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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremicccon3 25701 An open-below, closed-above interval is connected. (Contributed by FL, 30-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
t

Theoremicccon4 25702 An open interval is connected. (Contributed by FL, 30-May-2014.)
t

Theoremintvconlem1 25703 All the intervals of are connected. (Contributed by FL, 29-May-2014.)
t

Syntaxcder 25704 Extend class notation to include the derivative of a function.

Definitiondf-der 25705* Derivative of a function at . Meaningful when the domain of is an interval of , belongs to the domain of , the domain of is not and the values of are in .

Bourbaki doesn't explain why he requires the domain of be an interval. Here are some hints. The domain of is an interval, belongs to the domain of and guarantee is not an isolated point in (df-islpt 25584). We have (indif2 3412) but since is not an isolated point in and what is the condition required by trfil2 17582. And in this case the class is a filter. This latter condition is required by df-flimfrs 25579 and this definition is used by df-der 25705.

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

Theoremhdrmp 25706 Hard to describe. A picture can help. (Contributed by FL, 29-May-2014.)

Theoremisder 25707* The derivative of at point is the limit of the slope when tends to . Definition 1 of [BourbakiFVR] p. I.11. (Contributed by FL, 29-May-2014.)

18.13.45  Directed multi graphs

Syntaxcmgra 25708 Extend class notation with the class of directed multi graphs.

Definitiondf-mgra 25709* 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.)

Theoremismgra 25710 The predicate "is a directed multi graph". (Contributed by FL, 10-Jan-2008.)

18.13.46  Category and deductive system underlying "structure"

Syntaxcalg 25711 Extend class notation with the class of structures used by and .

Syntaxcdom_ 25712 Extend class notation with the function returning the function domain of a category.

Syntaxccod_ 25713 Extend class notation with the function returning the function codomain of a category.

Syntaxcid_ 25714 Extend class notation with the function returning the function identity of a category.

Syntaxco_ 25715 Extend class notation with the function returning the composition of morphisms of a category.

Definitiondf-alg 25716* and structure. Metamath for internal reasons doesn't like too large definitions. Then has been split giving birth to and . If has a real mathematical use, is only here to give relief to Metamath. (Contributed by FL, 24-Oct-2007.)

Definitiondf-dom_ 25717 Definition of . (Contributed by FL, 24-Oct-2007.)

Definitiondf-cod_ 25718 Definition of . (Contributed by FL, 26-Oct-2007.)

Definitiondf-id_ 25719 Definition of . (Contributed by FL, 26-Oct-2007.)

Definitiondf-cmpa 25720 Definition of . (Contributed by FL, 26-Oct-2007.)

Theoremisalg 25721 The predicate "has the structure required by and ." (Contributed by FL, 24-Oct-2007.)

Theorem1alg 25722 CatOLDegory has the structure required by and . (Contributed by FL, 30-Oct-2007.)

Theoremdomval 25723 Value of the domain function expressed with the function. (Contributed by FL, 24-Oct-2007.)

Theoremcodval 25724 Value of the function codomain expressed with the and functions. (Contributed by FL, 26-Oct-2007.)

Theoremidval 25725 Value of the identity function expressed with the and functions. (Contributed by FL, 26-Oct-2007.)

Theoremcmpval 25726 Value of the identity function expressed with the functions. (Contributed by FL, 26-Oct-2007.)

Theoremalgi 25727 "Axiomatic" properties of . (Contributed by FL, 24-Oct-2007.)

Theoremdoma 25728 is a mapping from the morphisms of to the objects of . (Contributed by FL, 24-Oct-2007.)

Theoremcoda 25729 is a mapping from the morphisms of to the objects of . (Contributed by FL, 26-Oct-2007.)

Theoremida 25730 is a mapping from the objects of to the morphisms of . (Contributed by FL, 26-Oct-2007.)

Theoremidmoa 25731 An identity is a morphism. (Contributed by FL, 10-Jan-2008.)

Theoremcmppfa 25732 is a partial operation on the morphisms of . (Contributed by FL, 26-Oct-2007.)

Theoremdcsda 25733 and have the same domain. (Contributed by FL, 10-Jan-2008.)

