id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
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proofwiki-23100 | Equivalence of Definitions of Natural Isomorphism between Covariant Functors | Let $\mathbf C$ and $\mathbf D$ be categories.
Let $F, G : \mathbf C \to \mathbf D$ be covariant functors.
{{TFAE|def=Natural Isomorphism}}
=== Definition 1 ===
{{:Definition:Natural Isomorphism between Covariant Functors/Definition 1}}
=== Definition 2 ===
{{:Definition:Natural Isomorphism between Covariant Functors/D... | Let $\operatorname{id}_F$ and $\operatorname{id}_G$ denote the identity natural transformations of $F$ and $G$ respectively. | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
{{TFAE|def=Natural Isomorphism}}
=== [[Definition:Natural Isomorphism between Covariant Functors/Definition 1|Definition 1]] ===
{{:Definition:Natural Is... | Let $\operatorname{id}_F$ and $\operatorname{id}_G$ denote the [[Definition:Identity Natural Transformation|identity natural transformations]] of $F$ and $G$ respectively. | Equivalence of Definitions of Natural Isomorphism between Covariant Functors | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Natural_Isomorphism_between_Covariant_Functors | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Natural_Isomorphism_between_Covariant_Functors | [
"Equivalence of Definitions of Natural Isomorphism between Covariant Functors",
"Natural Isomorphisms"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Natural Isomorphism between Covariant Functors/Definition 1",
"Definition:Natural Isomorphism between Covariant Functors/Definition 2"
] | [
"Definition:Identity Natural Transformation",
"Definition:Identity Natural Transformation"
] |
proofwiki-23101 | Holomorphic Elliptic Function is Constant | Let $\phi: \C \to \C$ be an elliptic function.
Let $\phi$ in addition be a holomorphic function.
Then $\phi$ is constant | {{tidy|minor stuff, good job}}
Let $\phi: \C \to \C$ be a holomorphic elliptic function.
Then there exist $\omega_1, \omega_2 \in \C$ such that:
:$\phi \paren z = \phi \paren {z + \omega_1} = \phi \paren {z + \omega_2}$
Let:
:$S = \set {a \omega_1 + b \omega_2 : a,b \in \closedint 0 1}$
Then $S$ is open and connected.
... | Let $\phi: \C \to \C$ be an [[Definition:Elliptic Function|elliptic function]].
Let $\phi$ in addition be a [[Definition:Holomorphic Function|holomorphic function]].
Then $\phi$ is [[Definition:Constant Mapping|constant]] | {{tidy|minor stuff, good job}}
Let $\phi: \C \to \C$ be a holomorphic elliptic function.
Then there exist $\omega_1, \omega_2 \in \C$ such that:
:$\phi \paren z = \phi \paren {z + \omega_1} = \phi \paren {z + \omega_2}$
Let:
:$S = \set {a \omega_1 + b \omega_2 : a,b \in \closedint 0 1}$
Then $S$ is open and conne... | Holomorphic Elliptic Function is Constant | https://proofwiki.org/wiki/Holomorphic_Elliptic_Function_is_Constant | https://proofwiki.org/wiki/Holomorphic_Elliptic_Function_is_Constant | [
"Elliptic Functions"
] | [
"Definition:Elliptic Function",
"Definition:Holomorphic Function",
"Definition:Constant Mapping"
] | [
"Maximum Modulus Principle",
"Liouville's Theorem (Complex Analysis)"
] |
proofwiki-23102 | Equivalence of Definitions of Natural Isomorphism between Covariant Functors/Definition 1 Implies Definition 2 | Let $\mathbf C$ and $\mathbf D$ be categories.
Let $F, G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\eta : F \to G$ be a natural transformation such that for all $X \in \mathbf C$, $\eta_X : FX \to GX$ is an isomorphism.
Then:
:$\eta$ is an an isomorphism from $F$ to $G$ in the functor category $\operatorna... | By definition of isomorphism:
:for each $X \in \mathbf C$ let $\xi_X : G X \to F X$ be the inverse of $\eta_X$
Let $f : Y \to X$ be a morphism of $\mathbf C$.
We have:
{{begin-eqn}}
{{eqn | l = \xi_X \circ Gf
| r = \paren{\xi_X \circ Gf} \circ \operatorname{id}_{GY}
| c = {{Defof|Identity Morphism}}
}}
{{e... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\eta : F \to G$ be a [[Definition:Natural Transformation|natural transformation]] such that for all $X \in \mathbf C$, $\eta_X : FX \to GX$ is an [[... | By definition of [[Definition:Isomorphism (Category Theory)|isomorphism]]:
:for each $X \in \mathbf C$ let $\xi_X : G X \to F X$ be the [[Definition:Inverse Morphism|inverse]] of $\eta_X$
Let $f : Y \to X$ be a [[Definition:Morphism (Category Theory)|morphism]] of $\mathbf C$.
We have:
{{begin-eqn}}
{{eqn | l = \xi... | Equivalence of Definitions of Natural Isomorphism between Covariant Functors/Definition 1 Implies Definition 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Natural_Isomorphism_between_Covariant_Functors/Definition_1_Implies_Definition_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Natural_Isomorphism_between_Covariant_Functors/Definition_1_Implies_Definition_2 | [
"Equivalence of Definitions of Natural Isomorphism between Covariant Functors"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Natural Transformation",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism",
"Definition:Functor Category",
"Definition:Natural Transformation",
"Definition:Vertical Composition of Natural Transformations",
"Definit... | [
"Definition:Isomorphism (Category Theory)",
"Definition:Inverse Morphism",
"Definition:Morphism",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Natural Isomorphism",
"Definition:Isomorphism",
"Definition:Functor Category"
] |
proofwiki-23103 | Equivalence of Definitions of Natural Isomorphism between Covariant Functors/Definition 2 Implies Definition 1 | Let $\mathbf C$ and $\mathbf D$ be categories.
Let $F, G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\eta : F \to G$ be a natural transformation such that $\eta$ is an isomorphism from $F$ to $G$ in the functor category $\operatorname{Funct}(\mathbf C, \mathbf D)$, that is:
:there exists a natural transforma... | For each $X \in \mathbf C$ we have:
{{begin-eqn}}
{{eqn | l = \xi_X \circ \eta_X
| r = \paren{\xi \circ \eta}_X
| c = {{Defof|Vertical Composition of Natural Transformations}}
}}
{{eqn | r = \paren{\operatorname{id}_F}_X
| c = {{Defof|Isomorphism (Category Theory)|Isomorphism}}
}}
{{eqn | r = \operat... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\eta : F \to G$ be a [[Definition:Natural Transformation|natural transformation]] such that $\eta$ is an [[Definition:Isomorphism|isomorphism]] from... | For each $X \in \mathbf C$ we have:
{{begin-eqn}}
{{eqn | l = \xi_X \circ \eta_X
| r = \paren{\xi \circ \eta}_X
| c = {{Defof|Vertical Composition of Natural Transformations}}
}}
{{eqn | r = \paren{\operatorname{id}_F}_X
| c = {{Defof|Isomorphism (Category Theory)|Isomorphism}}
}}
{{eqn | r = \operat... | Equivalence of Definitions of Natural Isomorphism between Covariant Functors/Definition 2 Implies Definition 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Natural_Isomorphism_between_Covariant_Functors/Definition_2_Implies_Definition_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Natural_Isomorphism_between_Covariant_Functors/Definition_2_Implies_Definition_1 | [
"Equivalence of Definitions of Natural Isomorphism between Covariant Functors"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Natural Transformation",
"Definition:Isomorphism",
"Definition:Functor Category",
"Definition:Natural Transformation",
"Definition:Vertical Composition of Natural Transformations",
"Definition:Identity Natural Transformation",
"Defin... | [
"Definition:Isomorphism (Category Theory)"
] |
proofwiki-23104 | Identity Morphism is Isomorphism | Let $\mathbf C$ be a metacategory.
Let $X$ be an object in $\mathbf C$.
Then:
:the identity morphism $\operatorname{id}_X$ of $X$ is an isomorphism in $\mathbf C$. | By definition of identity morphism:
:$\operatorname{dom} \operatorname{id}_X = X$
:$f \circ \operatorname{id}_X = f$
whenever $X$ is the domain of $f$.
In particular:
:$\operatorname{id}_X \circ \operatorname{id}_X = \operatorname{id}_X$
By definition of an isomorphism:
:a morphism $f: Y \to Z$ is an '''isomorphism''' ... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $X$ be an [[Definition:Object (Category Theory)|object]] in $\mathbf C$.
Then:
:the [[Definition:Identity Morphism|identity morphism]] $\operatorname{id}_X$ of $X$ is an [[Definition:Isomorphism (Category Theory)|isomorphism]] in $\mathbf C$. | By definition of [[Definition:Identity Morphism|identity morphism]]:
:$\operatorname{dom} \operatorname{id}_X = X$
:$f \circ \operatorname{id}_X = f$
whenever $X$ is the [[Definition:Domain (Category Theory)|domain]] of $f$.
In particular:
:$\operatorname{id}_X \circ \operatorname{id}_X = \operatorname{id}_X$
By d... | Identity Morphism is Isomorphism | https://proofwiki.org/wiki/Identity_Morphism_is_Isomorphism | https://proofwiki.org/wiki/Identity_Morphism_is_Isomorphism | [
"Isomorphisms (Category Theory)"
] | [
"Definition:Metacategory",
"Definition:Object (Category Theory)",
"Definition:Identity Morphism",
"Definition:Isomorphism (Category Theory)"
] | [
"Definition:Identity Morphism",
"Definition:Domain (Category Theory)",
"Definition:Isomorphism (Category Theory)",
"Definition:Morphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)",
"Category:Isomorphisms (Category Theory)"
] |
proofwiki-23105 | Left Adjoint Functor is Unique up to Natural Isomorphism | Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $\tuple {F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$.
Then:
:the left adjoint functor $F$ of $G$ is unique up to isomorphism in the functor category $\operatorname{Funct} \tuple{C, D}$.
That is:
:left adjoint functor $F$ is unique up... | Let $\tuple {F, G, \alpha}$ and $\tuple {H, G, \gamma}$ be adjunctions between $\mathbf C$ and $\mathbf D$.
From Characterization of Adjunction Using Unit of Adjunction:
:there exists a natural transformation $\eta: \operatorname {id}_{\mathbf D} \to GF$ such that:
::for each object $D$ in $\mathbf D$ the morphism $\et... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $\tuple {F, G, \alpha}$ be an [[Definition:Adjunction|adjunction]] between $\mathbf C$ and $\mathbf D$.
Then:
:the [[Definition:Left Adjoint Functor|left adjoint functor]] $F$ of $G$ is [[Definition:Unique up to Isom... | Let $\tuple {F, G, \alpha}$ and $\tuple {H, G, \gamma}$ be [[Definition:Adjunction|adjunctions]] between $\mathbf C$ and $\mathbf D$.
From [[Characterization of Adjunction Using Unit of Adjunction]]:
:there exists a [[Definition:Natural Transformation|natural transformation]] $\eta: \operatorname {id}_{\mathbf D} \to... | Left Adjoint Functor is Unique up to Natural Isomorphism | https://proofwiki.org/wiki/Left_Adjoint_Functor_is_Unique_up_to_Natural_Isomorphism | https://proofwiki.org/wiki/Left_Adjoint_Functor_is_Unique_up_to_Natural_Isomorphism | [
"Adjunctions",
"Natural Isomorphisms"
] | [
"Definition:Locally Small Category",
"Definition:Adjunction",
"Definition:Left Adjoint Functor",
"Definition:Unique up to Isomorphism",
"Definition:Functor Category",
"Definition:Left Adjoint Functor",
"Definition:Unique",
"Definition:Natural Isomorphism"
] | [
"Definition:Adjunction",
"Characterization of Adjunction Using Unit of Adjunction",
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Universal Morphism from Object to Functor",
"Definition:Object (Category Theory)",
"Definition:Functor/Cova... |
proofwiki-23106 | Right Adjoint Functor is Unique up to Natural Isomorphism | Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $\tuple {F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$.
Then:
:the right adjoint functor $G$ of $F$ is unique up to isomorphism in the functor category $\operatorname{Funct} \tuple{C, D}$.
That is:
:right adjoint functor $G$ is unique ... | Let $\tuple {F, G, \alpha}$ and $\tuple {F, H, \gamma}$ be adjunctions between $\mathbf C$ and $\mathbf D$.
From Characterization of Adjunction Using Counit of Adjunction:
:there exists a natural transformation $\xi: FG \to \operatorname {id}_{\mathbf C}$ such that:
::for each object $C$ in $\mathbf C$ the morphism $\x... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $\tuple {F, G, \alpha}$ be an [[Definition:Adjunction|adjunction]] between $\mathbf C$ and $\mathbf D$.
Then:
:the [[Definition:Right Adjoint Functor|right adjoint functor]] $G$ of $F$ is [[Definition:Unique up to Is... | Let $\tuple {F, G, \alpha}$ and $\tuple {F, H, \gamma}$ be [[Definition:Adjunction|adjunctions]] between $\mathbf C$ and $\mathbf D$.
From [[Characterization of Adjunction Using Counit of Adjunction]]:
:there exists a [[Definition:Natural Transformation|natural transformation]] $\xi: FG \to \operatorname {id}_{\mathb... | Right Adjoint Functor is Unique up to Natural Isomorphism | https://proofwiki.org/wiki/Right_Adjoint_Functor_is_Unique_up_to_Natural_Isomorphism | https://proofwiki.org/wiki/Right_Adjoint_Functor_is_Unique_up_to_Natural_Isomorphism | [
"Adjunctions",
"Natural Isomorphisms"
] | [
"Definition:Locally Small Category",
"Definition:Adjunction",
"Definition:Right Adjoint Functor",
"Definition:Unique up to Isomorphism",
"Definition:Functor Category",
"Definition:Right Adjoint Functor",
"Definition:Unique",
"Definition:Natural Isomorphism"
] | [
"Definition:Adjunction",
"Characterization of Adjunction Using Counit of Adjunction",
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Universal Morphism from Functor to Object",
"Definition:Functor/Covariant",
"Definition:Object (Category ... |
proofwiki-23107 | Vertical Composition of Natural Transformations is Associative | Let $\mathbf C$ and $\mathbf D$ be categories.
Let $F_1, F_2, F_3, F_4 : \mathbf C \to \mathbf D$ be functors.
Let $\eta: F_1 \to F_2$, $\xi: F_2 \to F_3$ and $\alpha: F_3 \to F_4$ be natural transformations.
Then:
:$\paren {\alpha \circ \xi} \circ \eta = \alpha \circ \paren {\xi \circ \eta}$
where $\circ$ denotes vert... | By definition of natural transformation:
:For every object $A$ of $\mathbf C$, we have morphisms of $\mathsf D$:
::$\eta_A: F_1 A \to F_2 A$
::$\xi_A: F_2 A \to F_3 A$
::$\alpha_A: F_3 A \to F_4 A$
We have:
{{begin-eqn}}
{{eqn | l = \paren {\paren {\alpha \circ \xi} \circ \eta}_A
| r = \paren {\alpha \circ \xi}_A... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Category|categories]].
Let $F_1, F_2, F_3, F_4 : \mathbf C \to \mathbf D$ be [[Definition:Functor|functors]].
Let $\eta: F_1 \to F_2$, $\xi: F_2 \to F_3$ and $\alpha: F_3 \to F_4$ be [[Definition:Natural Transformation|natural transformations]].
Then:
:$\paren {\alpha... | By definition of [[Definition:Natural Transformation|natural transformation]]:
:For every [[Definition:Object (Category Theory)|object]] $A$ of $\mathbf C$, we have [[Definition:Morphism (Category Theory)|morphisms]] of $\mathsf D$:
::$\eta_A: F_1 A \to F_2 A$
::$\xi_A: F_2 A \to F_3 A$
::$\alpha_A: F_3 A \to F_4 A$
... | Vertical Composition of Natural Transformations is Associative | https://proofwiki.org/wiki/Vertical_Composition_of_Natural_Transformations_is_Associative | https://proofwiki.org/wiki/Vertical_Composition_of_Natural_Transformations_is_Associative | [
"Natural Transformations"
] | [
"Definition:Category",
"Definition:Functor",
"Definition:Natural Transformation",
"Definition:Vertical Composition of Natural Transformations"
] | [
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Metacategory",
"Definition:Natural Transformation",
"Category:Natural Transformations"
] |
proofwiki-23108 | Composition of Isomorphisms is Isomorphism | Let $\mathbf C$ be a metacategory.
Let $f: A \to B$ and $g: B \to C$ be isomorphisms with inverses $f^{-1}: B \to A$ and $g^{-1}: C \to B$ respectively.
Then:
:the composition $g \circ f: A \to C$ is an isomorphism with inverse $f^{-1} \circ g^{-1}: C \to A$. | Let $\operatorname{id}_A, \operatorname{id}_B$ and $\operatorname{id}_C$ denote the identity morphisms of $A, B$ and $C$ respectively.
We have:
{{begin-eqn}}
{{eqn | l = \paren {f^{-1} \circ g^{-1} } \circ \paren {g \circ f}
| r = \paren {f^{-1} \circ \paren {g^{-1} \circ g} } \circ f
| c = Associativity of... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $f: A \to B$ and $g: B \to C$ be [[Definition:Isomorphism (Category Theory)|isomorphisms]] with [[Definition:Inverse Morphism|inverses]] $f^{-1}: B \to A$ and $g^{-1}: C \to B$ respectively.
Then:
:the [[Definition:Composite Morphism|composition]] $g... | Let $\operatorname{id}_A, \operatorname{id}_B$ and $\operatorname{id}_C$ denote the [[Definition:Identity Morphism|identity morphisms]] of $A, B$ and $C$ respectively.
We have:
{{begin-eqn}}
{{eqn | l = \paren {f^{-1} \circ g^{-1} } \circ \paren {g \circ f}
| r = \paren {f^{-1} \circ \paren {g^{-1} \circ g} } \... | Composition of Isomorphisms is Isomorphism | https://proofwiki.org/wiki/Composition_of_Isomorphisms_is_Isomorphism | https://proofwiki.org/wiki/Composition_of_Isomorphisms_is_Isomorphism | [
"Isomorphisms (Category Theory)"
] | [
"Definition:Metacategory",
"Definition:Isomorphism (Category Theory)",
"Definition:Inverse Morphism",
"Definition:Composition of Morphisms",
"Definition:Isomorphism (Category Theory)",
"Definition:Inverse Morphism"
] | [
"Definition:Identity Morphism",
"Definition:Metacategory",
"Definition:Metacategory",
"Definition:Isomorphism (Category Theory)",
"Definition:Inverse Morphism",
"Category:Isomorphisms (Category Theory)"
] |
proofwiki-23109 | Power Series Ring over Noetherian Ring is Noetherian | Let $A$ be a Noetherian ring.
Let $A \sqbrk x$ be the formal power series ring over $A$ in the single indeterminate $x$.
Then $A \sqbrk x$ is also a Noetherian ring. | Let $I$ be an ideal of $A \sqbrk x$.
We show that $I$ is finitely generated.
If $0 \ne g \in A \sqbrk x$, define $\order g$ as the largest non-negative integer $n$ such that $x^n$ divides $g$, and $\order 0 = \infty$.
{{explain|What does "divides" mean in this context? "Division" as such is defined on {{ProofWiki}} onl... | Let $A$ be a [[Definition:Noetherian Ring|Noetherian ring]].
Let $A \sqbrk x$ be the [[Definition:Ring of Formal Power Series|formal power series ring]] over $A$ in the single [[Definition:Indeterminate|indeterminate]] $x$.
Then $A \sqbrk x$ is also a [[Definition:Noetherian Ring|Noetherian ring]]. | Let $I$ be an [[Definition:Ideal of Ring|ideal]] of $A \sqbrk x$.
We show that $I$ is [[Definition:Finitely Generated Ideal of Ring|finitely generated]].
If $0 \ne g \in A \sqbrk x$, define $\order g$ as the largest [[Definition:Non-Negative Integer|non-negative integer]] $n$ such that $x^n$ divides $g$, and $\order ... | Power Series Ring over Noetherian Ring is Noetherian | https://proofwiki.org/wiki/Power_Series_Ring_over_Noetherian_Ring_is_Noetherian | https://proofwiki.org/wiki/Power_Series_Ring_over_Noetherian_Ring_is_Noetherian | [
"Noetherian Rings",
"Formal Power Series"
] | [
"Definition:Noetherian Ring",
"Definition:Ring of Formal Power Series",
"Definition:Indeterminate",
"Definition:Noetherian Ring"
] | [
"Definition:Ideal of Ring",
"Definition:Finitely Generated Ideal of Ring",
"Definition:Positive/Integer",
"Definition:Positive/Integer",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring",
"Definition:Noetherian Ring",
"Definition:Positive/Integer",
"Definition:Posi... |
proofwiki-23110 | Uncertainty is Maximal for Uniform Distribution | Let $X$ be a discrete probability distribution.
Let the uncertainty of $X$ be maximal.
Then $X$ is the (discrete) uniform distribution. | From Axiom $1$ of the Axioms of Uncertainty:
{{:Axiom:Axioms of Uncertainty}}
$H_n$ fulfils the following axiom:
:$\map {H_n} {p_1, p_2, \ldots, p_n}$ is a maximum when $p_1 = p_2 = \dotsb = p_n = \dfrac 1 n$
The result follows.
{{qed}} | Let $X$ be a [[Definition:Discrete Probability Distribution|discrete probability distribution]].
Let the [[Definition:Uncertainty|uncertainty]] of $X$ be [[Definition:Maximal|maximal]].
Then $X$ is the [[Definition:Discrete Uniform Distribution|(discrete) uniform distribution]]. | From [[Axiom:Axioms of Uncertainty/Axiom 1|Axiom $1$ of the Axioms of Uncertainty]]:
{{:Axiom:Axioms of Uncertainty}}
$H_n$ fulfils the following [[Definition:Axiom|axiom]]:
:$\map {H_n} {p_1, p_2, \ldots, p_n}$ is a [[Definition:Maximum Value|maximum]] when $p_1 = p_2 = \dotsb = p_n = \dfrac 1 n$
The result follow... | Uncertainty is Maximal for Uniform Distribution | https://proofwiki.org/wiki/Uncertainty_is_Maximal_for_Uniform_Distribution | https://proofwiki.org/wiki/Uncertainty_is_Maximal_for_Uniform_Distribution | [
"Discrete Uniform Distribution",
"Uncertainty"
] | [
"Definition:Discrete Probability Distribution",
"Definition:Uncertainty",
"Definition:Maximal",
"Definition:Uniform Distribution/Discrete"
] | [
"Axiom:Axioms of Uncertainty/Axiom 1",
"Definition:Axiom",
"Definition:Maximum Value of Real Function/Absolute"
] |
proofwiki-23111 | Equivalence of Definitions of Affine Algebraic Variety | Let $k$ be a field.
Let $n \ge 1$ be an integer.
{{TFAE|def = Affine Algebraic Variety}} | Let $X \subseteq k^n$ be an affine algebraic set. | Let $k$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $n \ge 1$ be an [[Definition:Integer|integer]].
{{TFAE|def = Affine Algebraic Variety}} | Let $X \subseteq k^n$ be an [[Definition:Affine Algebraic Set|affine algebraic set]]. | Equivalence of Definitions of Affine Algebraic Variety | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Affine_Algebraic_Variety | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Affine_Algebraic_Variety | [
"Algebraic Varieties"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Integer"
] | [
"Definition:Affine Algebraic Set"
] |
proofwiki-23112 | Product of Functors is Functor | Let $\mathbf C$, $\mathbf C'$, $\mathbf D$, $\mathbf D'$ be categories.
Let $F: \mathbf C \to \mathbf D$ and $G: \mathbf C' \to \mathbf D'$ be covariant functors.
Let $F \times G:\mathbf C \times \mathbf C' \to \mathbf D \times \mathbf D'$ denote the product of $F$ and $G$.
Then:
:$F \times G:\mathbf C \times \mathbf C... | === $F \times G$ Preserves Composite Morphsims ===
For any composable morphisms $\tuple{f_1, g_1}, \tuple{f_2, g_2} \in \mathbf C \times \mathbf C'$ we have:
{{begin-eqn}}
{{eqn | l = \map {\paren{F \times G} } {\tuple{f_2, g_2} \circ \tuple{f_1, g_1} }
| r = \map {\paren{F \times G} } {f_2 \circ f_1, g_2 \circ g... | Let $\mathbf C$, $\mathbf C'$, $\mathbf D$, $\mathbf D'$ be [[Definition:Category|categories]].
Let $F: \mathbf C \to \mathbf D$ and $G: \mathbf C' \to \mathbf D'$ be [[Definition:Covariant Functor|covariant functors]].
