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Theorem List for Metamath Proof Explorer - 24801-24900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-infp 24801 Definition of the infimum of an indexed class of extended reals. (Contributed by FL, 21-May-2012.)

Theoremsupbrr 24802* The supremum of a set of extended reals always exists. (Contributed by FL, 16-Apr-2012.)

Syntaxcfrf 24803 Extends class notation with Frechet's filter.

Definitiondf-frf 24804* Frechet's filter. Used to define the limit of a sequence. (Contributed by FL, 21-May-2012.)

Theorembsi2 24805* Membership to the set of closed intervals. (Contributed by FL, 29-May-2014.)

Theoremicof 24806 The set of closed-below, open-above intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 29-May-2014.)

Theorembsi3 24807* Membership to the set of closed-above, open-below intervals. (Contributed by FL, 29-May-2014.)

Theoremiocf 24808 The set of closed-below, open-above intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 29-May-2014.)

Theorembsi4 24809* Membership to the set of open-below, closed-above intervals. (Contributed by FL, 29-May-2014.)

16.11.43  ( RR ^ N ) and ( CC ^ N )

Syntaxcplcv 24810 Extends class notation with addition of complex vectors.

Definitiondf-addcv 24811* Addition of complex vectors in a space of dimension . Experimental. (Contributed by FL, 14-Sep-2013.)

Theoremcladdrv 24813 Closure of addition of complex vectors. (Contributed by FL, 14-Sep-2013.)

Theoremcladdrvr 24814 Closure of addition of real vectors. (Contributed by FL, 29-May-2014.)

Syntaxc0cv 24816 Extends class notation with null vector.

Definitiondf-nullcv 24817* The null vector in a space of dimension . Experimental. (Contributed by FL, 15-Sep-2013.)

Theoremisnullcv 24818* The null vector in a space of dimension . (Contributed by FL, 15-Sep-2013.)

Theoremzernpl 24819 The null vector is a complex vector. (Contributed by FL, 15-Sep-2013.)

Theoremvalvze 24820 Value of the complex vector at a specific coordinate. (Contributed by FL, 15-Sep-2013.)

Theoremaddidv2 24823 The null vector is a left identity for vector addition. (Contributed by FL, 15-Sep-2013.)

Theoremaddidrv2 24824 The null vector is a right identity for vector addition. (Contributed by FL, 15-Sep-2013.)

Theoremcnegvex2 24826* Existence of a left inverse for vector addition. (Contributed by FL, 15-Sep-2013.)

Theoremrnegvex2 24827* Existence of a left inverse for vector addition. (Contributed by FL, 29-May-2014.)

Theoremcnegvex2b 24828* Existence of a left inverse for vector addition. (Contributed by FL, 15-Sep-2013.)

Theoremrnegvex2b 24829* Existence of a left inverse for vector addition. (Contributed by FL, 29-May-2014.)

Syntaxcmcv 24830 Extends class notation with substraction of complex vectors.

Definitiondf-subcatv 24831* Substraction of complex vectors in a space of dimension . Experimental. (Contributed by FL, 15-Sep-2013.)

Theoremnegveud 24834* Existential uniqueness of vector negatives. (Contributed by FL, 15-Sep-2013.)

Theoremnegveudr 24835* Existential uniqueness of vector negatives. (Contributed by FL, 29-May-2014.)

Theoremissubcv 24836* Substraction of complex vectors in a space of dimension . (Contributed by FL, 15-Sep-2013.)

Theoremissubrv 24838* Addition of complex vectors. (Contributed by FL, 29-May-2014.)

Theoremsubclcvd 24839 Closure law for vector substraction. (Contributed by FL, 15-Sep-2013.)

Theoremsubclrvd 24840 Closure law for vector substraction. (Contributed by FL, 29-May-2014.)

Syntaxcnegcv 24841 Extends class notation with the negative of a complex vector.

Definitiondf-ucv 24842* Negative of a complex vector. Experimental. (Contributed by FL, 15-Sep-2013.)

Theoremisucv 24843 Negative of a complex vector. (Contributed by FL, 15-Sep-2013.)

Theoremisucvr 24844 Negative of a complex vector. (Contributed by FL, 29-May-2014.)

Syntaxcsmcv 24845 Extends class notation with scalar multiplication of complex vectors.

Definitiondf-mulcv 24846* Multiplication of complex vectors by a scalar in a space of dimension . Experimental. (Contributed by FL, 15-Sep-2013.)

Theoremismulcv 24847* Multiplication of complex vectors by a scalar in a space of dimension . (Contributed by FL, 15-Sep-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)

Theoremclsmulcv 24848 Closure of scalar multiplication. (Contributed by FL, 29-May-2014.)

Theoremclsmulrv 24849 Closure of scalar multiplication. (Contributed by FL, 29-May-2014.)

Theoremfnmulcv 24850 Functionality of scalar multiplication. (Contributed by FL, 29-May-2014.)

Theoremmulone 24851 Multiplication of a vector by 1. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Jan-2015.)

Theoremvecscmonto 24852 Vector addition is onto. (Contributed by FL, 29-May-2014.)