18.13.47  Deductive systems

Syntaxcded 25734 Extend class notation with the class of deductive systems.

Definitiondf-ded 25735* 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.)

Theoremisded 25736* The predicate "is a deductive system". (Contributed by FL, 24-Oct-2007.)

Theoremdedi 25737* Properties of a deductive system. (Contributed by FL, 24-Oct-2007.)

Theorem1ded 25738 Category is a deductive system. We can think of the morphism of Category as corresponding to . (Contributed by FL, 30-Oct-2007.)

Theoremstrded 25739 Structure of a deductive system. (Contributed by FL, 28-Oct-2007.)

Theoremrelded 25740 A deductive system is a relation. (Contributed by FL, 28-Oct-2007.)

Theoremreldded 25741 The domain of a deductive system is a relation. (Contributed by FL, 28-Oct-2007.)

Theoremrelrded 25742 The range of a deductive system is a relation. (Contributed by FL, 28-Oct-2007.)

Theoremdedalg 25743 A deductive system is an "algebra". (Contributed by FL, 28-Oct-2007.)

Theoremidosd 25744 The identity is a morphism which has the same object as its domain and its codomain. (Contributed by FL, 28-Oct-2007.)

Theoremcmppfd 25745 is only defined when the domain of is the codomain of . (Contributed by FL, 29-Oct-2007.)

Theoremdomcmpd 25746 When is defined its domain is the domain of . (Contributed by FL, 29-Oct-2007.)

Theoremcodcmpd 25747 When is defined its codomain is the codomain of . (Contributed by FL, 29-Oct-2007.)

Theoremrdmob 25748 The range of is the class of the objects. (Contributed by FL, 10-Jan-2008.)

Theoremrcmob 25749 The range of is the class of the objects. (Contributed by FL, 10-Jan-2008.)

Theoremaidm2 25750 The underlying directed multi graph of a deductive system. (Contributed by FL, 20-Sep-2009.)

Theoremdmrngcmp 25751 Domain and range of the domain of the composition. (Contributed by FL, 5-Oct-2009.)

18.13.48  Categories

SyntaxccatOLD 25752 Extend class notation with the class of categories.

Definitiondf-catOLD 25753* 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.)

TheoremiscatOLD 25754* The predicate "is a category". (Contributed by FL, 24-Oct-2007.)

Theoremcati 25755* Definitional properties of a category. (Contributed by FL, 24-Oct-2007.)

Theorem0alg 25756 Lemma for 0ded 25757. (Contributed by FL, 10-Jan-2008.)

Theorem0ded 25757 A deductive system with no object and no morphism. (Contributed by FL, 10-Jan-2008.)

Theorem0catOLD 25758 Category has no object and no morphism. (Contributed by FL, 10-Jan-2008.)

Theorem1cat 25759 Category has one object and one morphism. (Contributed by FL, 30-Oct-2007.)

Theoremstrcat 25760 Structure of a category. (Contributed by FL, 26-Oct-2007.)

Theoremrelcat 25761 A category is a relation. (Contributed by FL, 26-Oct-2007.)

Theoremreldcat 25762 The domain of a category is a relation. (Contributed by FL, 26-Oct-2007.)

Theoremrelrcat 25763 The range of a category is a relation. (Contributed by FL, 26-Oct-2007.)

Theoremcatded 25764 A category is a deductive system. (Contributed by FL, 26-Oct-2007.)

Theoremdomc 25765 The 1st "axiom" of a category: is a mapping from the morphisms of to the objects of . (Contributed by FL, 2-Jan-2008.)

Theoremcodc 25766 The 2nd "axiom" of a category is a mapping from the morphisms of to the objects of . (Contributed by FL, 2-Jan-2008.)

Theoremidc 25767 The 3rd "axiom" of a category is a mapping from the objects of to the morphisms of . (Contributed by FL, 5-Dec-2007.)

Theoremcmppfc 25768 The 4th "axiom" of a category: is a partial operation from the morphisms of to the morphisms of . (Contributed by FL, 10-Mar-2008.)