Let $F \times G:\mathbf C \times \mathbf C' \to \mathbf D \times \mathbf D'$ denote the [[Defini... | === $F \times G$ Preserves Composite Morphsims ===
For any [[Definition:Composite Morphism|composable morphisms]] $\tuple{f_1, g_1}, \tuple{f_2, g_2} \in \mathbf C \times \mathbf C'$ we have:
{{begin-eqn}}
{{eqn | l = \map {\paren{F \times G} } {\tuple{f_2, g_2} \circ \tuple{f_1, g_1} }
| r = \map {\paren{F \tim... | Product of Functors is Functor | https://proofwiki.org/wiki/Product_of_Functors_is_Functor | https://proofwiki.org/wiki/Product_of_Functors_is_Functor | [
"Products of Functors"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Product of Functors",
"Definition:Functor/Covariant"
] | [
"Definition:Composition of Morphisms",
"Definition:Composition of Morphisms"
] |
proofwiki-23113 | Functor Induced on Opposite Categories | Let $\mathbf C$, $\mathbf D$ be categories.
Let $\mathbf C^\text{op}$, $\mathbf D^\text{op}$ denote the dual categories of $\mathbf C$, $\mathbf D$ respectively.
Let $F: \mathbf C \to \mathbf D$ be a covariant functor.
Let $F^\text{op}: \mathbf C^\text{op} \to \mathbf D^\text{op}$ be defined as:
{{DefineFunctor
|ob = f... | === $F^\text{op}$ Preserves Composite Morphsims ===
For any composable morphisms $f_1^\text{op}, f_2^\text{op} \in \mathbf C^\text{op}$ we have:
{{begin-eqn}}
{{eqn | l = \map {F^\text{op} } {f_2^\text{op} \circ f_1^\text{op} }
| r = \map {F^\text{op} } {\paren{f_1 \circ f_2}^\text{op} }
| c = {{Defof|Dual ... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Category|categories]].
Let $\mathbf C^\text{op}$, $\mathbf D^\text{op}$ denote the [[Definition:Dual Category|dual categories]] of $\mathbf C$, $\mathbf D$ respectively.
Let $F: \mathbf C \to \mathbf D$ be a [[Definition:Covariant Functor|covariant functor]].
Let $F^\te... | === $F^\text{op}$ Preserves Composite Morphsims ===
For any [[Definition:Composite Morphism|composable morphisms]] $f_1^\text{op}, f_2^\text{op} \in \mathbf C^\text{op}$ we have:
{{begin-eqn}}
{{eqn | l = \map {F^\text{op} } {f_2^\text{op} \circ f_1^\text{op} }
| r = \map {F^\text{op} } {\paren{f_1 \circ f_2}^\t... | Functor Induced on Opposite Categories | https://proofwiki.org/wiki/Functor_Induced_on_Opposite_Categories | https://proofwiki.org/wiki/Functor_Induced_on_Opposite_Categories | [
"Functors",
"Dual Categories"
] | [
"Definition:Category",
"Definition:Dual Category",
"Definition:Functor/Covariant",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Functor/Covariant"
] | [
"Definition:Composition of Morphisms",
"Definition:Composition of Morphisms"
] |
proofwiki-23114 | Characterization of Hom Bifunctor with Left Functor as Composition of Functors | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$ be a locally small category.
Let $\mathbf D$ be a locally small category.
Let $L : \mathbf D \to \mathbf C$ be a covariant functor.
Let $\map {\operatorname{Hom}_{\mathbf C} } {L-, -} : \mathbf D^{\text{op} } \times \mathbf C \to \mathbf {Set}$ denote the hom... | Let $F$ denote the hom bifunctor with left functor $\map {\operatorname{Hom}_{\mathbf C} } {L-, -} : \mathbf D^{\text{op} } \times \mathbf C \to \mathbf {Set}$.
Let $G$ denote the hom bifunctor $\map {\operatorname{Hom}_{\mathbf C} } {-, -} : \mathbf D^{\text{op} } \times \mathbf C \to \mathbf {Set}$.
For each object $... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$ be a [[Definition:Locally Small Category|locally small category]].
Let $\mathbf D$ be a [[Definition:Locally Small Category|locally small category]].
Let $L : \mathbf D \to \mathbf C$ be a [[Definition:Covariant Functor|cova... | Let $F$ denote the [[Definition:Hom Bifunctor With Left Functor|hom bifunctor with left functor]] $\map {\operatorname{Hom}_{\mathbf C} } {L-, -} : \mathbf D^{\text{op} } \times \mathbf C \to \mathbf {Set}$.
Let $G$ denote the [[Definition:Hom Bifunctor|hom bifunctor]] $\map {\operatorname{Hom}_{\mathbf C} } {-, -} : ... | Characterization of Hom Bifunctor with Left Functor as Composition of Functors | https://proofwiki.org/wiki/Characterization_of_Hom_Bifunctor_with_Left_Functor_as_Composition_of_Functors | https://proofwiki.org/wiki/Characterization_of_Hom_Bifunctor_with_Left_Functor_as_Composition_of_Functors | [
"Bifunctors"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Hom Bifunctor With Left Functor",
"Definition:Functor/Covariant",
"Definition:Object Functor",
"Definition:Object (Category Theory)",
"Definition:Morph... | [
"Definition:Hom Bifunctor With Left Functor",
"Definition:Hom Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Morphism"
] |
proofwiki-23115 | Characterization of Hom Bifunctor with Right Functor as Composition of Functors | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$ be a locally small category.
Let $\mathbf D$ be a locally small category.
Let $R : \mathbf D \to \mathbf C$ be a covariant functor.
Let $\map {\operatorname{Hom}_{\mathbf D} } {-, R-} : \mathbf D^{\text{op} } \times \mathbf C \to \mathbf {Set}$ denote the hom... | Let $F$ denote the hom bifunctor with right functor $\map {\operatorname{Hom}_{\mathbf D} } {-, R-} : \mathbf D^\text{op} \times \mathbf C \to \mathbf {Set}$.
Let $G$ denote the hom bifunctor $\map {\operatorname{Hom}_{\mathbf D} } {-, -} : \mathbf D^\text{op} \times \mathbf C \to \mathbf {Set}$.
For each object $\tupl... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$ be a [[Definition:Locally Small Category|locally small category]].
Let $\mathbf D$ be a [[Definition:Locally Small Category|locally small category]].
Let $R : \mathbf D \to \mathbf C$ be a [[Definition:Covariant Functor|cova... | Let $F$ denote the [[Definition:Hom Bifunctor With Right Functor|hom bifunctor with right functor]] $\map {\operatorname{Hom}_{\mathbf D} } {-, R-} : \mathbf D^\text{op} \times \mathbf C \to \mathbf {Set}$.
Let $G$ denote the [[Definition:Hom Bifunctor|hom bifunctor]] $\map {\operatorname{Hom}_{\mathbf D} } {-, -} : \... | Characterization of Hom Bifunctor with Right Functor as Composition of Functors | https://proofwiki.org/wiki/Characterization_of_Hom_Bifunctor_with_Right_Functor_as_Composition_of_Functors | https://proofwiki.org/wiki/Characterization_of_Hom_Bifunctor_with_Right_Functor_as_Composition_of_Functors | [
"Bifunctors"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Hom Bifunctor With Right Functor",
"Definition:Identity Functor",
"Definition:Product of Functors",
"Definition:Hom Bifunctor",
"Definition:Composition... | [
"Definition:Hom Bifunctor With Right Functor",
"Definition:Hom Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Morphism"
] |
proofwiki-23116 | Equations of Motion with Constant Acceleration/Distance after Time in terms of Velocities | :$\mathbf s = \dfrac {\paren {\mathbf u + \mathbf v} t} 2$ | From Equations of Motion with Constant Acceleration: Velocity after Time:
:$(1): \quad \mathbf v = \mathbf u + \mathbf a t$
From Equations of Motion with Constant Acceleration: Distance after Time:
:$(2): \quad \mathbf s = \mathbf u t + \dfrac {\mathbf a t^2} 2$
Hence:
{{begin-eqn}}
{{eqn | n = 3
| l = \mathbf v ... | :$\mathbf s = \dfrac {\paren {\mathbf u + \mathbf v} t} 2$ | From [[Equations of Motion with Constant Acceleration/Velocity after Time|Equations of Motion with Constant Acceleration: Velocity after Time]]:
:$(1): \quad \mathbf v = \mathbf u + \mathbf a t$
From [[Equations of Motion with Constant Acceleration/Distance after Time|Equations of Motion with Constant Acceleration: Di... | Equations of Motion with Constant Acceleration/Distance after Time in terms of Velocities | https://proofwiki.org/wiki/Equations_of_Motion_with_Constant_Acceleration/Distance_after_Time_in_terms_of_Velocities | https://proofwiki.org/wiki/Equations_of_Motion_with_Constant_Acceleration/Distance_after_Time_in_terms_of_Velocities | [
"Equations of Motion with Constant Acceleration"
] | [] | [
"Equations of Motion with Constant Acceleration/Velocity after Time",
"Equations of Motion with Constant Acceleration/Distance after Time"
] |
proofwiki-23117 | Characterization of Bifunctor Induced By One-Variable Functors | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be categories.
Let $F_0: \mathbf C_0 \times \mathbf D_0 \to \mathbf E_0$ be an object functor.
Let $F_1: \mathbf C_1 \times \mathbf D_1 \to \mathbf E_1$ be a morphism functor.
For each $C \in \mathbf C_0$, let:
:$\paren{L_C}_0: \mathbf D_0 \to \mathbf E_0$ denote the object ... | === Necessary Condition ===
Let $F$ be a bifunctor.
{{:Characterization of Bifunctor Induced By One-Variable Functors/Necessary Condition|Necessary Condition}}{{qed|lemma}} | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
Let $F_0: \mathbf C_0 \times \mathbf D_0 \to \mathbf E_0$ be an [[Definition:Object Functor|object functor]].
Let $F_1: \mathbf C_1 \times \mathbf D_1 \to \mathbf E_1$ be a [[Definition:Morphism Functor|morphism functor]].
For each... | === [[Characterization of Bifunctor Induced By One-Variable Functors/Necessary Condition|Necessary Condition]] ===
Let $F$ be a [[Definition:Bifunctor|bifunctor]].
{{:Characterization of Bifunctor Induced By One-Variable Functors/Necessary Condition|Necessary Condition}}{{qed|lemma}} | Characterization of Bifunctor Induced By One-Variable Functors | https://proofwiki.org/wiki/Characterization_of_Bifunctor_Induced_By_One-Variable_Functors | https://proofwiki.org/wiki/Characterization_of_Bifunctor_Induced_By_One-Variable_Functors | [
"Characterization of Bifunctor Induced By One-Variable Functors",
"Bifunctors"
] | [
"Definition:Category",
"Definition:Object Functor",
"Definition:Morphism Functor",
"Definition:Object Functor",
"Definition:Morphism Functor",
"Definition:Object Functor",
"Definition:Morphism Functor",
"Definition:Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
... | [
"Characterization of Bifunctor Induced By One-Variable Functors/Necessary Condition",
"Definition:Bifunctor"
] |
proofwiki-23118 | Characterization of Natural Transformation Between Bifunctors | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be categories.
Let $F, G : \mathbf C \times \mathbf D \to \mathbf E$ be bifunctors from the product category $\mathbf C \times \mathbf D$ to the category $\mathbf E$.
For each object $C \in\mathbf C$ let:
:$F \tuple{C, -}$ and $G \tuple{C, -}$
denote the functors obtained by ... | === Necessary Condition ===
Let $\alpha : F \to G$ be a natural transformation.
{{:Characterization of Natural Transformation Between Bifunctors/Proof 1 Necessary Condition}}{{qed|lemma}}
=== Sufficient Condition ===
Let:
:$(1)\quad$ For each object $C \in \mathbf C : \alpha \tuple{C, -} : F \tuple{C, -} \to G \tuple{... | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \times \mathbf D \to \mathbf E$ be [[Definition:Bifunctor|bifunctors]] from the [[Definition:Product Category|product category]] $\mathbf C \times \mathbf D$ to the [[Definition:Category|category]] $\mathbf E$.
Fo... | === [[Characterization of Natural Transformation Between Bifunctors/Proof 1 Necessary Condition|Necessary Condition]] ===
Let $\alpha : F \to G$ be a [[Definition:Natural Transformation|natural transformation]].
{{:Characterization of Natural Transformation Between Bifunctors/Proof 1 Necessary Condition}}{{qed|lemma}... | Characterization of Natural Transformation Between Bifunctors/Proof 1 | https://proofwiki.org/wiki/Characterization_of_Natural_Transformation_Between_Bifunctors | https://proofwiki.org/wiki/Characterization_of_Natural_Transformation_Between_Bifunctors/Proof_1 | [
"Characterization of Natural Transformation Between Bifunctors",
"Bifunctors"
] | [
"Definition:Category",
"Definition:Bifunctor",
"Definition:Product Category",
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Object (Categ... | [
"Characterization of Natural Transformation Between Bifunctors/Proof 1 Necessary Condition",
"Definition:Natural Transformation",
"Characterization of Natural Transformation Between Bifunctors/Proof 1 Sufficient Condition",
"Definition:Object (Category Theory)",
"Definition:Natural Transformation",
"Defin... |
proofwiki-23119 | Characterization of Natural Transformation Between Bifunctors | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be categories.
Let $F, G : \mathbf C \times \mathbf D \to \mathbf E$ be bifunctors from the product category $\mathbf C \times \mathbf D$ to the category $\mathbf E$.
For each object $C \in\mathbf C$ let:
:$F \tuple{C, -}$ and $G \tuple{C, -}$
denote the functors obtained by ... | By definition of natural transformation:
:for all morphisms $f: C \to C' \in \mathbf C$ and $g: D \to D' \in \mathbf D$ the following diagrams commute:
::<nowiki>$\begin{xy}
<0em,0em>*+{\map F {C, D} } = "FCD",
<12em,0em>*+{\map F {C', D} } = "FC2D",
<0em,-6em>*+{\map G {C, D} } = "GCD",
<12em,-6em>*+{\map G {C', D} } ... | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \times \mathbf D \to \mathbf E$ be [[Definition:Bifunctor|bifunctors]] from the [[Definition:Product Category|product category]] $\mathbf C \times \mathbf D$ to the [[Definition:Category|category]] $\mathbf E$.
Fo... | By definition of [[Definition:Natural Transformation|natural transformation]]:
:for all [[Definition:Morphism (Category Theory)|morphisms]] $f: C \to C' \in \mathbf C$ and $g: D \to D' \in \mathbf D$ the following [[Definition:Commutative Diagram|diagrams commute]]:
::<nowiki>$\begin{xy}
<0em,0em>*+{\map F {C, D} } =... | Characterization of Natural Transformation Between Bifunctors/Proof 1 Necessary Condition | https://proofwiki.org/wiki/Characterization_of_Natural_Transformation_Between_Bifunctors | https://proofwiki.org/wiki/Characterization_of_Natural_Transformation_Between_Bifunctors/Proof_1_Necessary_Condition | [
"Characterization of Natural Transformation Between Bifunctors",
"Bifunctors"
] | [
"Definition:Category",
"Definition:Bifunctor",
"Definition:Product Category",
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Object (Categ... | [
"Definition:Natural Transformation",
"Definition:Morphism",
"Definition:Commutative Diagram",
"Definition:Morphism",
"Definition:Commutative Diagram",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Natural Transformation",
... |
proofwiki-23120 | Characterization of Natural Transformation Between Bifunctors | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be categories.
Let $F, G : \mathbf C \times \mathbf D \to \mathbf E$ be bifunctors from the product category $\mathbf C \times \mathbf D$ to the category $\mathbf E$.
For each object $C \in\mathbf C$ let:
:$F \tuple{C, -}$ and $G \tuple{C, -}$
denote the functors obtained by ... | By definition of natural transformation:
:for all morphisms $f: C \to C' \in \mathbf C$ and $g: D \to D' \in \mathbf D$ the following diagrams commute:
::<nowiki>$\begin{xy}
<0em,0em>*+{\map {F \tuple{-, D} } C} = "FCD",
<15em,0em>*+{\map {F \tuple{-, D} } {C'} } = "FC2D",
<0em,-6em>*+{\map {G \tuple{-, D} } C} = "GCD"... | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \times \mathbf D \to \mathbf E$ be [[Definition:Bifunctor|bifunctors]] from the [[Definition:Product Category|product category]] $\mathbf C \times \mathbf D$ to the [[Definition:Category|category]] $\mathbf E$.
Fo... | By definition of [[Definition:Natural Transformation|natural transformation]]:
:for all [[Definition:Morphism (Category Theory)|morphisms]] $f: C \to C' \in \mathbf C$ and $g: D \to D' \in \mathbf D$ the following [[Definition:Commutative Diagram|diagrams commute]]:
::<nowiki>$\begin{xy}
<0em,0em>*+{\map {F \tuple{-,... | Characterization of Natural Transformation Between Bifunctors/Proof 1 Sufficient Condition | https://proofwiki.org/wiki/Characterization_of_Natural_Transformation_Between_Bifunctors | https://proofwiki.org/wiki/Characterization_of_Natural_Transformation_Between_Bifunctors/Proof_1_Sufficient_Condition | [
"Characterization of Natural Transformation Between Bifunctors",
"Bifunctors"
] | [
"Definition:Category",
"Definition:Bifunctor",
"Definition:Product Category",
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Object (Categ... | [
"Definition:Natural Transformation",
"Definition:Morphism",
"Definition:Commutative Diagram",
"Definition:Morphism",
"Definition:Commutative Diagram",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Bifunctor",
"Definition:Bifunctor",
"Definition:Identity Morphism",
"Definition:Morphis... |
proofwiki-23121 | Characterization of Natural Transformation Between Bifunctors | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be categories.
Let $F, G : \mathbf C \times \mathbf D \to \mathbf E$ be bifunctors from the product category $\mathbf C \times \mathbf D$ to the category $\mathbf E$.
For each object $C \in\mathbf C$ let:
:$F \tuple{C, -}$ and $G \tuple{C, -}$
denote the functors obtained by ... | === Necessary Condition ===
Let $\alpha : F \to G$ be a natural transformation.
{{:Characterization of Natural Transformation Between Bifunctors/Proof 2 Necessary Condition}}{{qed|lemma}}
=== Sufficient Condition ===
Let:
:$(1)\quad$ For each object $C \in \mathbf C : \alpha \tuple{C, -} : F \tuple{C, -} \to G \tuple{... | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \times \mathbf D \to \mathbf E$ be [[Definition:Bifunctor|bifunctors]] from the [[Definition:Product Category|product category]] $\mathbf C \times \mathbf D$ to the [[Definition:Category|category]] $\mathbf E$.
Fo... | === [[Characterization of Natural Transformation Between Bifunctors/Proof 2 Necessary Condition|Necessary Condition]] ===
Let $\alpha : F \to G$ be a [[Definition:Natural Transformation|natural transformation]].
{{:Characterization of Natural Transformation Between Bifunctors/Proof 2 Necessary Condition}}{{qed|lemma}... | Characterization of Natural Transformation Between Bifunctors/Proof 2 | https://proofwiki.org/wiki/Characterization_of_Natural_Transformation_Between_Bifunctors | https://proofwiki.org/wiki/Characterization_of_Natural_Transformation_Between_Bifunctors/Proof_2 | [
"Characterization of Natural Transformation Between Bifunctors",
"Bifunctors"
] | [
"Definition:Category",
"Definition:Bifunctor",
"Definition:Product Category",
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Object (Categ... | [
"Characterization of Natural Transformation Between Bifunctors/Proof 2 Necessary Condition",
"Definition:Natural Transformation",
"Characterization of Natural Transformation Between Bifunctors/Proof 2 Sufficient Condition",
"Definition:Object (Category Theory)",
"Definition:Natural Transformation",
"Defin... |
proofwiki-23122 | Characterization of Natural Transformation Between Bifunctors | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be categories.
Let $F, G : \mathbf C \times \mathbf D \to \mathbf E$ be bifunctors from the product category $\mathbf C \times \mathbf D$ to the category $\mathbf E$.
For each object $C \in\mathbf C$ let:
:$F \tuple{C, -}$ and $G \tuple{C, -}$
denote the functors obtained by ... | By definition of natural transformation:
:for all morphisms $\tuple{f: C \to C', g:D \to D'} \in \mathbf C \times \mathbf D : \map G {f, g} \circ \alpha_{\tuple{C, D} } = \alpha_{\tuple{C', D'} } \circ \map F {f, g}$
In particular:
:$(3)\quad$ for all morphisms $\tuple{f: C \to C', \operatorname{id}_D:D \to D} \in \mat... | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \times \mathbf D \to \mathbf E$ be [[Definition:Bifunctor|bifunctors]] from the [[Definition:Product Category|product category]] $\mathbf C \times \mathbf D$ to the [[Definition:Category|category]] $\mathbf E$.
Fo... | By definition of [[Definition:Natural Transformation|natural transformation]]:
:for all [[Definition:Morphism (Category Theory)|morphisms]] $\tuple{f: C \to C', g:D \to D'} \in \mathbf C \times \mathbf D : \map G {f, g} \circ \alpha_{\tuple{C, D} } = \alpha_{\tuple{C', D'} } \circ \map F {f, g}$
In particular:
:$(3)\... | Characterization of Natural Transformation Between Bifunctors/Proof 2 Necessary Condition | https://proofwiki.org/wiki/Characterization_of_Natural_Transformation_Between_Bifunctors | https://proofwiki.org/wiki/Characterization_of_Natural_Transformation_Between_Bifunctors/Proof_2_Necessary_Condition | [
"Characterization of Natural Transformation Between Bifunctors",
"Bifunctors"
] | [
"Definition:Category",
"Definition:Bifunctor",
"Definition:Product Category",
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Object (Categ... | [
"Definition:Natural Transformation",
"Definition:Morphism",
"Definition:Morphism",
"Definition:Morphism",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Object (Category Theory)",
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Definition:Mor... |
proofwiki-23123 | Characterization of Natural Transformation Between Bifunctors | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be categories.
Let $F, G : \mathbf C \times \mathbf D \to \mathbf E$ be bifunctors from the product category $\mathbf C \times \mathbf D$ to the category $\mathbf E$.
For each object $C \in\mathbf C$ let:
:$F \tuple{C, -}$ and $G \tuple{C, -}$
denote the functors obtained by ... | By definition of natural transformation:
:for all morphisms $f: C \to C' \in \mathbf C$ and $g:D \to D' \in \mathbf D$:
::$\map {G \tuple{-, D} } f \circ \alpha \tuple{-, D}_C = \alpha \tuple{-, D}_{C'} \circ \map {F \tuple{-, D} } f$
::$\map {G \tuple{C, -} } g \circ \alpha \tuple{C, -}_D = \alpha \tuple{C, -}_{D'} \c... | Let $\mathbf C, \mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \times \mathbf D \to \mathbf E$ be [[Definition:Bifunctor|bifunctors]] from the [[Definition:Product Category|product category]] $\mathbf C \times \mathbf D$ to the [[Definition:Category|category]] $\mathbf E$.
Fo... | By definition of [[Definition:Natural Transformation|natural transformation]]:
:for all [[Definition:Morphism (Category Theory)|morphisms]] $f: C \to C' \in \mathbf C$ and $g:D \to D' \in \mathbf D$:
::$\map {G \tuple{-, D} } f \circ \alpha \tuple{-, D}_C = \alpha \tuple{-, D}_{C'} \circ \map {F \tuple{-, D} } f$
::$\m... | Characterization of Natural Transformation Between Bifunctors/Proof 2 Sufficient Condition | https://proofwiki.org/wiki/Characterization_of_Natural_Transformation_Between_Bifunctors | https://proofwiki.org/wiki/Characterization_of_Natural_Transformation_Between_Bifunctors/Proof_2_Sufficient_Condition | [
"Characterization of Natural Transformation Between Bifunctors",
"Bifunctors"
] | [
"Definition:Category",
"Definition:Bifunctor",
"Definition:Product Category",
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Object (Categ... | [
"Definition:Natural Transformation",
"Definition:Morphism",
"Definition:Morphism",
"Definition:Functor",
"Definition:Bifunctor",
"Definition:Bifunctor",
"Definition:Morphism",
"Definition:Natural Transformation"
] |
proofwiki-23124 | Circle is Locus of Equidistant Points from Point in Plane | Let $P$ be a point in the plane.
The locus of points which are equidistant from $P$ is the circle whose center is $P$. | {{Recall|Circle}}
:{{Definition:Circle}}
The result follows directly.
{{qed}} | Let $P$ be a [[Definition:Point|point]] in [[Definition:The Plane|the plane]].
