (Contributed by FL, 26-Oct-2007.)
 |-  id_  =  ( 1st  o.  2nd )
 
Definitiondf-cmpa 25823 Definition of  o_. (Contributed by FL, 26-Oct-2007.)
 |-  o_  =  ( 2nd  o.  2nd )
 
Theoremisalg 25824 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 25825 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 25826 Value of the domain function expressed with the  1st function. (Contributed by FL, 24-Oct-2007.)
 |-  D  =  ( dom_ `  T )   =>    |-  D  =  ( 1st `  ( 1st `  T ) )
 
Theoremcodval 25827 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 25828 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 25829 Value of the identity function expressed with the  2nd functions. (Contributed by FL, 26-Oct-2007.)
 |-  G  =  ( o_ `  T )   =>    |-  G  =  ( 2nd `  ( 2nd `  T ) )
 
Theoremalgi 25830 "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 25831  ( 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 25832  ( 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 25833  ( 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 25834 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 25835  ( 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 25836  ( 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 25837 Extend class notation with the class of deductive systems.
 class  Ded
 
Definitiondf-ded 25838* 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 25839* 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 25840* 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 25841 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 25842 Structure of a deductive system. (Contributed by FL, 28-Oct-2007.)
 |-  Ded  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
 
Theoremrelded 25843 A deductive system is a relation. (Contributed by FL, 28-Oct-2007.)
 |-  Rel  Ded
 
Theoremreldded 25844 The domain of a deductive system is a relation. (Contributed by FL, 28-Oct-2007.)
 |-  Rel  dom 
 Ded
 
Theoremrelrded 25845 The range of a deductive system is a relation. (Contributed by FL, 28-Oct-2007.)
 |-  Rel  ran 
 Ded
 
Theoremdedalg 25846 A deductive system is an "algebra". (Contributed by FL, 28-Oct-2007.)
 |-  ( T  e.  Ded  ->  T  e.  Alg  )
 
Theoremidosd 25847 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 25848  ( 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 25849 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 25850 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 25851 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 25852 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 25853 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 25854 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 25855 Extend class notation with the class of categories.
 class  Cat OLD
 
Definitiondf-catOLD 25856* 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 25857* 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 25858* 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 25859 Lemma for 0ded 25860. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Alg
 
Theorem0ded 25860 A deductive system with no object and no morphism. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Ded
 
Theorem0catOLD 25861 Category  0 has no object and no morphism. (Contributed by FL, 10-Jan-2008.)
 |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Cat OLD
 
Theorem1cat 25862 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 25863 Structure of a category. (Contributed by FL, 26-Oct-2007.)
 |-  Cat OLD  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
 
Theoremrelcat 25864 A category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  Cat
 OLD
 
Theoremreldcat 25865 The domain of a category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  dom 
 Cat OLD
 
Theoremrelrcat 25866 The range of a category is a relation. (Contributed by FL, 26-Oct-2007.)
 |-  Rel  ran 
 Cat OLD
 
Theoremcatded 25867 A category is a deductive system. (Contributed by FL, 26-Oct-2007.)
 |-  ( T  e.  Cat OLD  ->  T  e.  Ded )
 
Theoremdomc 25868 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 25869 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 25870 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 25871 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 25872 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 25873 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 25874 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 25875 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 25876 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 25877 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 25878 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 25879 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 25880 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 25881 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 25882 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 25883 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 25884 Composition is a function. (Contributed by FL, 5-Oct-2009.)
 |-  R  =  ( o_ `  T )   =>    |-  ( T  e.  Cat OLD 
 ->  Fun  R )
 
Theoremdualalg 25885 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 25886 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 25887 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 25888 Extend class notation with the function returning all the morphisms between two objects.
 class  hom
 
Definitiondf-homOLD 25889*  ( 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 25890* 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 25891* 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 25892 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 ) ) )
 
Theoremishomd 25893 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 )
 ) )
 
Theoremehm 25894 The elements of a homset are morphisms. JFM CAT1 th. 21. (Contributed by FL, 5-Dec-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  M  =  dom  ( dom_ `  T )   &    |-  H  =  ( hom `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O  /\  B  e.  O ) 
 ->  ( F  e.  ( H `  <. A ,  B >. )  ->  F  e.  M ) )
 
Theoremdehm 25895 Domain of an element of a homset. JFM CAT1 th. 23. (Contributed by FL, 5-Dec-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  D  =  ( dom_ `  T )   &    |-  H  =  ( hom `  T )   =>    |-  (
 ( T  e.  Cat OLD  /\  A  e.  O  /\  B  e.  O )  ->  ( F  e.  ( H `  <. A ,  B >. )  ->  ( D `  F )  =  A ) )
 
Theoremcehm 25896 Codomain of an element of a homset. JFM CAT1 th. 23. (Contributed by FL, 5-Dec-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  C  =  ( cod_ `  T )   &    |-  H  =  ( hom `  T )   =>    |-  (
 ( T  e.  Cat OLD  /\  A  e.  O  /\  B  e.  O )  ->  ( F  e.  ( H `  <. A ,  B >. )  ->  ( C `  F )  =  B ) )
 
Theoremmrdmcd 25897 A morphism belongs to the homset between its domain and its codomain. JFM CAT1 th. 22. (Contributed by FL, 1-Nov-2007.)
 |-  M  =  dom  ( dom_ `  T )   &    |-  H  =  ( hom `  T )   &    |-  D  =  (
 dom_ `  T )   &    |-  C  =  ( cod_ `  T )   =>    |-  ( T  e.  Cat OLD  ->  ( F  e.  M  ->  F  e.  ( H `  <. ( D `  F ) ,  ( C `  F ) >. ) ) )
 
Theoremeqidob 25898 When the identities are equal, the objects are equal. JFM CAT1 th. 45. (Contributed by FL, 24-Apr-2007.)
 |-  O  =  dom  J   &    |-  J  =  ( id_ `  C )   =>    |-  (
 ( C  e.  Cat OLD  /\  A  e.  O  /\  B  e.  O )  ->  ( ( J `  A )  =  ( J `  B )  ->  A  =  B )
 )
 
Theoremhomib 25899 The homset which  ( ( id_ `  T ) `  A
) belongs to. JFM CAT1 th. 55. (Contributed by FL, 5-Dec-2007.)
 |-  O  =  dom  ( id_ `  T )   &    |-  J  =  ( id_ `  T )   &    |-  H  =  ( hom `  T )   =>    |-  (
 ( T  e.  Cat OLD  /\  A  e.  O ) 
 ->  ( J `  A )  e.  ( H ` 
 <. A ,  A >. ) )
 
Theoremhine 25900 The homset  ( H `  <. A ,  A >. ) is not empty. JFM CAT1 th. 56. (Contributed by FL, 3-Jan-2008.)
 |-  O  =  dom  ( id_ `  T )   &    |-  H  =  ( hom `  T )   =>    |-  ( ( T  e.  Cat
 OLD  /\  A  e.  O )  ->  ( H `  <. A ,  A >. )  =/=  (/) )
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