Theoremidosc 25769 The 5th "axiom" of a category: identities are morphisms whose domains and codomains are equal. (Contributed by FL, 5-Dec-2007.)

Theoremcmppfcd 25770 The 6th "axiom" of a category: is only defined when the domain of equals the codomain of . (Contributed by FL, 10-Mar-2008.)

Theoremdomcmpc 25771 The 7th "axiom" of a category: when is defined its domain is the domain of . (Contributed by FL, 10-Mar-2008.)

Theoremcodcmpc 25772 The 8th "axiom" of a category: when is defined its codomain is the codomain of . (Contributed by FL, 10-Jan-2008.)

Theoremcmpasso 25773 The 9th "axiom" of a category: is associative. (Contributed by FL, 29-Oct-2007.)

Theoremcmpida 25774 The 10th "axiom" of a category: is a left neutral element. (Contributed by FL, 29-Oct-2007.)

Theoremcmpidb 25775 The 11th "axiom" of a category: is a right neutral element. (Contributed by FL, 24-Oct-2007.)

Theoremdmo 25776 The domain of a morphism is an object. (Contributed by FL, 2-Jan-2008.)

Theoremcdmo 25777 The codomain of a morphism is an object. (Contributed by FL, 2-Jan-2008.)

Theoremjdmo 25778 An identity is a morphism. (Contributed by FL, 10-Jan-2008.)

Theoremcmpmorp 25779 Conditions for a composite to be a morphism. (Contributed by FL, 10-Mar-2008.)

Theoremmorcat 25780 Two ways to define the set of the morphisms of a category. (Contributed by FL, 19-Sep-2009.)

Theoremcmppfc1 25781 Composition is a function. (Contributed by FL, 5-Oct-2009.)

Theoremdualalg 25782 The dual of a is a . (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremdualded 25783 The dual of a deductive system is a deductive system. (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
tpos

Theoremdualcat2 25784 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.)
tpos

18.13.49  Homsets

SyntaxchomOLD 25785 Extend class notation with the function returning all the morphisms between two objects.

Definitiondf-homOLD 25786* is a function which returns for each pair of objects the morphisms whose domain is and codomain . JFM CAT1 def. 6 (Contributed by FL, 6-May-2007.)

Theoremishoma 25787* Definition of . (Contributed by FL, 6-May-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)

Theoremishomb 25788* The homset . (Contributed by FL, 18-May-2007.)

Theoremishomc 25789 The predicate JFM vol. 1.2 p. 411 th. 18. (Contributed by FL, 30-Nov-2007.)

Theoremishomd 25790 The predicate JFM vol. 1.2 p. 411 th. 18. (Contributed by FL, 30-Nov-2007.)

Theoremehm 25791 The elements of a homset are morphisms. JFM CAT1 th. 21. (Contributed by FL, 5-Dec-2007.)

Theoremdehm 25792 Domain of an element of a homset. JFM CAT1 th. 23. (Contributed by FL, 5-Dec-2007.)

Theoremcehm 25793 Codomain of an element of a homset. JFM CAT1 th. 23. (Contributed by FL, 5-Dec-2007.)

Theoremmrdmcd 25794 A morphism belongs to the homset between its domain and its codomain. JFM CAT1 th. 22. (Contributed by FL, 1-Nov-2007.)

Theoremeqidob 25795 When the identities are equal, the objects are equal. JFM CAT1 th. 45. (Contributed by FL, 24-Apr-2007.)

Theoremhomib 25796 The homset which belongs to. JFM CAT1 th. 55. (Contributed by FL, 5-Dec-2007.)

Theoremhine 25797 The homset is not empty. JFM CAT1 th. 56. (Contributed by FL, 3-Jan-2008.)

Theoremcmphmia 25798 Composite of the member of a homset with the identity. JFM CAT1 th. 57 (Contributed by FL, 5-Dec-2007.)

Theoremcmphmib 25799 Composite of a member of a homset with the identity. JFM CAT1 th. 58 (Contributed by FL, 5-Dec-2007.)

Theoremiri 25800 Composite of an identity with itself. JFM CAT1 th. 59 (Contributed by FL, 5-Dec-2007.)

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