The [[Definition:Locus|locus]] of [[Definition:Point|points]] which are [[Definition:Equidistant|equidistant]] from $P$ is the [[Definition:Circle|circle]] whose [[Definition:Center of Circle|center]] is $P$. | {{Recall|Circle}}
:{{Definition:Circle}}
The result follows directly.
{{qed}} | Circle is Locus of Equidistant Points from Point in Plane | https://proofwiki.org/wiki/Circle_is_Locus_of_Equidistant_Points_from_Point_in_Plane | https://proofwiki.org/wiki/Circle_is_Locus_of_Equidistant_Points_from_Point_in_Plane | [
"Equidistant",
"Circles"
] | [
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Definition:Locus",
"Definition:Point",
"Definition:Equidistant",
"Definition:Circle",
"Definition:Circle/Center"
] | [] |
proofwiki-23125 | Sphere is Locus of Equidistant Points from Point in Space | Let $P$ be a point in space.
The locus of points which are equidistant from $P$ is the sphere whose center is $P$. | {{Recall|Sphere (Geometry)}}
:{{Definition:Sphere (Geometry)}}
The result follows directly.
{{qed}} | Let $P$ be a [[Definition:Point|point]] in [[Definition:Ordinary Space|space]].
The [[Definition:Locus|locus]] of [[Definition:Point|points]] which are [[Definition:Equidistant|equidistant]] from $P$ is the [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Circle|center]] is $P$. | {{Recall|Sphere (Geometry)}}
:{{Definition:Sphere (Geometry)}}
The result follows directly.
{{qed}} | Sphere is Locus of Equidistant Points from Point in Space | https://proofwiki.org/wiki/Sphere_is_Locus_of_Equidistant_Points_from_Point_in_Space | https://proofwiki.org/wiki/Sphere_is_Locus_of_Equidistant_Points_from_Point_in_Space | [
"Equidistant",
"Spheres"
] | [
"Definition:Point",
"Definition:Ordinary Space",
"Definition:Locus",
"Definition:Point",
"Definition:Equidistant",
"Definition:Sphere/Geometry",
"Definition:Circle/Center"
] | [] |
proofwiki-23126 | Rhombus is Equilateral but not Equiangular | Let $\Box ABCD$ be a rhombus.
Then while $\Box ABCD$ is equilateral, it is not generally the case that $\Box ABCD$ is also equiangular.
If $\Box ABCD$ is also equiangular, then it is a square | {{Recall|Rhombus|rhombus}}
{{:Definition:Rhombus}}
{{Recall|Square (Geometry)|square}}
{{:Definition:Square (Geometry)}}
The result follows.
{{qed}} | Let $\Box ABCD$ be a [[Definition:Rhombus|rhombus]].
Then while $\Box ABCD$ is [[Definition:Equilateral Polygon|equilateral]], it is not generally the case that $\Box ABCD$ is also [[Definition:Equiangular Polygon|equiangular]].
If $\Box ABCD$ is also [[Definition:Equiangular Polygon|equiangular]], then it is a [[Def... | {{Recall|Rhombus|rhombus}}
{{:Definition:Rhombus}}
{{Recall|Square (Geometry)|square}}
{{:Definition:Square (Geometry)}}
The result follows.
{{qed}} | Rhombus is Equilateral but not Equiangular | https://proofwiki.org/wiki/Rhombus_is_Equilateral_but_not_Equiangular | https://proofwiki.org/wiki/Rhombus_is_Equilateral_but_not_Equiangular | [
"Rhombi"
] | [
"Definition:Quadrilateral/Rhombus",
"Definition:Polygon/Equilateral",
"Definition:Polygon/Equiangular",
"Definition:Polygon/Equiangular",
"Definition:Quadrilateral/Square"
] | [] |
proofwiki-23127 | Equivalence Classes with Common Element are Equal | Let $\RR$ be an equivalence relation on a set $S$.
Let $\eqclass x \RR$ and $\eqclass y \RR$ denote the $\RR$-equivalence classes of $x$ and $y$ respectively.
Let $z \in S$ such that $z \in \eqclass x \RR$ and $z \in \eqclass y \RR$.
Then:
:$\eqclass x \RR = \eqclass y \RR$ | Let $z \in S$ such that $z \in \eqclass x \RR$ and $z \in \eqclass y \RR$ {{hypothesis}}.
By definition of set intersection:
:$z \in \eqclass x \RR \cap \eqclass y \RR$
{{AimForCont}} $\eqclass x \RR \ne \eqclass y \RR$.
From Equivalence Classes are Disjoint:
:$\eqclass x \RR \cap \eqclass y \RR = \O$
But this contradi... | Let $\RR$ be an [[Definition:Equivalence Relation|equivalence relation]] on a [[Definition:Set|set]] $S$.
Let $\eqclass x \RR$ and $\eqclass y \RR$ denote the [[Definition:Equivalence Class|$\RR$-equivalence classes]] of $x$ and $y$ respectively.
Let $z \in S$ such that $z \in \eqclass x \RR$ and $z \in \eqclass y \R... | Let $z \in S$ such that $z \in \eqclass x \RR$ and $z \in \eqclass y \RR$ {{hypothesis}}.
By definition of [[Definition:Set Intersection|set intersection]]:
:$z \in \eqclass x \RR \cap \eqclass y \RR$
{{AimForCont}} $\eqclass x \RR \ne \eqclass y \RR$.
From [[Equivalence Classes are Disjoint]]:
:$\eqclass x \RR \cap... | Equivalence Classes with Common Element are Equal | https://proofwiki.org/wiki/Equivalence_Classes_with_Common_Element_are_Equal | https://proofwiki.org/wiki/Equivalence_Classes_with_Common_Element_are_Equal | [
"Equivalence Classes"
] | [
"Definition:Equivalence Relation",
"Definition:Set",
"Definition:Equivalence Class"
] | [
"Definition:Set Intersection",
"Equivalence Classes are Disjoint",
"Definition:Contradiction",
"Proof by Contradiction"
] |
proofwiki-23128 | Characterization of Bifunctor Induced By One-Variable Functors/Necessary Condition | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be categories.
Let $F: \mathbf C \times \mathbf D \to \mathbf E$ be a bifunctor.
For each $C \in \mathbf C_0$, let:
:$\paren{L_C}_0: \mathbf D_0 \to \mathbf E_0$ denote the object functor defined by:
::For each $D \in \mathbf D_0 : \map {\paren{L_C}_0} D = \map {F_0} {C, D}$... | ==== $(1) : M_D: \mathbf C \to \mathbf E$ is a Covariant Functor ====
Let $f: A \to B, g: B \to C \in \mathbf C_1$.
We have:
{{begin-eqn}}
{{eqn | l = \map {M_D} {g \circ f}
| r = \map F {g \circ f, \operatorname{id}_D}
| c = Definition of $M_D$ morphism functor
}}
{{eqn | r = \map F {g \circ f, \operatorna... | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
Let $F: \mathbf C \times \mathbf D \to \mathbf E$ be a [[Definition:Bifunctor|bifunctor]].
For each $C \in \mathbf C_0$, let:
:$\paren{L_C}_0: \mathbf D_0 \to \mathbf E_0$ denote the [[Definition:Object Functor|object functor]] defi... | ==== $(1) : M_D: \mathbf C \to \mathbf E$ is a Covariant Functor ====
Let $f: A \to B, g: B \to C \in \mathbf C_1$.
We have:
{{begin-eqn}}
{{eqn | l = \map {M_D} {g \circ f}
| r = \map F {g \circ f, \operatorname{id}_D}
| c = Definition of $M_D$ [[Definition:Morphism Functor|morphism functor]]
}}
{{eqn | ... | Characterization of Bifunctor Induced By One-Variable Functors/Necessary Condition | https://proofwiki.org/wiki/Characterization_of_Bifunctor_Induced_By_One-Variable_Functors/Necessary_Condition | https://proofwiki.org/wiki/Characterization_of_Bifunctor_Induced_By_One-Variable_Functors/Necessary_Condition | [
"Characterization of Bifunctor Induced By One-Variable Functors"
] | [
"Definition:Category",
"Definition:Bifunctor",
"Definition:Object Functor",
"Definition:Morphism Functor",
"Definition:Object Functor",
"Definition:Morphism Functor",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Object (Category Theory)",
"Definition:Functor/C... | [
"Definition:Morphism Functor",
"Identity Morphism is Idempotent",
"Definition:Morphism Functor",
"Definition:Composition of Morphisms",
"Definition:Morphism Functor",
"Definition:Object Functor",
"Definition:Identity Morphism",
"Definition:Functor/Covariant",
"Definition:Morphism Functor",
"Identi... |
proofwiki-23129 | Characterization of Bifunctor Induced By One-Variable Functors/Sufficient Condition | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be categories.
Let $F_0: \mathbf C_0 \times \mathbf D_0 \to \mathbf E_0$ be an object functor.
Let $F_1: \mathbf C_1 \times \mathbf D_1 \to \mathbf E_1$ be a morphism functor.
For each $C \in \mathbf C_0$, let:
:$\paren{L_C}_0: \mathbf D_0 \to \mathbf E_0$ denote the object ... | For morphisms:
:$\tuple{f : C \to C', g : D \to D'}, \tuple{f' : C' \to C' ', g' : D' \to D' '} \in \mathbf C \times \mathbf D$
we have:
{{begin-eqn}}
{{eqn | l = F \paren{\tuple{f', g'} \circ \tuple{f, g} }
| r = F \tuple{f' \circ f, g' \circ g}
| c = {{Defof|Product Category|Composition in Product Categor... | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
Let $F_0: \mathbf C_0 \times \mathbf D_0 \to \mathbf E_0$ be an [[Definition:Object Functor|object functor]].
Let $F_1: \mathbf C_1 \times \mathbf D_1 \to \mathbf E_1$ be a [[Definition:Morphism Functor|morphism functor]].
For each... | For [[Definition:Morphism (Category Theory)|morphisms]]:
:$\tuple{f : C \to C', g : D \to D'}, \tuple{f' : C' \to C' ', g' : D' \to D' '} \in \mathbf C \times \mathbf D$
we have:
{{begin-eqn}}
{{eqn | l = F \paren{\tuple{f', g'} \circ \tuple{f, g} }
| r = F \tuple{f' \circ f, g' \circ g}
| c = {{Defof|Produ... | Characterization of Bifunctor Induced By One-Variable Functors/Sufficient Condition | https://proofwiki.org/wiki/Characterization_of_Bifunctor_Induced_By_One-Variable_Functors/Sufficient_Condition | https://proofwiki.org/wiki/Characterization_of_Bifunctor_Induced_By_One-Variable_Functors/Sufficient_Condition | [
"Characterization of Bifunctor Induced By One-Variable Functors"
] | [
"Definition:Category",
"Definition:Object Functor",
"Definition:Morphism Functor",
"Definition:Object Functor",
"Definition:Morphism Functor",
"Definition:Object Functor",
"Definition:Morphism Functor",
"Definition:Object",
"Definition:Functor/Covariant",
"Definition:Object",
"Definition:Functor... | [
"Definition:Morphism",
"Definition:Composition of Morphisms",
"Definition:Object (Category Theory)",
"Identity Morphism is Idempotent",
"Definition:Identity Morphism",
"Definition:Bifunctor"
] |
proofwiki-23130 | Characterization of Existence of Bifunctor Induced By One-Variable Functors | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be categories.
For each object $C \in \mathbf C$, let:
:$L_C: \mathbf D \to \mathbf E$ be a covariant functor.
For each object $D \in \mathbf D$, let:
:$M_D: \mathbf C \to \mathbf E$ be a covariant functor.
Then:
:there exists a bifunctor $F: \mathbf C \times \mathbf D \to ... | === Necessary Condition ===
Let $F: \mathbf C \times \mathbf D \to \mathbf E$ be a bifunctor such that:
::$(1)\quad$ For each object $C \in \mathbf C : F(C, -) = L_C$
::$(2)\quad$ For each object $D \in \mathbf D : F(-, D) = M_D$
{{:Characterization of Existence of Bifunctor Induced By One-Variable Functors/Necessary C... | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
For each [[Definition:Object (Category Theory)|object]] $C \in \mathbf C$, let:
:$L_C: \mathbf D \to \mathbf E$ be a [[Definition:Covariant Functor|covariant functor]].
For each [[Definition:Object (Category Theory)|object]] $D \in \... | === [[Characterization of Existence of Bifunctor Induced By One-Variable Functors/Necessary Condition|Necessary Condition]] ===
Let $F: \mathbf C \times \mathbf D \to \mathbf E$ be a [[Definition:Bifunctor|bifunctor]] such that:
::$(1)\quad$ For each [[Definition:Object (Category Theory)|object]] $C \in \mathbf C : F(... | Characterization of Existence of Bifunctor Induced By One-Variable Functors | https://proofwiki.org/wiki/Characterization_of_Existence_of_Bifunctor_Induced_By_One-Variable_Functors | https://proofwiki.org/wiki/Characterization_of_Existence_of_Bifunctor_Induced_By_One-Variable_Functors | [
"Characterization of Existence of Bifunctor Induced By One-Variable Functors",
"Bifunctors"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definiti... | [
"Characterization of Existence of Bifunctor Induced By One-Variable Functors/Necessary Condition",
"Definition:Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)"
] |
proofwiki-23131 | Characterization of Existence of Bifunctor Induced By One-Variable Functors/Necessary Condition | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be categories.
For each object $C \in \mathbf C$, let:
:$L_C: \mathbf D \to \mathbf E$ be a covariant functor.
For each object $D \in \mathbf D$, let:
:$M_D: \mathbf C \to \mathbf E$ be a covariant functor.
Let $F: \mathbf C \times \mathbf D \to \mathbf E$ be a bifunctor su... | By definition of substitution of first variable:
:for each object $C \in \mathbf C$:
::for each object $D \in \mathbf D : \map {L_C} D = \map F {C, D}$
::for each mophism $g \in \mathbf D : \map {L_C} g = \map F {\operatorname{id}_C, g}$
By definition of substitution of second variable with $D$:
:for each object $D \in... | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
For each [[Definition:Object (Category Theory)|object]] $C \in \mathbf C$, let:
:$L_C: \mathbf D \to \mathbf E$ be a [[Definition:Covariant Functor|covariant functor]].
For each [[Definition:Object (Category Theory)|object]] $D \in \... | By definition of [[Definition:Bifunctor/Substitution of First Variable|substitution of first variable]]:
:for each [[Definition:Object (Category Theory)|object]] $C \in \mathbf C$:
::for each [[Definition:Object (Category Theory)|object]] $D \in \mathbf D : \map {L_C} D = \map F {C, D}$
::for each [[Definition:Morphism... | Characterization of Existence of Bifunctor Induced By One-Variable Functors/Necessary Condition | https://proofwiki.org/wiki/Characterization_of_Existence_of_Bifunctor_Induced_By_One-Variable_Functors/Necessary_Condition | https://proofwiki.org/wiki/Characterization_of_Existence_of_Bifunctor_Induced_By_One-Variable_Functors/Necessary_Condition | [
"Characterization of Existence of Bifunctor Induced By One-Variable Functors"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Functor",
"Definiti... | [
"Definition:Bifunctor/Substitution of First Variable",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Bifunctor/Substitution of Second Variable",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definition:... |
proofwiki-23132 | Characterization of Existence of Bifunctor Induced By One-Variable Functors/Sufficient Condition | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be categories.
For each object $C \in \mathbf C$, let:
:$L_C: \mathbf D \to \mathbf E$ be a covariant functor.
For each object $D \in \mathbf D$, let:
:$M_D: \mathbf C \to \mathbf E$ be a covariant functor.
Let:
:for morphisms $f: C \to C' \in \mathbf C$ and $g: D \to D' \i... | By hypothesis, for identity morphisms $\operatorname{id}_C : C \to C \in \mathbf C$ and $\operatorname{id}_D : D \to D \in \mathbf D$:
:$\map {M_D} {\operatorname{id}_C}$ and $\map {L_C} {\operatorname{id}_D}$ are composable with:
::$\map {M_D} {\operatorname{id}_C} \circ \map {L_C} {\operatorname{id}_D} = \map {L_C} ... | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
For each [[Definition:Object (Category Theory)|object]] $C \in \mathbf C$, let:
:$L_C: \mathbf D \to \mathbf E$ be a [[Definition:Covariant Functor|covariant functor]].
For each [[Definition:Object (Category Theory)|object]] $D \in \... | By hypothesis, for [[Definition:Identity Morphism|identity morphisms]] $\operatorname{id}_C : C \to C \in \mathbf C$ and $\operatorname{id}_D : D \to D \in \mathbf D$:
:$\map {M_D} {\operatorname{id}_C}$ and $\map {L_C} {\operatorname{id}_D}$ are [[Definition:Composite Morphism|composable]] with:
::$\map {M_D} {\opera... | Characterization of Existence of Bifunctor Induced By One-Variable Functors/Sufficient Condition | https://proofwiki.org/wiki/Characterization_of_Existence_of_Bifunctor_Induced_By_One-Variable_Functors/Sufficient_Condition | https://proofwiki.org/wiki/Characterization_of_Existence_of_Bifunctor_Induced_By_One-Variable_Functors/Sufficient_Condition | [
"Characterization of Existence of Bifunctor Induced By One-Variable Functors"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Morphism",
"Definition:Bifunctor",
"Definition:Object (Category Theory)",
"Definition:Object (Category Theory)",
"Definit... | [
"Definition:Identity Morphism",
"Definition:Composition of Morphisms",
"Definition:Object (Category Theory)",
"Definition:Object Functor",
"Definition:Object (Category Theory)",
"Definition:Morphism Functor",
"Definition:Morphism",
"Definition:Well-Defined",
"Definition:Object (Category Theory)",
... |
proofwiki-23133 | Shell Theorem | Let $P$ be a charged particle in space.
Let $P$ generate a field whose vector quantity $\mathbf F$ at a position $\mathbf r$ {{WRT}} $P$ is given by the inverse square law:
:$\mathbf F = \dfrac {k Q} {r^2} \hat {\mathbf r}$
where:
:$\mathbf F$ is the vector field quantity
:$r := \size {\mathbf r}$ denotes the magnitude... | thumb600pxalt=The diagram shows a [[Definition:Sphere (Geometry)spherical shell centered at $C$ and with radius $R$ with a charge $Q$ distributed uniformly over its surface. The field from the charged shell will be measured at position vector $\mathbf r$ at a distance $r$ from $C$. A thin strip on the surface of the sp... | Let $P$ be a [[Definition:Charge (Physics)|charged]] [[Definition:Particle|particle]] in [[Definition:Ordinary Space|space]].
Let $P$ generate a [[Definition:Field (Physics)|field]] whose [[Definition:Vector Quantity|vector quantity]] $\mathbf F$ at a [[Definition:Position Vector|position]] $\mathbf r$ {{WRT}} $P$ is ... | [[File:ShellTheorem.png|thumb|600px|alt=The diagram shows a [[Definition:Sphere (Geometry)|spherical]] [[Definition:Lamina|shell]] centered at $C$ and with [[Definition:Radius of Sphere|radius]] $R$ with a [[Definition:Charge (Physics)|charge]] $Q$ distributed uniformly over its [[Definition:Surface of Sphere|surface]]... | Shell Theorem | https://proofwiki.org/wiki/Shell_Theorem | https://proofwiki.org/wiki/Shell_Theorem | [
"Physics",
"Fields (Physics)",
"Vector Fields",
"Inverse Square Law",
"Charges (Physics)",
"Gravity",
"Electrostatics",
"Vector Calculus"
] | [
"Definition:Charge (Physics)",
"Definition:Particle",
"Definition:Ordinary Space",
"Definition:Field (Physics)",
"Definition:Vector Quantity",
"Definition:Position Vector",
"Definition:Inverse Square Law",
"Definition:Vector",
"Definition:Magnitude",
"Definition:Distance between Points",
"Defini... | [
"File:ShellTheorem.png",
"Definition:Sphere/Geometry",
"Definition:Lamina",
"Definition:Sphere/Geometry/Radius",
"Definition:Charge (Physics)",
"Definition:Sphere/Geometry",
"Definition:Field (Physics)",
"Definition:Charge (Physics)",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Radiu... |
proofwiki-23134 | Euler Characteristic of Sphere | Let $S$ be a surface which is homeomorphic to the sphere.
Then the Euler characteristic of $S$ is given by:
:$\map \chi S = 2$ | From the Euler Polyhedron Formula:
:$V - E + F = 2$
where:
:$V$ is the number of vertices
:$E$ is the number of edges
:$F$ is the number of faces
of a convex polyhedron.
That is, the Euler characteristic of a convex polyhedron is $2$.
A convex polyhedron is topologically equivalent to the sphere.
From Euler Characteri... | Let $S$ be a [[Definition:Surface|surface]] which is [[Definition:Homeomorphism|homeomorphic]] to the [[Definition:Sphere (Geometry)|sphere]].
Then the [[Definition:Euler Characteristic of Surface|Euler characteristic]] of $S$ is given by:
:$\map \chi S = 2$ | From the [[Euler Polyhedron Formula]]:
:$V - E + F = 2$
where:
:$V$ is the number of [[Definition:Vertex of Polyhedron|vertices]]
:$E$ is the number of [[Definition:Edge of Polyhedron|edges]]
:$F$ is the number of [[Definition:Face of Polyhedron|faces]]
of a [[Definition:Convex Polyhedron|convex polyhedron]].
That is... | Euler Characteristic of Sphere | https://proofwiki.org/wiki/Euler_Characteristic_of_Sphere | https://proofwiki.org/wiki/Euler_Characteristic_of_Sphere | [
"Euler Characteristic",
"Spheres"
] | [
"Definition:Surface",
"Definition:Homeomorphism",
"Definition:Sphere/Geometry",
"Definition:Euler Characteristic of Surface"
] | [
"Euler Polyhedron Formula",
"Definition:Polyhedron/Vertex",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Face",
"Definition:Convex Polyhedron",
"Definition:Euler Characteristic of Surface",
"Definition:Convex Polyhedron",
"Definition:Convex Polyhedron",
"Definition:Topological Equivalence",
... |
proofwiki-23135 | Euler Number for Odd Index is Zero | Let $E_n$ be the $n$th Euler number.
Let $n$ be odd.
Then $E_n = 0$. | {{Recall|Euler Numbers}}
{{:Definition:Euler Numbers}}
From Secant Function is Even:
{{begin-eqn}}
{{eqn | l = \sech x
| r = \map \sech {-x}
| c = Secant Function is Even
}}
{{eqn | ll= \leadsto
| l = \sum_{n \mathop = 0}^\infty \frac {E_n x^n} {n!}
| r = \sum_{n \mathop = 0}^\infty \frac {E_n \... | Let $E_n$ be the $n$th [[Definition:Euler Numbers|Euler number]].
Let $n$ be [[Definition:Odd Integer|odd]].
Then $E_n = 0$. | {{Recall|Euler Numbers}}
{{:Definition:Euler Numbers}}
From [[Secant Function is Even]]:
{{begin-eqn}}
{{eqn | l = \sech x
| r = \map \sech {-x}
| c = [[Secant Function is Even]]
}}
{{eqn | ll= \leadsto
| l = \sum_{n \mathop = 0}^\infty \frac {E_n x^n} {n!}
| r = \sum_{n \mathop = 0}^\infty \... | Euler Number for Odd Index is Zero | https://proofwiki.org/wiki/Euler_Number_for_Odd_Index_is_Zero | https://proofwiki.org/wiki/Euler_Number_for_Odd_Index_is_Zero | [
"Euler Numbers"
] | [
"Definition:Euler Numbers",
"Definition:Odd Integer"
] | [
"Secant Function is Even",
"Secant Function is Even",
"Definition:Odd Integer"
] |
proofwiki-23136 | Probability of Occurrence of Complementary Event | Let $\Pr$ be a probability measure on an event space $\Sigma$.
Let $A \in \Sigma$.
The probability of the occurrence of the complementary event to $A$ can be evaluated as:
:$\map \Pr {\overline A} = 1 - \map \Pr A$
where $\overline A$ denotes the complementary event to $A$. | From Union of Event with Complement is Certainty:
:$\map \Pr {A \cup \overline A} = 1$
From Intersection of Event with Complement Can't Happen:
:$\map \Pr {A \cap \overline A} = 0$
Then:
{{begin-eqn}}
{{eqn | l = \map \Pr A + \map \Pr {\overline A}
| r = \map \Pr {A \cup \overline A}
| c = Probability of Un... | Let $\Pr$ be a [[Definition:Probability Measure|probability measure]] on an [[Definition:Event Space|event space]] $\Sigma$.
Let $A \in \Sigma$.