Theoremmulmulvec 24853 Connection between multiplication of complex numbers and scalar multiplication. (Contributed by FL, 29-May-2014.)

Theoremdistmlva 24854 Distribution of scalar multiplication over vector addition. (Contributed by FL, 29-May-2014.)

Theoremdistsava 24855 "Distribution" of scalar addition. (Contributed by FL, 29-May-2014.)

Theoremtcnvec 24856 Nuples of complex numbers has a structure of vector space. (Contributed by FL, 29-May-2014.)

Syntaxcdivcv 24857 Extends class notation with scalar division of complex vectors.

Definitiondf-divcv 24858* Division of a complex vector by a scalar in a space of dimension . Experimental. (Contributed by FL, 29-May-2014.)

Theoremisdivcv2 24859 Division of complex vectors by a scalar in a space of dimension . (Contributed by FL, 29-May-2014.)

Theoremdivclcvd 24860 Closure law for vector division by a scalar. (Contributed by FL, 29-May-2014.)

Theoremdivclrvd 24861 Closure law for vector division by a scalar. (Contributed by FL, 29-May-2014.)

16.11.44  Calculus

Syntaxcintvl 24862 Extend class notation to include intervals.

Definitiondf-intvl 24863 The intervals of . (Contributed by FL, 29-May-2014.)

Theoremintvlset 24864 The set of intervals is a set. (Contributed by FL, 29-May-2014.)

Theoremintrr 24865 An interval is a part of . (Contributed by FL, 29-May-2014.)

Theoremicccon2 24866 A closed-below, open-above interval is connected. (Contributed by FL, 30-May-2014.)
t

Theoremicccon3 24867 An open-below, closed-above interval is connected. (Contributed by FL, 30-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
t

Theoremicccon4 24868 An open interval is connected. (Contributed by FL, 30-May-2014.)
t

Theoremintvconlem1 24869 All the intervals of are connected. (Contributed by FL, 29-May-2014.)
t

Syntaxcder 24870 Extend class notation to include the derivative of a function.

Definitiondf-der 24871* 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 24750). We have (indif2 3319) but since is not an isolated point in and what is the condition required by trfil2 17414. And in this case the class is a filter. This latter condition is required by df-flimfrs 24745 and this definition is used by df-der 24871.

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 24872 Hard to describe. A picture can help. (Contributed by FL, 29-May-2014.)

Theoremisder 24873* 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.)

16.11.45  Directed multi graphs

Syntaxcmgra 24874 Extend class notation with the class of directed multi graphs.

Definitiondf-mgra 24875* 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 24876 The predicate "is a directed multi graph". (Contributed by FL, 10-Jan-2008.)

16.11.46  Category and deductive system underlying "structure"

Syntaxcalg 24877 Extend class notation with the class of structures used by and .

Syntaxcdom_ 24878 Extend class notation with the function returning the function domain of a category.

Syntaxccod_ 24879 Extend class notation with the function returning the function codomain of a category.

Syntaxcid_ 24880 Extend class notation with the function returning the function identity of a category.

Syntaxco_ 24881 Extend class notation with the function returning the composition of morphisms of a category.

Definitiondf-alg 24882* 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_ 24883 Definition of . (Contributed by FL, 24-Oct-2007.)

Definitiondf-cod_ 24884 Definition of . (Contributed by FL, 26-Oct-2007.)

Definitiondf-id_ 24885 Definition of . (Contributed by FL, 26-Oct-2007.)

Definitiondf-cmpa 24886 Definition of . (Contributed by FL, 26-Oct-2007.)

Theoremisalg 24887 The predicate "has the structure required by and ." (Contributed by FL, 24-Oct-2007.)

Theorem1alg 24888 CatOLDegory has the structure required by and . (Contributed by FL, 30-Oct-2007.)

Theoremdomval 24889 Value of the domain function expressed with the function. (Contributed by FL, 24-Oct-2007.)

Theoremcodval 24890 Value of the function codomain expressed with the and functions. (Contributed by FL, 26-Oct-2007.)

Theoremidval 24891 Value of the identity function expressed with the and functions. (Contributed by FL, 26-Oct-2007.)

Theoremcmpval 24892 Value of the identity function expressed with the functions. (Contributed by FL, 26-Oct-2007.)

Theoremalgi 24893 "Axiomatic" properties of . (Contributed by FL, 24-Oct-2007.)

Theoremdoma 24894 is a mapping from the morphisms of to the objects of . (Contributed by FL, 24-Oct-2007.)

Theoremcoda 24895 is a mapping from the morphisms of to the objects of . (Contributed by FL, 26-Oct-2007.)

Theoremida 24896 is a mapping from the objects of to the morphisms of . (Contributed by FL, 26-Oct-2007.)

Theoremidmoa 24897 An identity is a morphism. (Contributed by FL, 10-Jan-2008.)

Theoremcmppfa 24898 is a partial operation on the morphisms of . (Contributed by FL, 26-Oct-2007.)

Theoremdcsda 24899 and have the same domain. (Contributed by FL, 10-Jan-2008.)

16.11.47  Deductive systems

Syntaxcded 24900 Extend class notation with the class of deductive systems.

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