The [[Definition:Probability|probability]] of the [[Definition:Occurrence of Event|occurrence]] of the [[Definition:Complementary Event|complementary event]] to $A$ can be ... | From [[Union of Event with Complement is Certainty]]:
:$\map \Pr {A \cup \overline A} = 1$
From [[Intersection of Event with Complement Can't Happen]]:
:$\map \Pr {A \cap \overline A} = 0$
Then:
{{begin-eqn}}
{{eqn | l = \map \Pr A + \map \Pr {\overline A}
| r = \map \Pr {A \cup \overline A}
| c = [[Pro... | Probability of Occurrence of Complementary Event | https://proofwiki.org/wiki/Probability_of_Occurrence_of_Complementary_Event | https://proofwiki.org/wiki/Probability_of_Occurrence_of_Complementary_Event | [
"Complementary Events",
"Occurrences of Events"
] | [
"Definition:Probability Measure",
"Definition:Event Space",
"Definition:Probability",
"Definition:Event/Occurrence",
"Definition:Complementary Event",
"Definition:Complementary Event"
] | [
"Union of Event with Complement is Certainty",
"Intersection of Event with Complement Can't Happen",
"Probability of Union of Disjoint Events is Sum of Individual Probabilities"
] |
proofwiki-23137 | Condition for Differential to be Exact | The expression:
:$\ds \sum_i P_i \rd x_i$
where
:$\set {P_i}$ are real-valued functions over $\set {x_1, x_2, \ldots}$
is an '''exact differential''' {{iff}}:
:$\forall i, j : \dfrac {\partial P_i} {\partial x_j} = \dfrac {\partial P_j} {\partial x_i}$ | === Sufficient Case ===
Let:
:$\ds \sum_i P_i \rd x_i$
be an exact differential.
By definition, this is the total differential of some real-valued function $f$.
Thus:
:$\forall i : P_i = \dfrac {\partial f} {\partial x_i}$
Therefore:
{{begin-eqn}}
{{eqn | l = \frac {\partial P_i} {\partial x_j}
| r = \frac \parti... | The [[Definition:Expression|expression]]:
:$\ds \sum_i P_i \rd x_i$
where
:$\set {P_i}$ are [[Definition:Real-Valued Function|real-valued functions]] over $\set {x_1, x_2, \ldots}$
is an '''[[Definition:Exact Differential|exact differential]]''' {{iff}}:
:$\forall i, j : \dfrac {\partial P_i} {\partial x_j} = \dfrac {\... | === Sufficient Case ===
Let:
:$\ds \sum_i P_i \rd x_i$
be an [[Definition:Exact Differential|exact differential]].
By definition, this is the [[Definition:Total Differential|total differential]] of some [[Definition:Real-Valued Function|real-valued function]] $f$.
Thus:
:$\forall i : P_i = \dfrac {\partial f} {\part... | Condition for Differential to be Exact | https://proofwiki.org/wiki/Condition_for_Differential_to_be_Exact | https://proofwiki.org/wiki/Condition_for_Differential_to_be_Exact | [
"Exact Differentials"
] | [
"Definition:Expression",
"Definition:Real-Valued Function",
"Definition:Exact Differential"
] | [
"Definition:Exact Differential",
"Definition:Differential of Mapping/Real-Valued Function",
"Definition:Real-Valued Function",
"Clairaut's Theorem",
"Definition:Exact Differential",
"Definition:Exact Differential",
"Definition:Exact Differential",
"Definition:Exact Differential"
] |
proofwiki-23138 | Constant Functor is Covariant Functor | Let $\mathbf C, \mathbf D$ be a categories.
Let $D \in \mathbf D$ be an object of $\mathbf D$.
Let $F: \mathbf C \to \mathbf D$ denote the $D$-valued constant functor.
Then:
:$F$ is a covariant functor. | === Constant Functor Preserves Composition ===
Let $f_1: C_1 \to C_2, f_2: C_2 \to C_3$ be morphisms in $\mathbf C$.
We have:
{{begin-eqn}}
{{eqn | l = \map F {f_2 \circ f_1}
| r = \operatorname{id}_D
| c = {{Defof|Constant Functor}}
}}
{{eqn | r = \operatorname{id}_D \circ \operatorname{id}_D
| c = I... | Let $\mathbf C, \mathbf D$ be a [[Definition:Category|categories]].
Let $D \in \mathbf D$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf D$.
Let $F: \mathbf C \to \mathbf D$ denote the [[Definition:Constant Functor|$D$-valued constant functor]].
Then:
:$F$ is a [[Definition:Covariant Functor|cova... | === Constant Functor Preserves Composition ===
Let $f_1: C_1 \to C_2, f_2: C_2 \to C_3$ be [[Definition:Morphism (Category Theory)|morphisms]] in $\mathbf C$.
We have:
{{begin-eqn}}
{{eqn | l = \map F {f_2 \circ f_1}
| r = \operatorname{id}_D
| c = {{Defof|Constant Functor}}
}}
{{eqn | r = \operatorname{i... | Constant Functor is Covariant Functor | https://proofwiki.org/wiki/Constant_Functor_is_Covariant_Functor | https://proofwiki.org/wiki/Constant_Functor_is_Covariant_Functor | [
"Functors"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Constant Functor",
"Definition:Functor/Covariant"
] | [
"Definition:Morphism",
"Identity Morphism is Idempotent",
"Definition:Composition of Morphisms"
] |
proofwiki-23139 | Constant Natural Transformation is Natural Transformation | Let $\mathbf C, \mathbf D$ be a categories.
Let $f : D_1 \to D_2 \in \mathbf D$ be a morphism of $\mathbf D$.
Let $F_{D_1}: \mathbf C \to \mathbf D, F_{D_2}: \mathbf C \to \mathbf D$ denote the $D_1$, $D_2$-valued constant functors respectively.
Let $\eta: F_{D_1} \to F_{D_2}$ denote the $f$-valued constant natural tra... | Let $g: X \to Y \in \mathbf C$ be a morphism.
We have:
{{begin-eqn}}
{{eqn | l = f
| r = f \circ \operatorname{id}_{D_1}
| c = {{Defof|Identity Morphism}}
}}
{{eqn | r = \eta_Y \circ \operatorname{id}_{D_1}
| c = {{Defof|Constant Natural Transformation}}
}}
{{eqn | n = 1
| r = \eta_Y \circ F_{D_... | Let $\mathbf C, \mathbf D$ be a [[Definition:Category|categories]].
Let $f : D_1 \to D_2 \in \mathbf D$ be a [[Definition:Morphism (Category Theory)|morphism]] of $\mathbf D$.
Let $F_{D_1}: \mathbf C \to \mathbf D, F_{D_2}: \mathbf C \to \mathbf D$ denote the [[Definition:Constant Functor|$D_1$, $D_2$-valued constan... | Let $g: X \to Y \in \mathbf C$ be a [[Definition:Morphism (Category Theory)|morphism]].
We have:
{{begin-eqn}}
{{eqn | l = f
| r = f \circ \operatorname{id}_{D_1}
| c = {{Defof|Identity Morphism}}
}}
{{eqn | r = \eta_Y \circ \operatorname{id}_{D_1}
| c = {{Defof|Constant Natural Transformation}}
}}
... | Constant Natural Transformation is Natural Transformation | https://proofwiki.org/wiki/Constant_Natural_Transformation_is_Natural_Transformation | https://proofwiki.org/wiki/Constant_Natural_Transformation_is_Natural_Transformation | [
"Natural Transformations"
] | [
"Definition:Category",
"Definition:Morphism",
"Definition:Constant Functor",
"Definition:Constant Natural Transformation",
"Definition:Natural Transformation"
] | [
"Definition:Morphism",
"Definition:Natural Transformation"
] |
proofwiki-23140 | Excenter of Triangle is Point of Intersection of Bisectors of Angles | Let $T$ be a triangle.
Let $X$ be an excenter of $T$.
Then $X$ is the intersection of:
:the angle bisector of the interior angle at one of the vertices of $T$
:the angle bisectors of the exterior angles at the other two vertices of $T$. | :490px
Let $T = \triangle ABC$ be a triangle.
Let an excircle $\bigcirc DEF$ be constructed:
:tangent to $BC$ at $F$
:tangent to the production of $AC$ at $D$
:tangent to the production of $AB$ at $E$.
Let $X$ denote the center of $\bigcirc DEF$.
Let $r$ denote the radius of $\bigcirc DEF$.
By definition, $X$ is an exc... | Let $T$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $X$ be an [[Definition:Excenter of Triangle|excenter]] of $T$.
Then $X$ is the [[Definition:Intersection (Geometry)|intersection]] of:
:the [[Definition:Angle Bisector|angle bisector]] of the [[Definition:Interior Angle of Polygon|interior angle]] at one ... | :[[File:Excenter-of-Triangle-as-Intersection-of-Angles.png|490px]]
Let $T = \triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let an [[Definition:Excircle of Triangle|excircle]] $\bigcirc DEF$ be constructed:
:[[Definition:Tangent to Circle|tangent]] to $BC$ at $F$
:[[Definition:Tangent to Circle|tang... | Excenter of Triangle is Point of Intersection of Bisectors of Angles | https://proofwiki.org/wiki/Excenter_of_Triangle_is_Point_of_Intersection_of_Bisectors_of_Angles | https://proofwiki.org/wiki/Excenter_of_Triangle_is_Point_of_Intersection_of_Bisectors_of_Angles | [
"Excenters of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Excircle of Triangle/Excenter",
"Definition:Intersection (Geometry)",
"Definition:Angle Bisector",
"Definition:Polygon/Internal Angle",
"Definition:Polygon/Vertex",
"Definition:Angle Bisector",
"Definition:Polygon/External Angle",
"Definition:Polygon/Ver... | [
"File:Excenter-of-Triangle-as-Intersection-of-Angles.png",
"Definition:Triangle (Geometry)",
"Definition:Excircle of Triangle",
"Definition:Tangent Line/Circle",
"Definition:Tangent Line/Circle",
"Definition:Production",
"Definition:Tangent Line/Circle",
"Definition:Production",
"Definition:Circle/C... |
proofwiki-23141 | Field from Barycentric Charge Distribution | Let $\map \rho r$ be a barycentric charge density within space and centered at $\CC$ where $r$ is the distance from $\CC$.
Let the total amount of charge be finite.
:$\ds \int \map \rho {\mathbf r'} \rd V < \infty$
where $\mathbf r'$ ranges over the whole of space.
Let the field generated by point charge be given by th... | In general, the field generated by a charge density distribution at a position $\mathbf r$ is found by integrating the field generated at $\mathbf r$ by every charged volume element:
:$\ds \map {\mathbf F} {\mathbf r} = \int \rd \map {\mathbf F} {\mathbf r'}$
where $\mathbf r'$ is the position of a charged volume eleme... | Let $\map \rho r$ be a [[Definition:Barycentric Body|barycentric]] [[Definition:Charge (Physics)|charge]] [[Definition:Density (Physics)|density]] within [[Definition:Ordinary Space|space]] and centered at $\CC$ where $r$ is the [[Definition:Distance Between Points|distance]] from $\CC$.
Let the total amount of [[Defi... | In general, the [[Definition:Field (Physics)|field]] generated by a [[Definition:Charge (Physics)|charge]] [[Definition:Density (Physics)|density]] distribution at a [[Definition:Position Vector|position]] $\mathbf r$ is found by [[Definition:Integration|integrating]] the [[Definition:Field (Physics)|field]] generated ... | Field from Barycentric Charge Distribution | https://proofwiki.org/wiki/Field_from_Barycentric_Charge_Distribution | https://proofwiki.org/wiki/Field_from_Barycentric_Charge_Distribution | [
"Physics",
"Fields (Physics)",
"Vector Fields",
"Inverse Square Law",
"Charges (Physics)",
"Gravity",
"Electrostatics",
"Vector Calculus"
] | [
"Definition:Barycentric Body",
"Definition:Charge (Physics)",
"Definition:Density (Physics)",
"Definition:Ordinary Space",
"Definition:Distance/Points",
"Definition:Charge (Physics)",
"Definition:Ordinary Space",
"Definition:Field (Physics)",
"Definition:Point",
"Definition:Charge (Physics)",
"D... | [
"Definition:Field (Physics)",
"Definition:Charge (Physics)",
"Definition:Density (Physics)",
"Definition:Position Vector",
"Definition:Primitive (Calculus)/Integration",
"Definition:Field (Physics)",
"Definition:Charge (Physics)",
"Definition:Volume Element",
"Definition:Position Vector",
"Definit... |
proofwiki-23142 | Section Formula | Let $A$, $B$ and $P$ be collinear points.
Let $P$ divide $AB$ externally in the ratio $1 : k$.
Let the position vectors of $A$, $B$ and $P$ be $\mathbf a$, $\mathbf b$ and $\mathbf p$ respectively.
Then:
:$\mathbf p = \dfrac {k \mathbf a - \mathbf b} {k - 1}$ | {{ProofWanted|We surely have something similar for the internal division in ratio situation, but it wasn't called this. This and that need to be combined and rationalised.}} | Let $A$, $B$ and $P$ be [[Definition:Collinear Points|collinear points]].
Let $P$ [[Definition:External Division in Ratio|divide $AB$ externally in the ratio]] $1 : k$.
Let the [[Definition:Position Vector|position vectors]] of $A$, $B$ and $P$ be $\mathbf a$, $\mathbf b$ and $\mathbf p$ respectively.
Then:
:$\math... | {{ProofWanted|We surely have something similar for the internal division in ratio situation, but it wasn't called this. This and that need to be combined and rationalised.}} | Section Formula | https://proofwiki.org/wiki/Section_Formula | https://proofwiki.org/wiki/Section_Formula | [
"Division in Ratio",
"Named Theorems"
] | [
"Definition:Collinear/Points",
"Definition:Division in Ratio/Straight Line/External",
"Definition:Position Vector"
] | [] |
proofwiki-23143 | Characterization of Adjunction Using Right Adjuncts of Triple Compositions | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\mathbf D \times \mathbf C$ denote the product category of $\mathbf D$ with $\mathbf C$.
For each object $\tuple{D, C} \in ... | By definition of adjunction:
:the triple $\tuple {F, G, \alpha}$ is an adjunction
{{iff}}:
:$(1)$ for all $f: D_2 \to D_1 \in \mathbf D$ and $g: C_1 \to C_2 \in \mathbf C$:
::$\alpha_{\tuple{D_2,C_2}} \circ \map {\operatorname{Hom}_{\mathbf C}} {Ff, g} = \map {\operatorname{Hom}_{\mathbf D}} {f, Gg} \circ \alpha_{\tupl... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\mathbf D \time... | By definition of [[Definition:Adjunction|adjunction]]:
:the [[Definition:Triple|triple]] $\tuple {F, G, \alpha}$ is an [[Definition:Adjunction|adjunction]]
{{iff}}:
:$(1)$ for all $f: D_2 \to D_1 \in \mathbf D$ and $g: C_1 \to C_2 \in \mathbf C$:
::$\alpha_{\tuple{D_2,C_2}} \circ \map {\operatorname{Hom}_{\mathbf C}} {... | Characterization of Adjunction Using Right Adjuncts of Triple Compositions | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Right_Adjuncts_of_Triple_Compositions | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Right_Adjuncts_of_Triple_Compositions | [
"Characterizations of Adjunctions",
"Adjunctions"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Product Category",
"Definition:Object (Category Theory)",
"Definition:Bijection",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction",
"Definition:Commutative... | [
"Definition:Adjunction",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction",
"Definition:Commutative Diagram",
"Definition:Mapping",
"Definition:Mapping",
"Equality of Mappings",
"Definition:Composition of Mappings"
] |
proofwiki-23144 | Symmetric Regular Right Bipyramid is not Isotoxal Polyhedron | Let $\PP$ be a symmetric regular right bipyramid.
Then $\PP$ is ''not'' an isotoxal polyhedron.
The only counterexample is the regular octahdron. | First we note that the regular octahdron is an example of a regular polyhedron.
Hence from Regular Polyhedron is Isotoxal, the regular octahdron is {{afortiori}} an isotoxal polyhedron.
Otherwise, we note that the bases of the pyramids forming $\PP$ are polygons with $n$ sides, where $n \ne 4$.
Hence apices of the pyra... | Let $\PP$ be a [[Definition:Symmetric Regular Right Bipyramid|symmetric regular right bipyramid]].
Then $\PP$ is ''not'' an [[Definition:Isotoxal|isotoxal]] [[Definition:Polyhedron|polyhedron]].
The only [[Definition:Counterexample|counterexample]] is the [[Definition:Regular Octahedron|regular octahdron]]. | First we note that the [[Definition:Regular Octahedron|regular octahdron]] is an example of a [[Definition:Regular Polyhedron|regular polyhedron]].
Hence from [[Regular Polyhedron is Isotoxal]], the [[Definition:Regular Octahedron|regular octahdron]] is {{afortiori}} an [[Definition:Isotoxal|isotoxal]] [[Definition:Po... | Symmetric Regular Right Bipyramid is not Isotoxal Polyhedron | https://proofwiki.org/wiki/Symmetric_Regular_Right_Bipyramid_is_not_Isotoxal_Polyhedron | https://proofwiki.org/wiki/Symmetric_Regular_Right_Bipyramid_is_not_Isotoxal_Polyhedron | [
"Bipyramids",
"Isotoxal"
] | [
"Definition:Symmetric Regular Right Bipyramid",
"Definition:Isotoxal",
"Definition:Polyhedron",
"Definition:Counterexample",
"Definition:Octahedron/Regular"
] | [
"Definition:Octahedron/Regular",
"Definition:Regular Polyhedron",
"Regular Polyhedron is Isotoxal",
"Definition:Octahedron/Regular",
"Definition:Isotoxal",
"Definition:Polyhedron",
"Definition:Pyramid/Base",
"Definition:Pyramid",
"Definition:Polygon",
"Definition:Polygon/Side",
"Definition:Pyram... |
proofwiki-23145 | Symmetric Regular Right Bipyramid is not Isogonal Polyhedron | Let $\PP$ be a symmetric regular right bipyramid.
Then $\PP$ is ''not'' in general an isogonal polyhedron.
The only counterexample is the regular octahdron. | First we note that the regular octahdron is an example of a regular polyhedron.
Hence from Regular Polyhedron is Isogonal, the regular octahdron is {{afortiori}} an isogonal polyhedron.
Otherwise, we note that the bases of the pyramids forming $\PP$ are polygons with $n$ sides, where $n \ne 4$.
Hence apices of the pyra... | Let $\PP$ be a [[Definition:Symmetric Regular Right Bipyramid|symmetric regular right bipyramid]].
Then $\PP$ is ''not'' in general an [[Definition:Isogonal Polyhedron|isogonal polyhedron]].
The only [[Definition:Counterexample|counterexample]] is the [[Definition:Regular Octahedron|regular octahdron]]. | First we note that the [[Definition:Regular Octahedron|regular octahdron]] is an example of a [[Definition:Regular Polyhedron|regular polyhedron]].
Hence from [[Regular Polyhedron is Isogonal]], the [[Definition:Regular Octahedron|regular octahdron]] is {{afortiori}} an [[Definition:Isogonal Polyhedron|isogonal polyhe... | Symmetric Regular Right Bipyramid is not Isogonal Polyhedron | https://proofwiki.org/wiki/Symmetric_Regular_Right_Bipyramid_is_not_Isogonal_Polyhedron | https://proofwiki.org/wiki/Symmetric_Regular_Right_Bipyramid_is_not_Isogonal_Polyhedron | [
"Bipyramids",
"Isogonal Polyhedra"
] | [
"Definition:Symmetric Regular Right Bipyramid",
"Definition:Isogonal Polyhedron",
"Definition:Counterexample",
"Definition:Octahedron/Regular"
] | [
"Definition:Octahedron/Regular",
"Definition:Regular Polyhedron",
"Regular Polyhedron is Isogonal",
"Definition:Octahedron/Regular",
"Definition:Isogonal Polyhedron",
"Definition:Pyramid/Base",
"Definition:Pyramid",
"Definition:Polygon",
"Definition:Polygon/Side",
"Definition:Pyramid/Apex",
"Def... |
proofwiki-23146 | Equiangular Polygon is not necessarily Isogonal | Let $\PP$ be an equiangular polygon.
Then it is not necessarily the case that $\PP$ is also isogonal. | ;Proof by Counterexample
Consider the following equiangular octagon $ABCDEFGH$:
:320px
where:
:all interior angles are $\dfrac {3 \pi} 4$ radians, that is, $135 \degrees$
:sides $AB$ and $EF$ are the same length $r$
:sides $CD$ and $GH$ are the same length $s$
:sides $BC$, $DE$, $FG$ and $HA$ are all the same length $t... | Let $\PP$ be an [[Definition:Equiangular Polygon|equiangular polygon]].
Then it is not necessarily the case that $\PP$ is also [[Definition:Isogonal Polygon|isogonal]]. | ;[[Proof by Counterexample]]
Consider the following [[Definition:Equiangular Polygon|equiangular]] [[Definition:Octagon|octagon]] $ABCDEFGH$:
:[[File:Equiangular-Octagon.png|320px]]
where:
:all [[Definition:Interior Angle of Polygon|interior angles]] are $\dfrac {3 \pi} 4$ [[Definition:Radian|radians]], that is, $13... | Equiangular Polygon is not necessarily Isogonal | https://proofwiki.org/wiki/Equiangular_Polygon_is_not_necessarily_Isogonal | https://proofwiki.org/wiki/Equiangular_Polygon_is_not_necessarily_Isogonal | [
"Equiangular Polygons",
"Isogonal Polygons"
] | [
"Definition:Polygon/Equiangular",
"Definition:Isogonal Polygon"
] | [
"Proof by Counterexample",
"Definition:Polygon/Equiangular",
"Definition:Octagon",
"File:Equiangular-Octagon.png",
"Definition:Polygon/Internal Angle",
"Definition:Angular Measure/Radian",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Linear ... |
proofwiki-23147 | General Solution to Differential Equation gives rise to Family of Curves | The general solution to a differential equation gives rise to a family of curves.
A particular solution may be identified by interpretation of boundary conditions. | {{ProofWanted|The assertion is made in the source given, but I'm not sure whether this is true in absolute general}} | The [[Definition:General Solution to Differential Equation|general solution]] to a [[Definition:Differential Equation|differential equation]] gives rise to a [[Definition:Family of Curves|family of curves]].
A [[Definition:Particular Solution to Differential Equation|particular solution]] may be identified by interpre... | {{ProofWanted|The assertion is made in the source given, but I'm not sure whether this is true in absolute general}} | General Solution to Differential Equation gives rise to Family of Curves | https://proofwiki.org/wiki/General_Solution_to_Differential_Equation_gives_rise_to_Family_of_Curves | https://proofwiki.org/wiki/General_Solution_to_Differential_Equation_gives_rise_to_Family_of_Curves | [
"Families of Curves",
"Differential Equations"
] | [
"Definition:Differential Equation/Solution/General Solution",
"Definition:Differential Equation",
"Definition:Family of Curves",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Boundary Condition"
] | [] |
proofwiki-23148 | Constant Functor is Contravariant Functor | Let $\mathbf C, \mathbf D$ be a categories.
Let $D \in \mathbf D$ be an object of $\mathbf D$.
Let $F: \mathbf C \to \mathbf D$ denote the $D$-valued constant functor.
Then:
:$F$ is a contravariant functor. | === Constant Functor Reverses Composition ===
Let $f_1: C_1 \to C_2, f_2: C_2 \to C_3$ be morphisms in $\mathbf C$.
We have:
{{begin-eqn}}
{{eqn | l = \map F {f_2 \circ f_1}
| r = \operatorname{id}_D
| c = {{Defof|Constant Functor}}
}}
{{eqn | r = \operatorname{id}_D \circ \operatorname{id}_D
| c = Id... | Let $\mathbf C, \mathbf D$ be a [[Definition:Category|categories]].
Let $D \in \mathbf D$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf D$.
Let $F: \mathbf C \to \mathbf D$ denote the [[Definition:Constant Functor|$D$-valued constant functor]].
Then:
:$F$ is a [[Definition:Contravariant Functor|... | === Constant Functor Reverses Composition ===
Let $f_1: C_1 \to C_2, f_2: C_2 \to C_3$ be [[Definition:Morphism (Category Theory)|morphisms]] in $\mathbf C$.
We have:
{{begin-eqn}}
{{eqn | l = \map F {f_2 \circ f_1}
| r = \operatorname{id}_D
| c = {{Defof|Constant Functor}}
}}
{{eqn | r = \operatorname{id... | Constant Functor is Contravariant Functor | https://proofwiki.org/wiki/Constant_Functor_is_Contravariant_Functor | https://proofwiki.org/wiki/Constant_Functor_is_Contravariant_Functor | [
"Functors"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Constant Functor",
"Definition:Functor/Contravariant"
] | [
"Definition:Morphism",
"Identity Morphism is Idempotent",
"Definition:Composition of Morphisms"
] |
proofwiki-23149 | Diagonal Functor on Product Category is Covariant Functor | Let $\mathbf C$ be a category.
Let $\mathbf C \times \mathbf C$ be the product category of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the diagonal functor on product category.
Then:
:$\Delta$ is a covariant functor. | === $\Delta$ Preserves Composition ===
Let $f_1 : C_1 \to C_2, f_2:C_2 \to C_3 \in \mathbf C$ be morphisms.
We have:
{{begin-eqn}}
{{eqn | l = \map \Delta {f_2 \circ f_1}
| r = \tuple{f_2 \circ f_1, f_2 \circ f_1}
| c = {{Defof|Diagonal Functor on Product Category}}
}}
{{eqn | r = \tuple{f_2, f_2} \circ \tu... | Let $\mathbf C$ be a [[Definition:Category|category]].
Let $\mathbf C \times \mathbf C$ be the [[Definition:Product Category|product category]] of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the [[Definition:Diagonal Functor on Product Category|diagonal functor on product c... | === $\Delta$ Preserves Composition ===
Let $f_1 : C_1 \to C_2, f_2:C_2 \to C_3 \in \mathbf C$ be [[Definition:Morphism (Category Theory)|morphisms]].
We have:
{{begin-eqn}}
{{eqn | l = \map \Delta {f_2 \circ f_1}
| r = \tuple{f_2 \circ f_1, f_2 \circ f_1}
| c = {{Defof|Diagonal Functor on Product Categor... | Diagonal Functor on Product Category is Covariant Functor | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_is_Covariant_Functor | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_is_Covariant_Functor | [
"Diagonal Functors"
] | [
"Definition:Category",
"Definition:Product Category",
"Definition:Diagonal Functor/Product Category",
"Definition:Functor/Covariant"
] | [
"Definition:Morphism",
"Definition:Composition of Morphisms"
] |
proofwiki-23150 | Diagonal Functor is Covariant Functor | Let $\mathbf C$ be a category.
Let $\mathbf J$ be a small index category.
Let $\mathbf {C^J}$ denote the functor category $\mathbf J$ into $\mathbf C$.
Let $\Delta_{\mathbf J}: \mathbf C \to \mathbf {C^J}$ denote the diagonal functor on index category.
Then:
:$\Delta_{\mathbf J}$ is a covariant functor. | === $\Delta_{\mathbf J}$ Preserves Composition ===
Let $f_1 : C_1 \to C_2, f_2:C_2 \to C_3 \in \mathbf C$ be morphisms.
We have:
{{begin-eqn}}
{{eqn | q = \forall i \in \mathbf J
| l = \map {\Delta_{\mathbf J} } {f_2 \circ f_1}_i
| r = f_2 \circ f_1
| c = {{Defof|Diagonal Functor on Index Category}}
}... | Let $\mathbf C$ be a [[Definition:Category|category]].
Let $\mathbf J$ be a [[Definition:Small Category|small]] [[Definition:Index Category|index category]].
Let $\mathbf {C^J}$ denote the [[Definition:Functor Category|functor category]] $\mathbf J$ into $\mathbf C$.
Let $\Delta_{\mathbf J}: \mathbf C \to \mathbf {... | === $\Delta_{\mathbf J}$ Preserves Composition ===
Let $f_1 : C_1 \to C_2, f_2:C_2 \to C_3 \in \mathbf C$ be [[Definition:Morphism (Category Theory)|morphisms]].
We have:
{{begin-eqn}}
{{eqn | q = \forall i \in \mathbf J
| l = \map {\Delta_{\mathbf J} } {f_2 \circ f_1}_i
| r = f_2 \circ f_1
| c = {... | Diagonal Functor is Covariant Functor | https://proofwiki.org/wiki/Diagonal_Functor_is_Covariant_Functor | https://proofwiki.org/wiki/Diagonal_Functor_is_Covariant_Functor | [
"Diagonal Functors"
] | [
"Definition:Category",
"Definition:Small Category",
"Definition:Diagram (Category Theory)/Index Category",
"Definition:Functor Category",
"Definition:Diagonal Functor/Index Category",
"Definition:Functor/Covariant"
] | [
"Definition:Morphism",
"Definition:Composition of Morphisms"
] |
proofwiki-23151 | Characterization of Adjunction Using Left Adjuncts of Triple Compositions | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\mathbf D \times \mathbf C$ denote the product category of $\mathbf D$ with $\mathbf C$.
For each object $\tuple{D, C} \in ... | From Characterization of Adjunction Using Right Adjuncts of Triple Compositions:
:the triple $\tuple {F, G, \alpha}$ is an adjunction
{{iff}}:
:$(2):\quad$for every:
::$f:D_2 \to D_1 \in \mathbf D$
::$h:F D_1 \to C_1 \in \mathbf D$
::$g:C_1 \to C_2 \in \mathbf C$
:we have:
::$\map {\alpha_{\tuple{D_2, C_2} } } {g \circ... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\mathbf D \time... | From [[Characterization of Adjunction Using Right Adjuncts of Triple Compositions]]:
:the [[Definition:Triple|triple]] $\tuple {F, G, \alpha}$ is an [[Definition:Adjunction|adjunction]]
{{iff}}:
:$(2):\quad$for every:
::$f:D_2 \to D_1 \in \mathbf D$
::$h:F D_1 \to C_1 \in \mathbf D$
::$g:C_1 \to C_2 \in \mathbf C$
:we ... | Characterization of Adjunction Using Left Adjuncts of Triple Compositions | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Left_Adjuncts_of_Triple_Compositions | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Left_Adjuncts_of_Triple_Compositions | [
"Characterizations of Adjunctions",
"Adjunctions"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Product Category",
"Definition:Object (Category Theory)",
"Definition:Bijection",
"Definition:Object (Category Theory)",
"Definition:Inverse Mapping",
"Definition:Ordered Tuple as Ordered ... | [
"Characterization of Adjunction Using Right Adjuncts of Triple Compositions",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction",
"Characterization of Adjunction Using Right Adjuncts of Triple Compositions"
] |
proofwiki-23152 | Product of Category of Sets | Let $\mathbf{Sets}$ be the category of sets.
Let $I$ be an indexing set.
Let $\family {S_i}_{i\in I}$ be a family of sets.
Let $\ds S := \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i\in I}$.
Let $\sequence {\pr_i: S \to S_i}_{i\in I}$ be the family of the projections.
Then $S$ together wit... | {{ProofWanted}}
Category:Category of Sets
hd2hvj8b3wm1sp7uk698dbborhmwtdm | Let $\mathbf{Sets}$ be the [[Definition:Category of Sets|category of sets]].
Let $I$ be an [[Definition:Indexing Set|indexing set]].
Let $\family {S_i}_{i\in I}$ be a [[Definition:Indexed Family of Sets|family of sets]].
Let $\ds S := \prod_{i \mathop \in I} S_i$ be the [[Definition:Cartesian Product of Family|Carte... | {{ProofWanted}}
[[Category:Category of Sets]]
hd2hvj8b3wm1sp7uk698dbborhmwtdm | Product of Category of Sets | https://proofwiki.org/wiki/Product_of_Category_of_Sets | https://proofwiki.org/wiki/Product_of_Category_of_Sets | [
"Category of Sets"
] | [
"Definition:Category of Sets",
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Cartesian Product/Family of Sets",
"Definition:Indexing Set/Family",
"Definition:Projection (Mapping Theory)/Family of Sets",
"Definition:Product (Category Theory)/General Definition"
] | [
"Category:Category of Sets"
] |
proofwiki-23153 | Complete Countable Metric Space has Isolated Point | Let $\struct {X, d}$ be a countable complete metric space.
Then $X$ has an isolated point. | Write:
:$\ds X = \bigcup_{x \mathop \in X} \set x$
This is a countable union.
From Finite Subset of Metric Space is Closed, $\set x$ is closed in $X$ for each $x \in X$.
By the Baire Category Theorem, $\struct {X, d}$ is a Baire space.
By Baire Space is Non-Meager, $\struct {X, d}$ is not meager.
Hence for some $x_0 \i... | Let $\struct {X, d}$ be a [[Definition:Countable Set|countable]] [[Definition:Complete Metric Space|complete metric space]].
Then $X$ has an [[Definition:Isolated Point (Metric Space)|isolated point]]. | Write:
:$\ds X = \bigcup_{x \mathop \in X} \set x$
This is a [[Definition:Countable Union|countable union]].
From [[Finite Subset of Metric Space is Closed]], $\set x$ is [[Definition:Closed Set|closed]] in $X$ for each $x \in X$.
By the [[Baire Category Theorem]], $\struct {X, d}$ is a [[Definition:Baire Space|Bair... | Complete Countable Metric Space has Isolated Point | https://proofwiki.org/wiki/Complete_Countable_Metric_Space_has_Isolated_Point | https://proofwiki.org/wiki/Complete_Countable_Metric_Space_has_Isolated_Point | [
"Metric Spaces",
"Isolated Points"
] | [
"Definition:Countable Set",
"Definition:Complete Metric Space",
"Definition:Isolated Point (Metric Space)"
] | [
"Definition:Set Union/Countable Union",
"Finite Subset of Metric Space is Closed",
"Definition:Closed Set",
"Baire Category Theorem",
"Definition:Baire Space",
"Baire Space is Non-Meager",
"Definition:Meager Space",
"Definition:Nowhere Dense",
"Definition:Interior (Topology)",
"Definition:Open Set... |
proofwiki-23154 | Continuous Real Function Vanishing Almost Everywhere is Zero Function | Let $I$ be an open interval.
Let $\lambda$ be the Lebesgue measure on $I$.
Let $f : I \to \R$ be a continuous function be such that:
:$\map f x = 0$ for almost all $x \in I$.
Then:
:$\map f x = 0$ for all $x \in I$. | {{AimForCont}} there exists $x_0 \in I$ such that:
:$\map f {x_0} \ne 0$
so that:
:$\cmod {\map f {x_0} } > 0$
Since $f$ is continuous function at $x_0$, there exists $\delta > 0$ such that:
:for all $x \in \openint {x_0 - \delta} {x_0 + \delta} \cap I$, we have:
::$\cmod {\map f x - \map f {x_0} } < \dfrac {\cmod {\ma... | Let $I$ be an [[Definition:Open Real Interval|open interval]].
Let $\lambda$ be the [[Definition:Lebesgue Measure|Lebesgue measure]] on $I$.
Let $f : I \to \R$ be a [[Definition:Continuous Real Function|continuous function]] be such that:
:$\map f x = 0$ for [[Definition:Almost All|almost all]] $x \in I$.
Then:
:$... | {{AimForCont}} there exists $x_0 \in I$ such that:
:$\map f {x_0} \ne 0$
so that:
:$\cmod {\map f {x_0} } > 0$
Since $f$ is [[Definition:Continuous Real Function|continuous function]] at $x_0$, there exists $\delta > 0$ such that:
:for all $x \in \openint {x_0 - \delta} {x_0 + \delta} \cap I$, we have:
::$\cmod {\map ... | Continuous Real Function Vanishing Almost Everywhere is Zero Function | https://proofwiki.org/wiki/Continuous_Real_Function_Vanishing_Almost_Everywhere_is_Zero_Function | https://proofwiki.org/wiki/Continuous_Real_Function_Vanishing_Almost_Everywhere_is_Zero_Function | [
"Lebesgue Measure",
"Continuous Real Functions"
] | [
"Definition:Real Interval/Open",
"Definition:Lebesgue Measure",
"Definition:Continuous Real Function",
"Definition:Almost All"
] | [
"Definition:Continuous Real Function",
"Reverse Triangle Inequality",
"Measure of Interval is Length",
"Definition:Almost All",
"Category:Lebesgue Measure",
"Category:Continuous Real Functions"
] |
proofwiki-23155 | Integral of Modulus of Continuous Real Function is Zero iff Zero Function | Let $I$ be an open interval.
Let $\lambda$ be the Lebesgue measure on $I$.
Let $f : I \to \R$ be a continuous real function such that:
:$\ds \int_I \size f \rd \lambda = 0$
Then $f = 0$. | From Measurable Function Zero A.E. iff Absolute Value has Zero Integral, we have:
:$\map f x = 0$ for $\lambda$-almost all $x \in I$.
From Continuous Real Function Vanishing Almost Everywhere is Zero Function, we have that:
:$\map f x = 0$ for all $x \in I$.
{{qed}}
Category:Integrals of Integrable Functions
Category:C... | Let $I$ be an [[Definition:Open Interval|open interval]].
Let $\lambda$ be the [[Definition:Lebesgue Measure|Lebesgue measure]] on $I$.
Let $f : I \to \R$ be a [[Definition:Continuous Real Function|continuous real function]] such that:
:$\ds \int_I \size f \rd \lambda = 0$
Then $f = 0$. | From [[Measurable Function Zero A.E. iff Absolute Value has Zero Integral]], we have:
:$\map f x = 0$ for [[Definition:Almost All|$\lambda$-almost all]] $x \in I$.
From [[Continuous Real Function Vanishing Almost Everywhere is Zero Function]], we have that:
:$\map f x = 0$ for all $x \in I$.
{{qed}}
[[Category:Integr... | Integral of Modulus of Continuous Real Function is Zero iff Zero Function | https://proofwiki.org/wiki/Integral_of_Modulus_of_Continuous_Real_Function_is_Zero_iff_Zero_Function | https://proofwiki.org/wiki/Integral_of_Modulus_of_Continuous_Real_Function_is_Zero_iff_Zero_Function | [
"Integrals of Positive Measurable Functions",
"Integrals of Integrable Functions",
"Integrals of Integrable Functions",
"Continuous Real Functions"
] | [
"Definition:Interval/Ordered Set/Open",
"Definition:Lebesgue Measure",
"Definition:Continuous Real Function"
] | [
"Measurable Function Zero A.E. iff Absolute Value has Zero Integral",
"Definition:Almost All",
"Continuous Real Function Vanishing Almost Everywhere is Zero Function",
"Category:Integrals of Integrable Functions",
"Category:Continuous Real Functions"
] |
proofwiki-23156 | P-Sequence Space is Strictly Increasing in p | Let $\GF \in \set {\R, \C}$.
Let $p, q \in \hointr 1 \infty$ be such that $1 \le p < q$.
Let $\map {\ell_p} {\N, \GF}$ and $\map {\ell_q} {\N, \GF}$ be the $p$ and $q$-sequence spaces.
Then:
:$\map {\ell_p} {\N, \GF} \subsetneq \map {\ell_q} {\N, \GF}$ | We first show that $\map {\ell_p} {\N, \GF} \subseteq \map {\ell_q} {\N, \GF}$.
Let $\phi \in \map {\ell_p} {\N, \GF}$.
From Terms in Convergent Series Converge to Zero, we have $\map \phi n \to 0$ as $n \to \infty$.
Hence there exists $N \in \N$ such that $\cmod {\map \phi n} < 1$ for $n \ge N$.
Then we have:
:$\cmod... | Let $\GF \in \set {\R, \C}$.
Let $p, q \in \hointr 1 \infty$ be such that $1 \le p < q$.
Let $\map {\ell_p} {\N, \GF}$ and $\map {\ell_q} {\N, \GF}$ be the [[Definition:P-Sequence Space|$p$ and $q$-sequence spaces]].
Then:
:$\map {\ell_p} {\N, \GF} \subsetneq \map {\ell_q} {\N, \GF}$ | We first show that $\map {\ell_p} {\N, \GF} \subseteq \map {\ell_q} {\N, \GF}$.
Let $\phi \in \map {\ell_p} {\N, \GF}$.
From [[Terms in Convergent Series Converge to Zero]], we have $\map \phi n \to 0$ as $n \to \infty$.
Hence there exists $N \in \N$ such that $\cmod {\map \phi n} < 1$ for $n \ge N$.
Then we have:... | P-Sequence Space is Strictly Increasing in p | https://proofwiki.org/wiki/P-Sequence_Space_is_Strictly_Increasing_in_p | https://proofwiki.org/wiki/P-Sequence_Space_is_Strictly_Increasing_in_p | [
"P-Sequence Spaces"
] | [
"Definition:P-Sequence Space"
] | [
"Terms in Convergent Series Converge to Zero",
"Harmonic Series is Divergent",
"Convergence of P-Series",
"Category:P-Sequence Spaces"
] |
proofwiki-23157 | Nth Forward Difference is Constant iff Function is Polynomial of Degree N/Corollary | Let $y = \map f x$ be a real function.
Let $\Delta y$ denote the forward difference operator on $y$.
Let $f$ be a polynomial function of degree $n$.
Then $\Delta^{n + 1} y$ is zero for all $y$. | By definition of $n$th forward difference operator:
{{begin-eqn}}
{{eqn | l = \map {\Delta^{n + 1} f} {x_i}
| r = \map \Delta {\map {\Delta^n f} {x_i} }
| c =
}}
{{eqn | r = \Delta^n \map f {x_{i + 1} } - \Delta^n \map f {x_i}
| c =
}}
{{end-eqn}}
From Nth Forward Difference is Constant iff Function... | Let $y = \map f x$ be a [[Definition:Real Function|real function]].
Let $\Delta y$ denote the [[Definition:Forward Difference Operator|forward difference operator]] on $y$.
Let $f$ be a [[Definition:Polynomial over Real Numbers|polynomial function]] of [[Definition:Degree of Polynomial|degree]] $n$.
Then $\Delta^{n... | By definition of [[Definition:kth Forward Difference Operator|$n$th forward difference operator]]:
{{begin-eqn}}
{{eqn | l = \map {\Delta^{n + 1} f} {x_i}
| r = \map \Delta {\map {\Delta^n f} {x_i} }
| c =
}}
{{eqn | r = \Delta^n \map f {x_{i + 1} } - \Delta^n \map f {x_i}
| c =
}}
{{end-eqn}}
Fro... | Nth Forward Difference is Constant iff Function is Polynomial of Degree N/Corollary | https://proofwiki.org/wiki/Nth_Forward_Difference_is_Constant_iff_Function_is_Polynomial_of_Degree_N/Corollary | https://proofwiki.org/wiki/Nth_Forward_Difference_is_Constant_iff_Function_is_Polynomial_of_Degree_N/Corollary | [
"Nth Forward Difference is Constant iff Function is Polynomial of Degree N"
] | [
"Definition:Real Function",
"Definition:Finite Difference Operator/Forward Difference",
"Definition:Polynomial/Real Numbers",
"Definition:Degree of Polynomial",
"Definition:Zero (Number)"
] | [
"Definition:kth Forward Difference Operator",
"Nth Forward Difference is Constant iff Function is Polynomial of Degree N"
] |
proofwiki-23158 | Existence of Null Sequence not in any P-Sequence Space | Let $\GF \in \set {\R, \C}$.
For each $p \in \hointr 1 \infty$, let $\map {\ell_p} {\N, \GF}$ be the $p$-sequence space.
Then there exists $\phi : \N \to \GF$ such that $\map \phi n \to 0$ as $n \to \infty$, yet:
:$\ds \phi \not \in \bigcup_{1 \le p < \infty} \map {\ell_p} {\N, \GF}$ | Define $\phi : \N \to \R$ by:
:$\ds \map \phi n = \frac 1 {\map \log n}$ for $n \in \N$.
Let $p \in \hointr 1 \infty$.
From Order of Natural Logarithm Function, we have:
:$\ds \map \log n \le p n^{1/p}$ for each $n \in \N$.
Hence:
:$\ds \frac 1 {\map \log n} \ge \frac 1 p n^{-1/p}$
We therefore have:
:$\ds \frac 1 {\p... | Let $\GF \in \set {\R, \C}$.
For each $p \in \hointr 1 \infty$, let $\map {\ell_p} {\N, \GF}$ be the [[Definition:P-Sequence Space|$p$-sequence space]].
Then there exists $\phi : \N \to \GF$ such that $\map \phi n \to 0$ as $n \to \infty$, yet:
:$\ds \phi \not \in \bigcup_{1 \le p < \infty} \map {\ell_p} {\N, \GF}$ | Define $\phi : \N \to \R$ by:
:$\ds \map \phi n = \frac 1 {\map \log n}$ for $n \in \N$.
Let $p \in \hointr 1 \infty$.
From [[Order of Natural Logarithm Function]], we have:
:$\ds \map \log n \le p n^{1/p}$ for each $n \in \N$.
Hence:
:$\ds \frac 1 {\map \log n} \ge \frac 1 p n^{-1/p}$
We therefore have:
:$\ds \fr... | Existence of Null Sequence not in any P-Sequence Space | https://proofwiki.org/wiki/Existence_of_Null_Sequence_not_in_any_P-Sequence_Space | https://proofwiki.org/wiki/Existence_of_Null_Sequence_not_in_any_P-Sequence_Space | [
"P-Sequence Space",
"P-Sequence Spaces",
"P-Sequence Spaces"
] | [
"Definition:P-Sequence Space"
] | [
"Order of Natural Logarithm Function",
"Harmonic Series is Divergent",
"Category:P-Sequence Spaces"
] |
proofwiki-23159 | Characterization of Adjunction Using Right Adjuncts of Compositions | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\mathbf D \times \mathbf C$ denote the product category of $\mathbf D$ with $\mathbf C$.
For each object $\tuple{D, C} \in ... | From Characterization of Adjunction Using Right Adjuncts of Triple Compositions:
:the triple $\tuple {F, G, \alpha}$ is an adjunction
{{iff}}:
:$(3):\quad$ for every $f:D_2 \to D_1 \in \mathbf D$, $h:FD_1 \to C_1 \in \mathbf C$ and $g:C_1 \to C_2 \in \mathbf C$:
::$\map {\alpha_{\tuple{D_2, C_2} } } {g \circ h \circ Ff... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\mathbf D \time... | From [[Characterization of Adjunction Using Right Adjuncts of Triple Compositions]]:
:the [[Definition:Triple|triple]] $\tuple {F, G, \alpha}$ is an [[Definition:Adjunction|adjunction]]
{{iff}}:
:$(3):\quad$ for every $f:D_2 \to D_1 \in \mathbf D$, $h:FD_1 \to C_1 \in \mathbf C$ and $g:C_1 \to C_2 \in \mathbf C$:
::$\m... | Characterization of Adjunction Using Right Adjuncts of Compositions | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Right_Adjuncts_of_Compositions | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Right_Adjuncts_of_Compositions | [
"Characterizations of Adjunctions",
"Adjunctions"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Product Category",
"Definition:Object (Category Theory)",
"Definition:Bijection",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction",
"Definition:Commutative... | [
"Characterization of Adjunction Using Right Adjuncts of Triple Compositions",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction"
] |
proofwiki-23160 | Barycentric Coordinates are Unique iff Vertices are Affinely Independent | Let $V = \tuple {v_0, v_1, \ldots, v_k}$ be a set of $k + 1$ vertices in $\mathbb R^k$.
Let $x$ be any point in $\mathbb R^k$.
Let $\tuple {\lambda_0, \lambda_1, \ldots, \lambda_k}$ be the barycentric coordinates of $x$ {{WRT}} the vertices.
Then the barycentric coordinates are unique {{iff}} $V$ is affinely independen... | === Necessary Case ===
Let $V$ be affinely independent.
{{AimForCont}} $\tuple {\beta_0, \beta_1, \ldots, \beta_k}$ are other barycentric coordinates for the point $x$ where:
:$\exists p \in \closedint 0 k : \beta_p \ne \lambda_p$
Then:
{{begin-eqn}}
{{eqn | l = x
| r = \sum_{i \mathop = 0}^k \lambda_i v_i
}}
{{e... | Let $V = \tuple {v_0, v_1, \ldots, v_k}$ be a set of $k + 1$ [[Definition:Vertex (Geometry)|vertices]] in $\mathbb R^k$.
Let $x$ be any [[Definition:Point|point]] in $\mathbb R^k$.
Let $\tuple {\lambda_0, \lambda_1, \ldots, \lambda_k}$ be the [[Definition:Barycentric Coordinates|barycentric coordinates]] of $x$ {{WRT... | === Necessary Case ===
Let $V$ be [[Definition:Affinely Independent|affinely independent]].
{{AimForCont}} $\tuple {\beta_0, \beta_1, \ldots, \beta_k}$ are other [[Definition:Barycentric Coordinates|barycentric coordinates]] for the [[Definition:Point|point]] $x$ where:
:$\exists p \in \closedint 0 k : \beta_p \ne \l... | Barycentric Coordinates are Unique iff Vertices are Affinely Independent | https://proofwiki.org/wiki/Barycentric_Coordinates_are_Unique_iff_Vertices_are_Affinely_Independent | https://proofwiki.org/wiki/Barycentric_Coordinates_are_Unique_iff_Vertices_are_Affinely_Independent | [
"Barycentric Coordinates",
"Affine Geometry"
] | [
"Definition:Vertex (Geometry)",
"Definition:Point",
"Definition:Barycentric Coordinates",
"Definition:Vertex (Geometry)",
"Definition:Barycentric Coordinates",
"Definition:Unique",
"Definition:Affinely Dependent/Independent"
] | [
"Definition:Affinely Dependent/Independent",
"Definition:Barycentric Coordinates",
"Definition:Point",
"Linear Combination of Indexed Summations",
"Definition:Summation/Summand",
"Definition:Summation/Indexed",
"Definition:Summation",
"Definition:Vertex (Geometry)",
"Definition:Linear Combination",
... |
proofwiki-23161 | Cyclic Group is Finitely Generated | Let $\CC$ be a cyclic group.
Then $\CC$ is finitely generated. | Let $\CC$ be a cyclic group.
Then by definition of cyclic group:
:$\exists g \in \CC: \CC = \gen g$
where $\gen g$ is the generator of $\CC$.
As $\set g$ is a finite set the result follows by definition of finitely generated group.
{{qed}} | Let $\CC$ be a [[Definition:Cyclic Group|cyclic group]].
Then $\CC$ is [[Definition:Finitely Generated Group|finitely generated]]. | Let $\CC$ be a [[Definition:Cyclic Group|cyclic group]].
Then by definition of [[Definition:Cyclic Group|cyclic group]]:
:$\exists g \in \CC: \CC = \gen g$
where $\gen g$ is the [[Definition:Generator of Group|generator]] of $\CC$.
As $\set g$ is a [[Definition:Finite Set|finite set]] the result follows by definiti... | Cyclic Group is Finitely Generated | https://proofwiki.org/wiki/Cyclic_Group_is_Finitely_Generated | https://proofwiki.org/wiki/Cyclic_Group_is_Finitely_Generated | [
"Finitely Generated Groups",
"Cyclic Groups"
] | [
"Definition:Cyclic Group",
"Definition:Finitely Generated Group"
] | [
"Definition:Cyclic Group",
"Definition:Cyclic Group",
"Definition:Generator of Group",
"Definition:Finite Set",
"Definition:Finitely Generated Group"
] |
proofwiki-23162 | Characterization of Adjunction Using Right Adjuncts of Morphisms | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\mathbf D \times \mathbf C$ denote the product category of $\mathbf D$ with $\mathbf C$.
For each object $\tuple{D, C} \in ... | From Characterization of Adjunction Using Right Adjuncts of Compositions:
:the triple $\tuple {F, G, \alpha}$ is an adjunction
{{iff}}:
:$(3)\quad$ for every $h:FD \to C_1 \in \mathbf C$ and $g:C_1 \to C_2 \in \mathbf C$
::$\qquad\qquad\map {\alpha_{\tuple{D, C_2} } } {g \circ h} = Gg \circ \map {\alpha_{\tuple{D, C_1}... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\mathbf D \time... | From [[Characterization of Adjunction Using Right Adjuncts of Compositions]]:
:the [[Definition:Triple|triple]] $\tuple {F, G, \alpha}$ is an [[Definition:Adjunction|adjunction]]
{{iff}}:
:$(3)\quad$ for every $h:FD \to C_1 \in \mathbf C$ and $g:C_1 \to C_2 \in \mathbf C$
::$\qquad\qquad\map {\alpha_{\tuple{D, C_2} } }... | Characterization of Adjunction Using Right Adjuncts of Morphisms | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Right_Adjuncts_of_Morphisms | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Right_Adjuncts_of_Morphisms | [
"Characterizations of Adjunctions",
"Adjunctions"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Product Category",
"Definition:Object (Category Theory)",
"Definition:Bijection",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction",
"Definition:Commutative... | [
"Characterization of Adjunction Using Right Adjuncts of Compositions",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction"
] |
proofwiki-23163 | Characterization of Adjunction Using Left Adjuncts of Compositions | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\mathbf D \times \mathbf C$ denote the product category of $\mathbf D$ with $\mathbf C$.
For each object $\tuple{D, C} \in ... | From Characterization of Adjunction Using Right Adjuncts of Compositions:
:the triple $\tuple {F, G, \alpha}$ is an adjunction
{{iff}}:
:$(3):\quad$ for every $h:FD \to C_1 \in \mathbf C$ and $g:C_1 \to C_2 \in \mathbf C$
::$\qquad\qquad\map {\alpha_{\tuple{D, C_2} } } {g \circ h} = Gg \circ \map {\alpha_{\tuple{D, C_1... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\mathbf D \time... | From [[Characterization of Adjunction Using Right Adjuncts of Compositions]]:
:the [[Definition:Triple|triple]] $\tuple {F, G, \alpha}$ is an [[Definition:Adjunction|adjunction]]
{{iff}}:
:$(3):\quad$ for every $h:FD \to C_1 \in \mathbf C$ and $g:C_1 \to C_2 \in \mathbf C$
::$\qquad\qquad\map {\alpha_{\tuple{D, C_2} } ... | Characterization of Adjunction Using Left Adjuncts of Compositions | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Left_Adjuncts_of_Compositions | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Left_Adjuncts_of_Compositions | [
"Characterizations of Adjunctions",
"Adjunctions"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Product Category",
"Definition:Object (Category Theory)",
"Definition:Bijection",
"Definition:Object (Category Theory)",
"Definition:Inverse Mapping",
"Definition:Ordered Tuple as Ordered ... | [
"Characterization of Adjunction Using Right Adjuncts of Compositions",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction",
"Characterization of Adjunction Using Right Adjuncts of Compositions"
] |
proofwiki-23164 | Characterization of Adjunction Using Left Adjuncts of Morphisms | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\mathbf D \times \mathbf C$ denote the product category of $\mathbf D$ with $\mathbf C$.
For each object $\tuple{D, C} \in ... | From Characterization of Adjunction Using Right Adjuncts of Compositions:
:the triple $\tuple {F, G, \alpha}$ is an adjunction
{{iff}}:
:$(3):\quad$ for every $h:D \to G C_1 \in \mathbf D$ and $g:C_1 \to C_2 \in \mathbf C$:
::$\qquad\qquad\map {\beta_{\tuple{D, C_2} } } {G g \circ h} = g \circ \map {\beta_{\tuple{D, C_... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\mathbf D \time... | From [[Characterization of Adjunction Using Right Adjuncts of Compositions]]:
:the [[Definition:Triple|triple]] $\tuple {F, G, \alpha}$ is an [[Definition:Adjunction|adjunction]]
{{iff}}:
:$(3):\quad$ for every $h:D \to G C_1 \in \mathbf D$ and $g:C_1 \to C_2 \in \mathbf C$:
::$\qquad\qquad\map {\beta_{\tuple{D, C_2} }... | Characterization of Adjunction Using Left Adjuncts of Morphisms | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Left_Adjuncts_of_Morphisms | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Left_Adjuncts_of_Morphisms | [
"Characterizations of Adjunctions",
"Adjunctions"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Product Category",
"Definition:Object (Category Theory)",
"Definition:Bijection",
"Definition:Object (Category Theory)",
"Definition:Inverse Mapping",
"Definition:Ordered Tuple as Ordered ... | [
"Characterization of Adjunction Using Right Adjuncts of Compositions",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction"
] |
proofwiki-23165 | Floor Function/Examples/Floor of 3.2 | :$\floor {3 \cdotp 2} = 3$ | We have that:
:$3 \le 3 \cdotp 2 < 4$
Hence $3$ is the floor of $3 \cdotp 2$ by definition.
{{qed}} | :$\floor {3 \cdotp 2} = 3$ | We have that:
:$3 \le 3 \cdotp 2 < 4$
Hence $3$ is the [[Definition:Floor Function|floor]] of $3 \cdotp 2$ by definition.
{{qed}} | Floor Function/Examples/Floor of 3.2 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_3.2 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_3.2 | [
"Examples of Floor Function"
] | [] | [
"Definition:Floor Function"
] |
proofwiki-23166 | Floor Function/Examples/Floor of 5 | :$\floor 5 = 5$ | We have that $5$ is an integer.
Thus this is a specific example of Real Number is Integer iff equals Floor:
$\floor x = x \iff x \in \Z$
{{qed}} | :$\floor 5 = 5$ | We have that $5$ is an [[Definition:Integer|integer]].
Thus this is a specific example of [[Real Number is Integer iff equals Floor]]:
$\floor x = x \iff x \in \Z$
{{qed}} | Floor Function/Examples/Floor of 5 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_5 | https://proofwiki.org/wiki/Floor_Function/Examples/Floor_of_5 | [
"Examples of Floor Function"
] | [] | [
"Definition:Integer",
"Real Number is Integer iff equals Floor"
] |
proofwiki-23167 | Universal Approximation Theorem | Let $I_n$ be the $n$-dimensional unit hypercube $\closedint 0 1^n$.
Let $\map C {I_n}$ be the space of continuous functions on $I_n$, equipped with the supremum norm:
:$\ds \norm f = \sup_{x \mathop \in I_n} \size {\map f x}$
Let $\sigma: \R \to \R$ be a continuous sigmoidal function.
That is, $\sigma$ satisfies:
:$\ds... | Let $S$ be the linear subspace of $\map C {I_n}$ spanned by functions of the form $\map \sigma {y^\intercal x + \theta}$.
We aim to show that the closure $\bar S = \map C {I_n}$.
We proceed by Proof by Contradiction.
{{AimForCont}} $\bar S$ is a proper subspace of $\map C {I_n}$, that is $\bar S \subsetneq \map C {I_n}... | Let $I_n$ be the $n$-dimensional unit hypercube $\closedint 0 1^n$.
Let $\map C {I_n}$ be the space of [[Definition:Continuous Real Function|continuous functions]] on $I_n$, equipped with the [[Definition:Supremum Norm|supremum norm]]:
:$\ds \norm f = \sup_{x \mathop \in I_n} \size {\map f x}$
Let $\sigma: \R \to \R$... | Let $S$ be the linear subspace of $\map C {I_n}$ spanned by functions of the form $\map \sigma {y^\intercal x + \theta}$.
We aim to show that the [[Definition:Closure of Set|closure]] $\bar S = \map C {I_n}$.
We proceed by [[Proof by Contradiction]].
{{AimForCont}} $\bar S$ is a proper subspace of $\map C {I_n}$, th... | Universal Approximation Theorem | https://proofwiki.org/wiki/Universal_Approximation_Theorem | https://proofwiki.org/wiki/Universal_Approximation_Theorem | [
"Universal Approximation Theorem",
"Artificial Intelligence",
"Neural Networks",
"Functional Analysis"
] | [
"Definition:Continuous Real Function",
"Definition:Supremum Norm",
"Definition:Sigmoid Function",
"Definition:Everywhere Dense"
] | [
"Definition:Closure of Set",
"Proof by Contradiction",
"Hahn-Banach Theorem",
"Riesz Representation Theorem",
"Definition:Discriminatory Function",
"Definition:Characteristic Function",
"Lebesgue's Dominated Convergence Theorem",
"Hahn-Banach Theorem"
] |
proofwiki-23168 | Force Field is Conservative iff Potential Energy Exists | Let $\map {\mathbf F} {\mathbf r}$ be a force field.
Then a potential energy function $\map P {\mathbf r}$ exists for $\map {\mathbf F} {\mathbf r}$ {{Iff}} $\map {\mathbf F} {\mathbf r}$ is conservative. | === Sufficient Case ===
We have that a potential energy function $\map P {\mathbf r}$ exists (not necessarily unique).
From the definition of potential energy:
:$\ds \map P {\mathbf r_2} - \map P {\mathbf r_1} = \int_{\mathbf r_1}^{\mathbf r_2} \map {\mathbf F} {\mathbf r} \cdot \rd \mathbf l$
Because $\map P {\mathbf ... | Let $\map {\mathbf F} {\mathbf r}$ be a [[Definition:Force Field|force field]].
Then a [[Definition:Potential Energy|potential energy]] [[Definition:Real-Valued Function|function]] $\map P {\mathbf r}$ exists for $\map {\mathbf F} {\mathbf r}$ {{Iff}} $\map {\mathbf F} {\mathbf r}$ is [[Definition:Conservative Vector ... | === Sufficient Case ===
We have that a [[Definition:Potential Energy|potential energy]] [[Definition:Real-Valued Function|function]] $\map P {\mathbf r}$ exists (not necessarily [[Definition:Unique|unique]]).
From the definition of [[Definition:Potential Energy|potential energy]]:
:$\ds \map P {\mathbf r_2} - \map P... | Force Field is Conservative iff Potential Energy Exists | https://proofwiki.org/wiki/Force_Field_is_Conservative_iff_Potential_Energy_Exists | https://proofwiki.org/wiki/Force_Field_is_Conservative_iff_Potential_Energy_Exists | [
"Conservative Vector Fields",
"Conservative Forces",
"Potential Energy",
"Vector Analysis",
"Vector Calculus",
"Physics"
] | [
"Definition:Force Field",
"Definition:Potential Energy",
"Definition:Real-Valued Function",
"Definition:Conservative Vector Field"
] | [
"Definition:Potential Energy",
"Definition:Real-Valued Function",
"Definition:Unique",
"Definition:Potential Energy",
"Definition:Mapping",
"Definition:Path (Topology)/Endpoint",
"Definition:Contour Integral",
"Definition:Path (Topology)/Endpoint",
"Definition:Contour",
"Definition:Conservative Ve... |
proofwiki-23169 | Angular Speed of Rotation of Earth | The angular speed $\omega$ of rotation of Earth is approximately:
:$7.29 \times 10^{-5} \, \mathrm {rad \, s}^{-1}$ | {{tidy|minor stuff}}
By definition of angular speed,
:$\omega = \frac{\theta}{t}$
where $\theta$ is the angle traversed in time $t$.
In one complete rotation of Earth, the angle traversed is
:$2\pi \,\mathrm{rad}$
Also, the time required for one such rotation relative to the fixed stars is approximately one sidereal da... | The [[Definition:Angular Speed|angular speed]] $\omega$ of [[Definition:Space Rotation|rotation]] of [[Definition:Earth|Earth]] is approximately:
:$7.29 \times 10^{-5} \, \mathrm {rad \, s}^{-1}$ | {{tidy|minor stuff}}
By definition of [[Definition:Angular Speed|angular speed]],
:$\omega = \frac{\theta}{t}$
where $\theta$ is the [[Definition:Angle|angle]] traversed in time $t$.
In one complete [[Definition:Space Rotation|rotation]] of [[Definition:Earth|Earth]], the angle traversed is
:$2\pi \,\mathrm{rad}$
A... | Angular Speed of Rotation of Earth | https://proofwiki.org/wiki/Angular_Speed_of_Rotation_of_Earth | https://proofwiki.org/wiki/Angular_Speed_of_Rotation_of_Earth | [
"Earth"
] | [
"Definition:Angular Speed",
"Definition:Rotation (Geometry)/Space",
"Definition:Earth"
] | [
"Definition:Angular Speed",
"Definition:Angle",
"Definition:Rotation (Geometry)/Space",
"Definition:Earth",
"Definition:Sidereal Day"
] |
proofwiki-23170 | Kottman Constant of Infinite-Dimensional Hilbert Space is Root 2 | Let $\GF \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be an infinite-dimensional Hilbert space over $\GF$.
Let $\map K \HH$ be the Kottman constant of $\HH$.
Then $\map K \HH = \sqrt 2$. | First consider the case $\GF = \R$.
Let $\norm {\, \cdot \,}$ be the inner product norm on $\HH$.
Let $B_\HH^-$ be the closed unit ball of $\struct {\HH, \norm {\, \cdot \,} }$.
We first show that $\map K \HH \ge \sqrt 2$.
Since $\HH$ is infinite-dimensional, there exists a countable linearly independent subset $S$ of... | Let $\GF \in \set {\R, \C}$.
Let $\struct {\HH, \innerprod \cdot \cdot}$ be an [[Definition:Infinite-Dimensional Vector Space|infinite-dimensional]] [[Definition:Hilbert Space|Hilbert space]] over $\GF$.
Let $\map K \HH$ be the [[Definition:Kottman Constant of Banach Space|Kottman constant]] of $\HH$.
Then $\map K ... | First consider the case $\GF = \R$.
Let $\norm {\, \cdot \,}$ be the [[Definition:Inner Product Norm|inner product norm]] on $\HH$.
Let $B_\HH^-$ be the [[Definition:Closed Unit Ball|closed unit ball]] of $\struct {\HH, \norm {\, \cdot \,} }$.
We first show that $\map K \HH \ge \sqrt 2$.
Since $\HH$ is [[Definiti... | Kottman Constant of Infinite-Dimensional Hilbert Space is Root 2 | https://proofwiki.org/wiki/Kottman_Constant_of_Infinite-Dimensional_Hilbert_Space_is_Root_2 | https://proofwiki.org/wiki/Kottman_Constant_of_Infinite-Dimensional_Hilbert_Space_is_Root_2 | [
"Hilbert Spaces"
] | [
"Definition:Infinite-Dimensional Vector Space",
"Definition:Hilbert Space",
"Definition:Kottman Constant of Banach Space"
] | [
"Definition:Inner Product Norm",
"Definition:Closed Unit Ball",
"Definition:Infinite-Dimensional Vector Space",
"Definition:Countable Set",
"Definition:Linearly Independent/Set",
"Gram-Schmidt Orthogonalization",
"Definition:Orthonormal Subset",
"Pythagoras's Theorem (Inner Product Space)",
"Definit... |
proofwiki-23171 | Euler-Maclaurin Summation Formula/Formulation 1 | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map f k
| r = \int_0^n \map f x \rd x + \sum_{k \mathop = 1}^\infty \frac {B_k} {k!} \paren {\map {f^{\paren {k - 1} } } n - \paren {-1}^k \map {f^{\paren {k - 1} } } 0}
| c =
}}
{{eqn | r = \int_0^n \map f x \rd x
| c =
}}
{{eqn | o =
|... | {{ProofWanted|I'll let someone do this who's better at this than me}} | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map f k
| r = \int_0^n \map f x \rd x + \sum_{k \mathop = 1}^\infty \frac {B_k} {k!} \paren {\map {f^{\paren {k - 1} } } n - \paren {-1}^k \map {f^{\paren {k - 1} } } 0}
| c =
}}
{{eqn | r = \int_0^n \map f x \rd x
| c =
}}
{{eqn | o =
|... | {{ProofWanted|I'll let someone do this who's better at this than me}} | Euler-Maclaurin Summation Formula/Formulation 1 | https://proofwiki.org/wiki/Euler-Maclaurin_Summation_Formula/Formulation_1 | https://proofwiki.org/wiki/Euler-Maclaurin_Summation_Formula/Formulation_1 | [
"Euler-Maclaurin Summation Formula",
"Integral Calculus",
"Series"
] | [
"Definition:Derivative/Higher Derivatives/Higher Order",
"Definition:Bernoulli Numbers"
] | [] |
proofwiki-23172 | Characterization of Adjunction Using Unit of Adjunction | Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be covariant functors.
Then:
:there exists an adjunction $\tuple{F, G, \alpha}$ between $\mathbf C$ and $\mathbf D$
{{iff}}:
:there exists a natural transformation $\eta: \operatorname {id}_{\math... | === Necessary Condition ===
Let $\tuple{F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$.
From Adjunction Induces Unit of Adjunction:
:the unit of adjunction $\eta$ is a natural transformation $\eta: \operatorname {id}_{\mathbf D} \to GF$ such that:
::for each object $D$ of $\mathbf D: \eta_{_D} = \m... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Then:
:there exists an [[Definition:Adjunction|adjunction]] $\tuple{F, G, \alpha}$ between $\mathbf... | === Necessary Condition ===
Let $\tuple{F, G, \alpha}$ be an [[Definition:Adjunction|adjunction]] between $\mathbf C$ and $\mathbf D$.
From [[Adjunction Induces Unit of Adjunction]]:
:the [[Definition:Unit of Adjunction|unit of adjunction]] $\eta$ is a [[Definition:Natural Transformation|natural transformation]] $\e... | Characterization of Adjunction Using Unit of Adjunction | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Unit_of_Adjunction | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Unit_of_Adjunction | [
"Characterizations of Adjunctions",
"Adjunctions"
] | [
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Adjunction",
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Universal Morphism from Object to Functor",
"Definition:Object (Category Theory)",
"Definition:... | [
"Definition:Adjunction",
"Adjunction Induces Unit of Adjunction",
"Definition:Unit of Adjunction",
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Morphism of Unit of Adjunction is Universal",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Univ... |
proofwiki-23173 | Bertrand's Ballot Theorem | Consider an election involving two candidates $A$ and $B$, which $A$ has won.
Let $p$ be the number of votes that $A$ received, and $q < p$ be the number of votes that $B$ received.
We assume that $A$ did not win by default, and that at least one vote was cast.
We put all $p + q$ votes in a ballot box and draw them at ... | Given that $A$ wins, the claim that $\map {T_B} k < \map {T_A} k$ for all $k$ is equivalent to the claim that there exists no $k$ with $\map {T_B} k = \map {T_A} k$.
That is, at no point in the count are candidates $A$ and $B$ tying.
We therefore aim to compute:
:$\map \Pr {\map {T_A} k \ne \map {T_B} k \text { for al... | Consider an election involving two candidates $A$ and $B$, which $A$ has won.
Let $p$ be the number of votes that $A$ received, and $q < p$ be the number of votes that $B$ received.
We assume that $A$ did not win by default, and that at least one vote was cast.
We put all $p + q$ votes in a ballot box and draw them ... | Given that $A$ wins, the claim that $\map {T_B} k < \map {T_A} k$ for all $k$ is equivalent to the claim that there exists no $k$ with $\map {T_B} k = \map {T_A} k$.
That is, at no point in the count are candidates $A$ and $B$ tying.
We therefore aim to compute:
:$\map \Pr {\map {T_A} k \ne \map {T_B} k \text { for ... | Bertrand's Ballot Theorem | https://proofwiki.org/wiki/Bertrand's_Ballot_Theorem | https://proofwiki.org/wiki/Bertrand's_Ballot_Theorem | [
"Combinatorics"
] | [
"Definition:Uniform Distribution/Discrete",
"Definition:Probability",
"Definition:Inequality"
] | [
"Law of Total Probability",
"Intersection with Subset is Subset",
"Definition:Uniform Distribution/Discrete",
"Definition:String",
"Definition:Mapping",
"Definition:Bijection",
"Definition:Injection",
"Definition:String",
"Definition:Surjection",
"Definition:Bijection",
"Category:Combinatorics"
... |
proofwiki-23174 | Expected Number of Cards Drawn Before First Ace | Consider a standard deck of $52$ cards.
We draw cards from the deck at uniform random without replacement.
The expected number of cards drawn ''before'' the first ace is $9.6$.
Hence the expected number of cards drawn before and including the first ace is $10.6$. | Note that there are $4$ aces in the deck.
We number the $48$ other cards $1, 2, \ldots, 48$.
Label the aces by $A_1, A_2, A_3, A_4$.
Define the indicator random variable $I_i$ by:
:$\ds I_i = \begin{cases}1 & \text {card } i \text { is drawn before the first ace} \\ 0 & \text{otherwise}\end{cases}$
The number of cards... | Consider a standard [[Definition:Deck of Cards|deck]] of $52$ [[Definition:Card|cards]].
We draw [[Definition:Card|cards]] from the [[Definition:Deck of Cards|deck]] at [[Definition:Discrete Uniform Distribution|uniform random]] [[Definition:Sampling without Replacement|without replacement]].
The [[Definition:Expect... | Note that there are $4$ [[Definition:Ace|aces]] in the [[Definition:Deck of Cards|deck]].
We number the $48$ other [[Definition:Card|cards]] $1, 2, \ldots, 48$.
Label the [[Definition:Ace|aces]] by $A_1, A_2, A_3, A_4$.
Define the [[Definition:Indicator Random Variable|indicator random variable]] $I_i$ by:
:$\ds I_... | Expected Number of Cards Drawn Before First Ace | https://proofwiki.org/wiki/Expected_Number_of_Cards_Drawn_Before_First_Ace | https://proofwiki.org/wiki/Expected_Number_of_Cards_Drawn_Before_First_Ace | [
"Combinatorics",
"Cards"
] | [
"Definition:Deck of Cards",
"Definition:Deck of Cards/Card",
"Definition:Deck of Cards/Card",
"Definition:Deck of Cards",
"Definition:Uniform Distribution/Discrete",
"Definition:Random Sample (Statistics)/Without Replacement",
"Definition:Expectation",
"Definition:Deck of Cards/Card",
"Definition:Ac... | [
"Definition:Ace",
"Definition:Deck of Cards",
"Definition:Deck of Cards/Card",
"Definition:Ace",
"Definition:Indicator Random Variable",
"Definition:Deck of Cards/Card",
"Definition:Ace",
"Definition:Expectation",
"Expectation is Linear",
"Definition:Event",
"Definition:Ace",
"Definition:Permu... |
proofwiki-23175 | Fredholm Alternative/Simple Form | Let $\mathbf A$ be a square matrix.
Then either:
:$(1): \quad \mathbf v = \mathbf A \mathbf v + \mathbf b$ has a solution $\mathbf v$ for all $\mathbf b$
or:
:$(2): \quad \mathbf v = \mathbf A \mathbf v$ has a non-zero solution.
$(1)$ holds when $\mathbf A - \mathbf I$ is invertible
$(2)$ holds when $\mathbf A - \mathb... | {{ProofWanted}}
{{Namedfor|Erik Ivar Fredholm}} | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]].
Then either:
:$(1): \quad \mathbf v = \mathbf A \mathbf v + \mathbf b$ has a [[Definition:Solution to Equation|solution]] $\mathbf v$ for all $\mathbf b$
or:
:$(2): \quad \mathbf v = \mathbf A \mathbf v$ has a non-[[Definition:Zero Matrix|zero]] [[Defini... | {{ProofWanted}}
{{Namedfor|Erik Ivar Fredholm}} | Fredholm Alternative/Simple Form | https://proofwiki.org/wiki/Fredholm_Alternative/Simple_Form | https://proofwiki.org/wiki/Fredholm_Alternative/Simple_Form | [
"Fredholm Alternative",
"Matrix Theory"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Fiber of Truth/Solution",
"Definition:Zero Matrix",
"Definition:Fiber of Truth/Solution",
"Definition:Nonsingular Matrix",
"Definition:Singular Matrix"
] | [] |
proofwiki-23176 | Fredholm Alternative/General Form | Let $\mathbf A$ be a self-adjoint compact operator over an infinite-dimensional Hilbert space.
{{help|The source work I am using does not go into detail of exactly what it is}} | {{ProofWanted}}
{{Namedfor|Erik Ivar Fredholm}} | Let $\mathbf A$ be a [[Definition:Self-Adjoint Operator|self-adjoint]] [[Definition:Compact Operator|compact operator]] over an [[Definition:Infinite-Dimensional Vector Space|infinite-dimensional]] [[Definition:Hilbert Space|Hilbert space]].
{{help|The source work I am using does not go into detail of exactly what it ... | {{ProofWanted}}
{{Namedfor|Erik Ivar Fredholm}} | Fredholm Alternative/General Form | https://proofwiki.org/wiki/Fredholm_Alternative/General_Form | https://proofwiki.org/wiki/Fredholm_Alternative/General_Form | [
"Fredholm Alternative",
"Functional Analysis"
] | [
"Definition:Hermitian Operator",
"Definition:Compact Linear Transformation",
"Definition:Infinite-Dimensional Vector Space",
"Definition:Hilbert Space"
] | [] |
proofwiki-23177 | Characterization of Adjunction Using Counit of Adjunction | Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be covariant functors.
Then:
:there exists an adjunction $\tuple{F, G, \alpha}$ between $\mathbf C$ and $\mathbf D$
{{iff}}:
:there exists a natural transformation $\xi: FG \to \operatorname {id}_... | === Necessary Condition ===
Let $\tuple{F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$.
Let $\beta$ denote the inverse of the natural isomorphism $\alpha$
From Adjunction Induces Counit of Adjunction:
:the counit of adjunction $\eta$ is a natural transformation $\xi: FG \to \operatorname {id}_{\mat... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Then:
:there exists an [[Definition:Adjunction|adjunction]] $\tuple{F, G, \alpha}$ between $\mathbf... | === Necessary Condition ===
Let $\tuple{F, G, \alpha}$ be an [[Definition:Adjunction|adjunction]] between $\mathbf C$ and $\mathbf D$.
Let $\beta$ denote the [[Definition:Inverse Natural Isomorphism between Covariant Functors|inverse]] of the [[Definition:Natural Isomorphism|natural isomorphism]] $\alpha$
From [[Ad... | Characterization of Adjunction Using Counit of Adjunction | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Counit_of_Adjunction | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Counit_of_Adjunction | [
"Characterizations of Adjunctions",
"Adjunctions",
"Universal Morphisms"
] | [
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Adjunction",
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Universal Morphism from Functor to Object",
"Definition:Functor/Covariant",
"Definition:Object ... | [
"Definition:Adjunction",
"Definition:Natural Isomorphism between Covariant Functors/Inverse",
"Definition:Natural Isomorphism",
"Adjunction Induces Counit of Adjunction",
"Definition:Counit of Adjunction",
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Morphism of Counit ... |
proofwiki-23178 | Universal Morphism from Object to Functor is Unique up to Isomorphism | Let $\mathbf C$ and $\mathbf D$ be metacategories.
Let $C$ be an object of $\mathbf C$.
Let $F: \mathbf D \to \mathbf C$ be a covariant functor.
Let $\tuple{R_1, u_1}$ and $\tuple{R_2, u_2}$ be a universal morphisms from $C$ to $F$.
Then:
:there exists an isomorphism $i: R_1 \to R_2$ such that $u_2 = Fi \circ u_1$ | By definition of universal morphisms from object to functor:
:$\tuple{R_1, u_1}$ and $\tuple{R_2, u_2}$ are initial objects of the functor under object comma category $\paren{C \downarrow F}$
From Initial Object is Unique:
:$\tuple{R_1, u_1}$ and $\tuple{R_2, u_2}$ are isomorphic in the functor under object comma categ... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Metacategory|metacategories]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $F: \mathbf D \to \mathbf C$ be a [[Definition:Covariant Functor|covariant functor]].
Let $\tuple{R_1, u_1}$ and $\tuple{R_2, u_2}$ be a [[Definition:Univer... | By definition of [[Definition:Universal Morphism from Object to Functor|universal morphisms from object to functor]]:
:$\tuple{R_1, u_1}$ and $\tuple{R_2, u_2}$ are [[Definition:Initial Object|initial objects]] of the [[Definition:Functor Under Object Comma Category|functor under object comma category]] $\paren{C \down... | Universal Morphism from Object to Functor is Unique up to Isomorphism | https://proofwiki.org/wiki/Universal_Morphism_from_Object_to_Functor_is_Unique_up_to_Isomorphism | https://proofwiki.org/wiki/Universal_Morphism_from_Object_to_Functor_is_Unique_up_to_Isomorphism | [
"Universal Morphisms"
] | [
"Definition:Metacategory",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Universal Morphism from Object to Functor",
"Definition:Isomorphism (Category Theory)"
] | [
"Definition:Universal Morphism from Object to Functor",
"Definition:Initial Object",
"Definition:Comma Category/Functor Under Object",
"Initial Object is Unique",
"Definition:Isomorphism (Category Theory)",
"Definition:Comma Category/Functor Under Object",
"Isomorphism of Functor Under Object Comma Cate... |
proofwiki-23179 | Universal Morphism from Functor to Object is Unique up to Isomorphism | Let $\mathbf C$ and $\mathbf D$ be metacategories.
Let $C$ be an object of $\mathbf C$.
Let $F: \mathbf D \to \mathbf C$ be a covariant functor.
Let $\tuple{R_1, u_1}$ and $\tuple{R_2, u_2}$ be a universal morphisms from $F$ to $C$.
Then:
:there exists an isomorphism $i: R_2 \to R_1$ such that $u_2 = u_1 \circ Fi$ | By definition of universal morphisms from functor to object:
:$\tuple{R_1, u_1}$ and $\tuple{R_2, u_2}$ are terminal objects of the functor over object comma category $\paren{F \downarrow C}$
From Terminal Object is Unique:
:$\tuple{R_1, u_1}$ and $\tuple{R_2, u_2}$ are isomorphic in the functor over object comma categ... | Let $\mathbf C$ and $\mathbf D$ be [[Definition:Metacategory|metacategories]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $F: \mathbf D \to \mathbf C$ be a [[Definition:Covariant Functor|covariant functor]].
Let $\tuple{R_1, u_1}$ and $\tuple{R_2, u_2}$ be a [[Definition:Univer... | By definition of [[Definition:Universal Morphism from Functor to Object|universal morphisms from functor to object]]:
:$\tuple{R_1, u_1}$ and $\tuple{R_2, u_2}$ are [[Definition:Terminal Object|terminal objects]] of the [[Definition:Functor Over Object Comma Category|functor over object comma category]] $\paren{F \down... | Universal Morphism from Functor to Object is Unique up to Isomorphism | https://proofwiki.org/wiki/Universal_Morphism_from_Functor_to_Object_is_Unique_up_to_Isomorphism | https://proofwiki.org/wiki/Universal_Morphism_from_Functor_to_Object_is_Unique_up_to_Isomorphism | [
"Universal Morphisms"
] | [
"Definition:Metacategory",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Universal Morphism from Functor to Object",
"Definition:Isomorphism (Category Theory)"
] | [
"Definition:Universal Morphism from Functor to Object",
"Definition:Terminal Object",
"Definition:Comma Category/Functor Over Object",
"Terminal Object is Unique",
"Definition:Isomorphism (Category Theory)",
"Definition:Comma Category/Functor Under Object",
"Isomorphism of Functor Over Object Comma Cate... |
proofwiki-23180 | Isomorphism of Functor Under Object Comma Category | Let $\mathbf C$, $\mathbf D$ be categories.
Let $C$ be an object of $\mathbf C$.
Let $F : \mathbf D \to \mathbf C$ be a covariant functor.
Let $\paren{C \downarrow F}$ denote the comma category $F$ under $C$.
Let $h : \tuple{D_1, f_1} \to \tuple{D_2, f_2}$ be a morphism in $\paren{C \downarrow F}$.
Then:
:$h$ is an iso... | By definition of morphisms in $\paren{C \downarrow F}$:
:$h:D_1 \to D_2 \in \mathbf D$ is a morphism:
::$(1):\quad f_2 = F h \circ f_1$
By definition of isomorphism:
:$h$ is an isomorphism in $\paren{C \downarrow F}$
{{iff}}
:$(2): \quad \exists g : \tuple{D_2, f_2} \to \tuple{D_1, f_1} \in \paren{C \downarrow F}: g \c... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Category|categories]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $F : \mathbf D \to \mathbf C$ be a [[Definition:Covariant Functor|covariant functor]].
Let $\paren{C \downarrow F}$ denote the [[Definition:Functor Under Object Comm... | By definition of [[Definition:Functor Under Object Comma Category|morphisms in $\paren{C \downarrow F}$]]:
:$h:D_1 \to D_2 \in \mathbf D$ is a [[Definition:Morphism (Category Theory)|morphism]]:
::$(1):\quad f_2 = F h \circ f_1$
By definition of [[Definition:Isomorphism (Category Theory)|isomorphism]]:
:$h$ is an [[D... | Isomorphism of Functor Under Object Comma Category | https://proofwiki.org/wiki/Isomorphism_of_Functor_Under_Object_Comma_Category | https://proofwiki.org/wiki/Isomorphism_of_Functor_Under_Object_Comma_Category | [
"Isomorphisms",
"Comma Categories"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Comma Category/Functor Under Object",
"Definition:Morphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)"
] | [
"Definition:Comma Category/Functor Under Object",
"Definition:Morphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)",
"Definition:Comma Category/Functor Under Object",
"Definition:Comma Category/Functor Under Object",
"Definition:Isomorphism (Category Theory)",... |
proofwiki-23181 | Isomorphism of Functor Over Object Comma Category | Let $\mathbf C$, $\mathbf D$ be categories.
Let $C$ be an object of $\mathbf C$.
Let $G : \mathbf E \to \mathbf C$ be a covariant functor.
Let $\paren{G \downarrow C}$ denote the comma category $G$ over $C$.
Let $k : \tuple{E_1, g_1} \to \tuple{E_2, g_2}$ be a morphism in $\paren{G \downarrow C}$.
Then:
:$k$ is an isom... | By definition of morphisms in $\paren{G \downarrow C}$:
:$k:E_1 \to E_2 \in \mathbf E$ is a morphism:
::$(1):\quad g_1 = g_2 \circ G k$
By definition of isomorphism:
:$h$ is an isomorphism in $\paren{G \downarrow C}$
{{iff}}
:$(2): \quad \exists l : \tuple{E_2, g_2} \to \tuple{E_1, g_1} \in \paren{G \downarrow C}: l \c... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Category|categories]].
Let $C$ be an [[Definition:Object (Category Theory)|object]] of $\mathbf C$.
Let $G : \mathbf E \to \mathbf C$ be a [[Definition:Covariant Functor|covariant functor]].
Let $\paren{G \downarrow C}$ denote the [[Definition:Functor Over Object Comma... | By definition of [[Definition:Functor Under Object Comma Category|morphisms in $\paren{G \downarrow C}$]]:
:$k:E_1 \to E_2 \in \mathbf E$ is a [[Definition:Morphism (Category Theory)|morphism]]:
::$(1):\quad g_1 = g_2 \circ G k$
By definition of [[Definition:Isomorphism (Category Theory)|isomorphism]]:
:$h$ is an [[D... | Isomorphism of Functor Over Object Comma Category | https://proofwiki.org/wiki/Isomorphism_of_Functor_Over_Object_Comma_Category | https://proofwiki.org/wiki/Isomorphism_of_Functor_Over_Object_Comma_Category | [
"Isomorphisms",
"Comma Categories"
] | [
"Definition:Category",
"Definition:Object (Category Theory)",
"Definition:Functor/Covariant",
"Definition:Comma Category/Functor Over Object",
"Definition:Morphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)"
] | [
"Definition:Comma Category/Functor Under Object",
"Definition:Morphism",
"Definition:Isomorphism (Category Theory)",
"Definition:Isomorphism (Category Theory)",
"Definition:Comma Category/Functor Under Object",
"Definition:Comma Category/Functor Under Object",
"Definition:Isomorphism (Category Theory)",... |
proofwiki-23182 | Characterization of Adjunction Using Right Adjuncts of Commutative Squares | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\mathbf D \times \mathbf C$ denote the product category of $\mathbf D$ with $\mathbf C$.
For each object $\tuple{D, C} \in ... | From Characterization of Adjunction Using Right Adjuncts of Compositions:
:the triple $\tuple {F, G, \alpha}$ is an adjunction
{{iff}}:
:$(2)\quad$ for every $h:FD \to C_1 \in \mathbf C$ and $g:C_1 \to C_2 \in \mathbf C$
::$\qquad\qquad\map {\alpha_{\tuple{D, C_2} } } {g \circ h} = Gg \circ \map {\alpha_{\tuple{D, C_1}... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\mathbf D \time... | From [[Characterization of Adjunction Using Right Adjuncts of Compositions]]:
:the [[Definition:Triple|triple]] $\tuple {F, G, \alpha}$ is an [[Definition:Adjunction|adjunction]]
{{iff}}:
:$(2)\quad$ for every $h:FD \to C_1 \in \mathbf C$ and $g:C_1 \to C_2 \in \mathbf C$
::$\qquad\qquad\map {\alpha_{\tuple{D, C_2} } }... | Characterization of Adjunction Using Right Adjuncts of Commutative Squares | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Right_Adjuncts_of_Commutative_Squares | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Right_Adjuncts_of_Commutative_Squares | [
"Characterizations of Adjunctions",
"Adjunctions"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Product Category",
"Definition:Object (Category Theory)",
"Definition:Bijection",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction",
"Definition:Commutative... | [
"Characterization of Adjunction Using Right Adjuncts of Compositions",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction",
"Characterization of Adjunction Using Right Adjuncts of Compositions"
] |
proofwiki-23183 | Characterization of Adjunction Using Left Adjuncts of Commutative Squares | Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\mathbf D \times \mathbf C$ denote the product category of $\mathbf D$ with $\mathbf C$.
For each object $\tuple{D, C} \in ... | From Characterization of Adjunction Using Right Adjuncts of Commutative Squares:
:the triple $\tuple {F, G, \alpha}$ is an adjunction
{{iff}}:
:$(2):\quad$for every:
::$f:D_1 \to D_2 \in \mathbf D$
::$g:C_1 \to C_2 \in \mathbf C$
::$h:FD_1 \to C_1 \in \mathbf C$
::$k:FD_2 \to C_2 \in \mathbf C$
:we have:
::$k \circ Ff ... | Let $\mathbf {Set}$ be the [[Definition:Category of Sets|category of sets]].
Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F : \mathbf D \to \mathbf C$, $G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\mathbf D \time... | From [[Characterization of Adjunction Using Right Adjuncts of Commutative Squares]]:
:the [[Definition:Triple|triple]] $\tuple {F, G, \alpha}$ is an [[Definition:Adjunction|adjunction]]
{{iff}}:
:$(2):\quad$for every:
::$f:D_1 \to D_2 \in \mathbf D$
::$g:C_1 \to C_2 \in \mathbf C$
::$h:FD_1 \to C_1 \in \mathbf C$
::$k:... | Characterization of Adjunction Using Left Adjuncts of Commutative Squares | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Left_Adjuncts_of_Commutative_Squares | https://proofwiki.org/wiki/Characterization_of_Adjunction_Using_Left_Adjuncts_of_Commutative_Squares | [
"Characterizations of Adjunctions",
"Adjunctions"
] | [
"Definition:Category of Sets",
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Product Category",
"Definition:Object (Category Theory)",
"Definition:Bijection",
"Definition:Object (Category Theory)",
"Definition:Inverse Mapping",
"Definition:Ordered Tuple as Ordered ... | [
"Characterization of Adjunction Using Right Adjuncts of Commutative Squares",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction",
"Characterization of Adjunction Using Right Adjuncts of Commutative Squares"
] |
proofwiki-23184 | Left Whiskered Composite of Natural Transformation is Natural Transformation | Let $\mathbf C$, $\mathbf D$ and $\mathbf B$ be categories.
Let $F, G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\eta: F \to G$ be a natural transformation.
Let $H : \mathbf B \to \mathbf C$ be a covariant functor.
Let $\eta H: F \circ H \to G \circ H$ denote the left whiskered composite of $\eta$ by $H$.
T... | Let $\operatorname{id}_{\mathbf D}$ denote the identity functor of $\mathbf D$
Let $\operatorname{id}_{\mathbf D} \eta H : \operatorname{id}_{\mathbf D} \mathop \circ F \circ H \to \operatorname{id}_{\mathbf D} \mathop \circ G \circ H$ denote the double-sided whiskered composite of $\eta$ by $\operatorname{id}_{\mathbf... | Let $\mathbf C$, $\mathbf D$ and $\mathbf B$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\eta: F \to G$ be a [[Definition:Natural Transformation|natural transformation]].
Let $H : \mathbf B \to \mathbf C$ be a [[Definition... | Let $\operatorname{id}_{\mathbf D}$ denote the [[Definition:Identity Functor|identity functor]] of $\mathbf D$
Let $\operatorname{id}_{\mathbf D} \eta H : \operatorname{id}_{\mathbf D} \mathop \circ F \circ H \to \operatorname{id}_{\mathbf D} \mathop \circ G \circ H$ denote the [[Definition:Double-sided Whiskered Comp... | Left Whiskered Composite of Natural Transformation is Natural Transformation | https://proofwiki.org/wiki/Left_Whiskered_Composite_of_Natural_Transformation_is_Natural_Transformation | https://proofwiki.org/wiki/Left_Whiskered_Composite_of_Natural_Transformation_is_Natural_Transformation | [
"Whiskered Composites"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Natural Transformation",
"Definition:Functor/Covariant",
"Definition:Whiskered Composite of Natural Transformation/Left Whiskered",
"Definition:Natural Transformation"
] | [
"Definition:Identity Functor",
"Definition:Whiskered Composite of Natural Transformation/Double-sided Whiskered",
"Double-sided Whiskered Composite of Natural Transformation is Natural Transformation",
"Definition:Natural Transformation",
"Left Whiskered Composite of Natural Transformation is Double-sided W... |
proofwiki-23185 | Square Matrix has Full Rank iff Nonsingular | Let $\mathbf A$ be an $n \times n$ square matrix.
Then:
:$\mathbf A$ is of full rank
{{iff}}:
:$\mathbf A$ is nonsingular. | By the definition of full rank:
:$\mathbf A$ is of full rank {{iff}} $\map \rho {\mathbf A} = n$
where $\map \rho {\mathbf A}$ denotes the rank of $\mathbf A$.
By the definition of rank:
:$\map \rho {\mathbf A} = n$ {{iff}} the rows of $\mathbf A$ are linearly independent
By Square Matrix has Linearly Dependent Rows if... | Let $\mathbf A$ be an $n \times n$ [[Definition:Square Matrix|square matrix]].
Then:
:$\mathbf A$ is of [[Definition:Full Rank|full rank]]
{{iff}}:
:$\mathbf A$ is [[Definition:Nonsingular Matrix|nonsingular]]. | By the definition of [[Definition:Full Rank|full rank]]:
:$\mathbf A$ is of [[Definition:Full Rank|full rank]] {{iff}} $\map \rho {\mathbf A} = n$
where $\map \rho {\mathbf A}$ denotes the [[Definition:Rank of Matrix|rank]] of $\mathbf A$.
By the definition of [[Definition:Rank of Matrix/Definition 3|rank]]:
:$\map \... | Square Matrix has Full Rank iff Nonsingular | https://proofwiki.org/wiki/Square_Matrix_has_Full_Rank_iff_Nonsingular | https://proofwiki.org/wiki/Square_Matrix_has_Full_Rank_iff_Nonsingular | [
"Square Matrices",
"Full Rank",
"Nonsingular Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Full Rank",
"Definition:Nonsingular Matrix"
] | [
"Definition:Full Rank",
"Definition:Full Rank",
"Definition:Rank/Matrix",
"Definition:Rank of Matrix/Definition 3",
"Definition:Matrix/Row",
"Definition:Linearly Independent/Set/Real Vector Space",
"Square Matrix has Linearly Dependent Rows iff Determinant is Zero",
"Definition:Matrix/Row",
"Definit... |
proofwiki-23186 | Right Whiskered Composite of Natural Transformation is Natural Transformation | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be categories.
Let $F, G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\eta: F \to G$ be a natural transformation.
Let $K : \mathbf D \to \mathbf E$ be a covariant functor.
Let $K \eta : K \circ F \to K \circ G$ denote the right whiskered composite of $\eta$ by $K$.... | Let $\operatorname{id}_{\mathbf C}$ denote the identity functor of $\mathbf C$
Let $K \eta \operatorname{id}_{\mathbf C}: K \circ F \circ \operatorname{id}_{\mathbf C} \to K \circ G \circ \operatorname{id}_{\mathbf C}$ denote the double-sided whiskered composite of $\eta$ by $K$ and $\operatorname{id}_{\mathbf C}$.
Fro... | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\eta: F \to G$ be a [[Definition:Natural Transformation|natural transformation]].
Let $K : \mathbf D \to \mathbf E$ be a [[Definition... | Let $\operatorname{id}_{\mathbf C}$ denote the [[Definition:Identity Functor|identity functor]] of $\mathbf C$
Let $K \eta \operatorname{id}_{\mathbf C}: K \circ F \circ \operatorname{id}_{\mathbf C} \to K \circ G \circ \operatorname{id}_{\mathbf C}$ denote the [[Definition:Double-sided Whiskered Composite of Natural ... | Right Whiskered Composite of Natural Transformation is Natural Transformation | https://proofwiki.org/wiki/Right_Whiskered_Composite_of_Natural_Transformation_is_Natural_Transformation | https://proofwiki.org/wiki/Right_Whiskered_Composite_of_Natural_Transformation_is_Natural_Transformation | [
"Whiskered Composites"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Natural Transformation",
"Definition:Functor/Covariant",
"Definition:Whiskered Composite of Natural Transformation/Right Whiskered",
"Definition:Natural Transformation"
] | [
"Definition:Identity Functor",
"Definition:Whiskered Composite of Natural Transformation/Double-sided Whiskered",
"Double-sided Whiskered Composite of Natural Transformation is Natural Transformation",
"Definition:Natural Transformation",
"Right Whiskered Composite of Natural Transformation is Double-sided ... |
proofwiki-23187 | Double-sided Whiskered Composite of Natural Transformation is Natural Transformation | Let $\mathbf C$, $\mathbf D$, $\mathbf E$ and $\mathbf B$ be categories.
Let $F, G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\eta: F \to G$ be a natural transformation.
Let $H : \mathbf B \to \mathbf C$ and $K : \mathbf D \to \mathbf E$ be a covariant functors.
Let $K \eta H: K \circ F \circ H \to K \circ ... | We have:
{{begin-eqn}}
{{eqn | q = \forall f: X \to Y \in \mathbf B
| l = \paren{K \eta H}_Y \circ \paren{K \circ F \circ H} f
| r = K \eta_{HY} \circ \paren{K \circ F \circ H} f
| c = {{Defof|Double-sided Whiskered Composite of Natural Transformation|Double-sided Whiskered Composite}}
}}
{{eqn | r = ... | Let $\mathbf C$, $\mathbf D$, $\mathbf E$ and $\mathbf B$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\eta: F \to G$ be a [[Definition:Natural Transformation|natural transformation]].
Let $H : \mathbf B \to \mathbf C$ and ... | We have:
{{begin-eqn}}
{{eqn | q = \forall f: X \to Y \in \mathbf B
| l = \paren{K \eta H}_Y \circ \paren{K \circ F \circ H} f
| r = K \eta_{HY} \circ \paren{K \circ F \circ H} f
| c = {{Defof|Double-sided Whiskered Composite of Natural Transformation|Double-sided Whiskered Composite}}
}}
{{eqn | r = ... | Double-sided Whiskered Composite of Natural Transformation is Natural Transformation | https://proofwiki.org/wiki/Double-sided_Whiskered_Composite_of_Natural_Transformation_is_Natural_Transformation | https://proofwiki.org/wiki/Double-sided_Whiskered_Composite_of_Natural_Transformation_is_Natural_Transformation | [
"Whiskered Composites"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Natural Transformation",
"Definition:Functor/Covariant",
"Definition:Whiskered Composite of Natural Transformation/Double-sided Whiskered",
"Definition:Natural Transformation"
] | [
"Definition:Natural Transformation"
] |
proofwiki-23188 | Left Whiskered Composite of Natural Transformation is Double-sided Whiskered Composite | Let $\mathbf C$, $\mathbf D$ and $\mathbf B$ be categories.
Let $F, G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\eta: F \to G$ be a natural transformation.
Let $H : \mathbf B \to \mathbf C$ be a covariant functor.
Let $\eta H: F \circ H \to G \circ H$ denote the left whiskered composite of $\eta$ by $H$.
L... | We have:
{{begin-eqn}}
{{eqn | q = \forall B \in \mathbf B
| l = \paren{\operatorname{id}_{\mathbf D} \eta H}_B
| r = \operatorname{id}_{\mathbf D} \eta_{HB}
| c = {{Defof|Double-sided Whiskered Composite of Natural Transformation|Double-sided Whiskered Composite}}
}}
{{eqn | r = \eta_{HB}
| c =... | Let $\mathbf C$, $\mathbf D$ and $\mathbf B$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\eta: F \to G$ be a [[Definition:Natural Transformation|natural transformation]].
Let $H : \mathbf B \to \mathbf C$ be a [[Definition... | We have:
{{begin-eqn}}
{{eqn | q = \forall B \in \mathbf B
| l = \paren{\operatorname{id}_{\mathbf D} \eta H}_B
| r = \operatorname{id}_{\mathbf D} \eta_{HB}
| c = {{Defof|Double-sided Whiskered Composite of Natural Transformation|Double-sided Whiskered Composite}}
}}
{{eqn | r = \eta_{HB}
| c =... | Left Whiskered Composite of Natural Transformation is Double-sided Whiskered Composite | https://proofwiki.org/wiki/Left_Whiskered_Composite_of_Natural_Transformation_is_Double-sided_Whiskered_Composite | https://proofwiki.org/wiki/Left_Whiskered_Composite_of_Natural_Transformation_is_Double-sided_Whiskered_Composite | [
"Whiskered Composites"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Natural Transformation",
"Definition:Functor/Covariant",
"Definition:Whiskered Composite of Natural Transformation/Left Whiskered",
"Definition:Identity Functor",
"Definition:Whiskered Composite of Natural Transformation/Double-sided Whi... | [
"Category:Whiskered Composites"
] |
proofwiki-23189 | Right Whiskered Composite of Natural Transformation is Double-sided Whiskered Composite | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be categories.
Let $F, G : \mathbf C \to \mathbf D$ be covariant functors.
Let $\eta: F \to G$ be a natural transformation.
Let $K : \mathbf D \to \mathbf E$ be a covariant functor.
Let $K \eta: K \circ F \to K \circ G$ denote the right whiskered composite of $\eta$ by $K$.
... | We have:
{{begin-eqn}}
{{eqn | q = \forall C \in \mathbf C
| l = \paren{K \eta \operatorname{id}_{\mathbf C} }_C
| r = K \eta_{\operatorname{id}_{\mathbf C} C}
| c = {{Defof|Double-sided Whiskered Composite of Natural Transformation|Double-sided Whiskered Composite}}
}}
{{eqn | r = K \eta_C
| c ... | Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be [[Definition:Category|categories]].
Let $F, G : \mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\eta: F \to G$ be a [[Definition:Natural Transformation|natural transformation]].
Let $K : \mathbf D \to \mathbf E$ be a [[Definition... | We have:
{{begin-eqn}}
{{eqn | q = \forall C \in \mathbf C
| l = \paren{K \eta \operatorname{id}_{\mathbf C} }_C
| r = K \eta_{\operatorname{id}_{\mathbf C} C}
| c = {{Defof|Double-sided Whiskered Composite of Natural Transformation|Double-sided Whiskered Composite}}
}}
{{eqn | r = K \eta_C
| c ... | Right Whiskered Composite of Natural Transformation is Double-sided Whiskered Composite | https://proofwiki.org/wiki/Right_Whiskered_Composite_of_Natural_Transformation_is_Double-sided_Whiskered_Composite | https://proofwiki.org/wiki/Right_Whiskered_Composite_of_Natural_Transformation_is_Double-sided_Whiskered_Composite | [
"Whiskered Composites"
] | [
"Definition:Category",
"Definition:Functor/Covariant",
"Definition:Natural Transformation",
"Definition:Functor/Covariant",
"Definition:Whiskered Composite of Natural Transformation/Right Whiskered",
"Definition:Identity Functor",
"Definition:Whiskered Composite of Natural Transformation/Double-sided Wh... | [
"Category:Whiskered Composites"
] |
proofwiki-23190 | Characterization of Adjunction via Triangle Identities | Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be covariant functors.
Then:
:there exists an adjunction $\tuple{F, G, \alpha}$ between $\mathbf C$ and $\mathbf D$
{{iff}}:
:there exist natural transformations $\eta: \operatorname {id}_{\mathbf... | === Necessary Condition ===
Let $\tuple{F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$.
Let $\eta: \operatorname {id}_{\mathbf D} \to GF$ denote the unit of adjunction $\alpha$.
Let $\xi: FG \to \operatorname {id}_{\mathbf C}$ denote the counit of adjunction $\alpha$.
{{:Characterization of Adjunct... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Then:
:there exists an [[Definition:Adjunction|adjunction]] $\tuple{F, G, \alpha}$ between $\mathbf... | === [[Characterization of Adjunction via Triangle Identities/Necessary Condition|Necessary Condition]] ===
Let $\tuple{F, G, \alpha}$ be an [[Definition:Adjunction|adjunction]] between $\mathbf C$ and $\mathbf D$.
Let $\eta: \operatorname {id}_{\mathbf D} \to GF$ denote the [[Definition:Unit of Adjunction|unit of ad... | Characterization of Adjunction via Triangle Identities | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities | [
"Characterization of Adjunction via Triangle Identities",
"Characterizations of Adjunctions"
] | [
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Adjunction",
"Definition:Natural Transformation",
"Definition:Triangle Identities",
"Definition:Unit of Adjunction",
"Definition:Counit of Adjunction"
] | [
"Characterization of Adjunction via Triangle Identities/Necessary Condition",
"Definition:Adjunction",
"Definition:Unit of Adjunction",
"Definition:Counit of Adjunction"
] |
proofwiki-23191 | Characterization of Adjunction via Triangle Identities/Necessary Condition | Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be covariant functors.
Let $\tuple{F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$.
Let $\eta$ denote the unit of adjunction $\alpha$
Let $\xi$ denote the counit of adjunction ... | By definition of unit and counit of adjunction:
:$\eta$ and $\xi$ are natural transformations.
==== Proof of $G \xi \circ \eta G = \operatorname{id}_G$ ====
We have:
{{begin-eqn}}
{{eqn | q = \forall C \in \mathbf C
| l = \paren{G \xi \circ \eta G}_C
| r = \paren{G \xi}_C \circ \paren{\eta G}_C
| c = ... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\tuple{F, G, \alpha}$ be an [[Definition:Adjunction|adjunction]] between $\mathbf C$ and $\mat... | By definition of [[Definition:Unit of Adjunction|unit]] and [[Definition:Counit of Adjunction|counit of adjunction]]:
:$\eta$ and $\xi$ are [[Definition:Natural Transformation|natural transformations]].
==== Proof of $G \xi \circ \eta G = \operatorname{id}_G$ ====
We have:
{{begin-eqn}}
{{eqn | q = \forall C \in \ma... | Characterization of Adjunction via Triangle Identities/Necessary Condition | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities/Necessary_Condition | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities/Necessary_Condition | [
"Characterization of Adjunction via Triangle Identities"
] | [
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Adjunction",
"Definition:Unit of Adjunction",
"Definition:Counit of Adjunction",
"Definition:Whiskered Composite of Natural Transformation/Right Whiskered",
"Definition:Whiskered Composite of Natural Transformation/Left Whi... | [
"Definition:Unit of Adjunction",
"Definition:Counit of Adjunction",
"Definition:Natural Transformation",
"Characterization of Adjunction Using Right Adjuncts of Compositions",
"Definition:Object (Category Theory)",
"Definition:Inverse Mapping",
"Characterization of Adjunction Using Left Adjuncts of Comp... |
proofwiki-23192 | Characterization of Adjunction via Triangle Identities/Sufficient Condition | Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be covariant functors.
Let $\eta: \operatorname {id}_{\mathbf D} \to GF$ and $\xi: FG \to \operatorname {id}_{\mathbf C}$ be natural transformations such that:
:$G \xi \circ \eta G = \operatorname... | For objects $D \in \mathbf D$ and $C \in \mathbf C$ let:
:$\alpha_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf C} } {FD, C} \to \map {\mathrm {Hom}_{\mathbf D} } {D, GC}$
:$\beta_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf D} } {D, GC} \to \map {\mathrm {Hom}_{\mathbf C} } {FD, C}$
be the mappings defined by:
:$\foral... | Let $\mathbf C$, $\mathbf D$ be [[Definition:Locally Small Category|locally small categories]].
Let $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be [[Definition:Covariant Functor|covariant functors]].
Let $\eta: \operatorname {id}_{\mathbf D} \to GF$ and $\xi: FG \to \operatorname {id}_{\mathbf C}$ ... | For [[Definition:Object (Category Theory)|objects]] $D \in \mathbf D$ and $C \in \mathbf C$ let:
:$\alpha_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf C} } {FD, C} \to \map {\mathrm {Hom}_{\mathbf D} } {D, GC}$
:$\beta_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf D} } {D, GC} \to \map {\mathrm {Hom}_{\mathbf C} } {FD, ... | Characterization of Adjunction via Triangle Identities/Sufficient Condition | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities/Sufficient_Condition | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities/Sufficient_Condition | [
"Characterization of Adjunction via Triangle Identities"
] | [
"Definition:Locally Small Category",
"Definition:Functor/Covariant",
"Definition:Natural Transformation",
"Definition:Whiskered Composite of Natural Transformation/Right Whiskered",
"Definition:Whiskered Composite of Natural Transformation/Left Whiskered",
"Definition:Identity Natural Transformation",
"... | [
"Definition:Object (Category Theory)",
"Definition:Mapping",
"Characterization of Adjunction Using Right Adjuncts of Morphisms",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Adjunction",
"Definition:Object (Category Theory)",
"Definition:Bijection",
"Definition:Object (Categor... |
proofwiki-23193 | Characterization of Adjunction via Triangle Identities/Lemma 1 | :for objects $D \in \mathbf D$ and $C \in \mathbf C$:
::$\map {\alpha_{\tuple{D, FD} } } {\operatorname{id}_{FD} } = \eta_D$
::$\map {\beta_{\tuple{GC, C} } } {\operatorname{id}_{GC} } = \xi_C$ | We have:
{{begin-eqn}}
{{eqn | l = \map {\alpha_{\tuple{D, FD} } } {\operatorname{id}_{FD} }
| r = G \operatorname{id}_{FD} \circ \eta_D
| c = Definition of $\alpha_{\tuple{D, FD} }$
}}
{{eqn | r = \operatorname{id}_{GFD} \circ \eta_D
| c = {{Defof|Covariant Functor}}
}}
{{eqn | r = \eta_D
| c =... | :for [[Definition:Object (Category Theory)|objects]] $D \in \mathbf D$ and $C \in \mathbf C$:
::$\map {\alpha_{\tuple{D, FD} } } {\operatorname{id}_{FD} } = \eta_D$
::$\map {\beta_{\tuple{GC, C} } } {\operatorname{id}_{GC} } = \xi_C$ | We have:
{{begin-eqn}}
{{eqn | l = \map {\alpha_{\tuple{D, FD} } } {\operatorname{id}_{FD} }
| r = G \operatorname{id}_{FD} \circ \eta_D
| c = Definition of $\alpha_{\tuple{D, FD} }$
}}
{{eqn | r = \operatorname{id}_{GFD} \circ \eta_D
| c = {{Defof|Covariant Functor}}
}}
{{eqn | r = \eta_D
| c =... | Characterization of Adjunction via Triangle Identities/Lemma 1 | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities/Lemma_1 | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities/Lemma_1 | [
"Characterization of Adjunction via Triangle Identities"
] | [
"Definition:Object (Category Theory)"
] | [
"Category:Characterization of Adjunction via Triangle Identities"
] |
proofwiki-23194 | Characterization of Adjunction via Triangle Identities/Lemma 2 | :for objects $D \in \mathbf D$ and $C \in \mathbf C$:
::$\alpha_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf C} } {FD, C} \to \map {\mathrm {Hom}_{\mathbf D} } {D, GC}$ is a bijection
:with inverse:
::$\beta_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf D} } {D, GC} \to \map {\mathrm {Hom}_{\mathbf C} } {FD, C}$ | Let $D \in \mathbf D$ and $C \in \mathbf C$ be objects.
Let $g: FD \to C \in \mathbf C$ and $f: D \to GC \in \mathbf D$ be morphisms
By definition of natural transformation:
:$(1): \quad GF f \circ \eta_D = \eta_{GC} \circ f$
and:
:$(2): \quad g \circ \xi_{FD} = \xi_C \circ FG g$
We have:
{{begin-eqn}}
{{eqn | l = \map... | :for [[Definition:Object (Category Theory)|objects]] $D \in \mathbf D$ and $C \in \mathbf C$:
::$\alpha_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf C} } {FD, C} \to \map {\mathrm {Hom}_{\mathbf D} } {D, GC}$ is a [[Definition:Bijection|bijection]]
:with [[Definition:Inverse Mapping|inverse]]:
::$\beta_{\tuple{D, C}}:... | Let $D \in \mathbf D$ and $C \in \mathbf C$ be [[Definition:Object (Category Theory)|objects]].
Let $g: FD \to C \in \mathbf C$ and $f: D \to GC \in \mathbf D$ be [[Definition:Morphism (Category Theory)|morphisms]]
By definition of [[Definition:Natural Transformation|natural transformation]]:
:$(1): \quad GF f \circ... | Characterization of Adjunction via Triangle Identities/Lemma 2 | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities/Lemma_2 | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities/Lemma_2 | [
"Characterization of Adjunction via Triangle Identities"
] | [
"Definition:Object (Category Theory)",
"Definition:Bijection",
"Definition:Inverse Mapping"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Natural Transformation",
"Definition:Object (Category Theory)",
"Definition:Bijection",
"Definition:Inverse Mapping",
"Category:Characterization of Adjunction via Triangle Identities"
] |
proofwiki-23195 | Characterization of Adjunction via Triangle Identities/Lemma 3 | :for objects $D \in \mathbf D, C \in \mathbf C$ and morphism $g:FD \to C \in \mathbf C$:
::$\map {\alpha_{\tuple{D, C} } } {g} = Gg \circ \map {\alpha_{\tuple{D, FD} } } {\operatorname{id}_{FD}}$ | ==== Lemma 1 ====
{{:Characterization of Adjunction via Triangle Identities/Lemma 1}}{{qed|lemma}}
We have:
{{begin-eqn}}
{{eqn | l = \map {\alpha_{\tuple{D, C} } } {g}
| r = Gg \circ \eta_D
| c = Definition of $\alpha_{\tuple{D, C} }$
}}
{{eqn | r = Gg \circ \map {\alpha_{\tuple{D, FD} } } {\operatorname{i... | :for [[Definition:Object (Category Theory)|objects]] $D \in \mathbf D, C \in \mathbf C$ and [[Definition:Morphism (Category Theory)|morphism]] $g:FD \to C \in \mathbf C$:
::$\map {\alpha_{\tuple{D, C} } } {g} = Gg \circ \map {\alpha_{\tuple{D, FD} } } {\operatorname{id}_{FD}}$ | ==== [[Characterization of Adjunction via Triangle Identities/Lemma 1|Lemma 1]] ====
{{:Characterization of Adjunction via Triangle Identities/Lemma 1}}{{qed|lemma}}
We have:
{{begin-eqn}}
{{eqn | l = \map {\alpha_{\tuple{D, C} } } {g}
| r = Gg \circ \eta_D
| c = Definition of $\alpha_{\tuple{D, C} }$
}}
... | Characterization of Adjunction via Triangle Identities/Lemma 3 | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities/Lemma_3 | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities/Lemma_3 | [
"Characterization of Adjunction via Triangle Identities"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism"
] | [
"Characterization of Adjunction via Triangle Identities/Lemma 1",
"Characterization of Adjunction via Triangle Identities/Lemma 1",
"Category:Characterization of Adjunction via Triangle Identities"
] |
proofwiki-23196 | Characterization of Adjunction via Triangle Identities/Lemma 4 | :for objects $D_1, D_2 \in \mathbf D$ and morphism $f:D_1 \to D_2 \in \mathbf D$:
::$\map {\alpha_{\tuple{D_1, FD_2} } } {Ff} = \map {\alpha_{\tuple{D_2, FD_2} } } {\operatorname{id}_{FD_2}} \circ f$ | ==== Lemma 1 ====
{{:Characterization of Adjunction via Triangle Identities/Lemma 1}}{{qed|lemma}}
By definition of natural transformation:
:$(1): \quad GF f \circ \eta_{D_1} = \eta_{D_2} \circ f$
We have:
{{begin-eqn}}
{{eqn | l = \map {\alpha_{\tuple{D_1, FD_2} } } {Ff}
| r = GF f \circ \eta_{D_1}
| c = D... | :for [[Definition:Object (Category Theory)|objects]] $D_1, D_2 \in \mathbf D$ and [[Definition:Morphism (Category Theory)|morphism]] $f:D_1 \to D_2 \in \mathbf D$:
::$\map {\alpha_{\tuple{D_1, FD_2} } } {Ff} = \map {\alpha_{\tuple{D_2, FD_2} } } {\operatorname{id}_{FD_2}} \circ f$ | ==== [[Characterization of Adjunction via Triangle Identities/Lemma 1|Lemma 1]] ====
{{:Characterization of Adjunction via Triangle Identities/Lemma 1}}{{qed|lemma}}
By definition of [[Definition:Natural Transformation|natural transformation]]:
:$(1): \quad GF f \circ \eta_{D_1} = \eta_{D_2} \circ f$
We have:
{{beg... | Characterization of Adjunction via Triangle Identities/Lemma 4 | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities/Lemma_4 | https://proofwiki.org/wiki/Characterization_of_Adjunction_via_Triangle_Identities/Lemma_4 | [
"Characterization of Adjunction via Triangle Identities"
] | [
"Definition:Object (Category Theory)",
"Definition:Morphism"
] | [
"Characterization of Adjunction via Triangle Identities/Lemma 1",
"Definition:Natural Transformation",
"Characterization of Adjunction via Triangle Identities/Lemma 1",
"Category:Characterization of Adjunction via Triangle Identities"
] |
proofwiki-23197 | Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products | Let $\mathbf C$ be a locally small category.
Let $\mathbf C \times \mathbf C$ be the product category of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the diagonal functor.
Then $\Delta$ has a right adjoint {{iff}} $\mathbf C$ has all binary products.
Moreover, the right adjoint... | === Necessary Condition ===
{{:Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products/Necessary Condition}}{{qed|lemma}} | Let $\mathbf C$ be a [[Definition:Locally Small Category|locally small category]].
Let $\mathbf C \times \mathbf C$ be the [[Definition:Product Category|product category]] of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the [[Definition:Diagonal Functor on Product Category|di... | === [[Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products/Necessary Condition|Necessary Condition]] ===
{{:Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products/Necessary Condition}}{{qed|lemma}} | Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Right_Adjoint_Iff_Category_has_Binary_Products | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Right_Adjoint_Iff_Category_has_Binary_Products | [
"Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products",
"Diagonal Functors",
"Adjunctions",
"Products (Category Theory)"
] | [
"Definition:Locally Small Category",
"Definition:Product Category",
"Definition:Diagonal Functor/Product Category",
"Definition:Right Adjoint Functor",
"Definition:Product (Category Theory)/Binary Product",
"Definition:Right Adjoint Functor",
"Definition:Product Functor"
] | [
"Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products/Necessary Condition"
] |
proofwiki-23198 | Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts | Let $\mathbf C$ be a locally small category.
Let $\mathbf C \times \mathbf C$ be the product category of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the diagonal functor.
Then $\Delta$ has a left adjoint {{iff}} $\mathbf C$ has all binary coproducts.
Moreover, the left adjoint... | === Necessary Condition ===
{{:Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts/Necessary Condition}}{{qed|Lemma}} | Let $\mathbf C$ be a [[Definition:Locally Small Category|locally small category]].
Let $\mathbf C \times \mathbf C$ be the [[Definition:Product Category|product category]] of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the [[Definition:Diagonal Functor on Product Category|di... | === [[Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts/Necessary Condition|Necessary Condition]] ===
{{:Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts/Necessary Condition}}{{qed|Lemma}} | Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Left_Adjoint_Iff_Category_has_Binary_Coproducts | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Left_Adjoint_Iff_Category_has_Binary_Coproducts | [
"Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts",
"Diagonal Functors",
"Adjunctions",
"Coproducts"
] | [
"Definition:Locally Small Category",
"Definition:Product Category",
"Definition:Diagonal Functor/Product Category",
"Definition:Left Adjoint Functor",
"Definition:Coproduct",
"Definition:Left Adjoint Functor",
"Definition:Coproduct Functor"
] | [
"Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts/Necessary Condition"
] |
proofwiki-23199 | Euler's Continued Fraction Formula/Corollary 1 | :$\dfrac 1 A - \dfrac 1 B + \dfrac 1 C - \dfrac 1 D + \dfrac 1 E - \cdots = \cfrac 1 {A + \cfrac {A^2} {B - A + \cfrac {B^2} {C - B + \cfrac {C^2} {D - C + \cfrac {D^2} {E - D + \cfrac {\ddots} {\ddots} } } } } }$ | From Euler's Continued Fraction Formula, we have:
{{begin-eqn}}
{{eqn | l = a_0 + a_0 a_1 + a_0 a_1 a_2 + a_0 a_1 a_2 a_3 + \cdots + a_0 a_1 a_2 a_3 \cdots a_n
| r = a_0 \paren {1 + a_1 \paren {1 + a_2 \paren { 1 + a_3 \paren {\cdots + a_n } } } }
| c =
}}
{{eqn | r = \cfrac {a_0} {1 - \cfrac {a_1} {1 + a_... | :$\dfrac 1 A - \dfrac 1 B + \dfrac 1 C - \dfrac 1 D + \dfrac 1 E - \cdots = \cfrac 1 {A + \cfrac {A^2} {B - A + \cfrac {B^2} {C - B + \cfrac {C^2} {D - C + \cfrac {D^2} {E - D + \cfrac {\ddots} {\ddots} } } } } }$ | From [[Euler's Continued Fraction Formula]], we have:
{{begin-eqn}}
{{eqn | l = a_0 + a_0 a_1 + a_0 a_1 a_2 + a_0 a_1 a_2 a_3 + \cdots + a_0 a_1 a_2 a_3 \cdots a_n
| r = a_0 \paren {1 + a_1 \paren {1 + a_2 \paren { 1 + a_3 \paren {\cdots + a_n } } } }
| c =
}}
{{eqn | r = \cfrac {a_0} {1 - \cfrac {a_1} {1... | Euler's Continued Fraction Formula/Corollary 1 | https://proofwiki.org/wiki/Euler's_Continued_Fraction_Formula/Corollary_1 | https://proofwiki.org/wiki/Euler's_Continued_Fraction_Formula/Corollary_1 | [
"Euler's Continued Fraction Formula"
] | [] | [
"Euler's Continued Fraction Formula"
] |
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