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 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-20300 | Locally Euclidean Space is Locally Path-Connected | Let $M$ be a locally Euclidean space of some dimension $d$.
Then $M$ is locally path-connected. | Let $m \in M$ be arbitrary.
From Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls:
:there exists a local basis $\family{U_n}_{n \in \N}$ of $m$ where each $U_n$ is the homeomorphic image of an open ball of $\R^d$.
For all $n \in \N$, let:
:$U_n = \phi_n \sqbrk {B_n}$
where $B_n$ is an open b... | Let $M$ be a [[Definition:Locally Euclidean Space|locally Euclidean space]] of some dimension $d$.
Then $M$ is [[Definition:Locally Path-Connected Space|locally path-connected]]. | Let $m \in M$ be arbitrary.
From [[Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls]]:
:there exists a [[Definition:Local Basis|local basis]] $\family{U_n}_{n \in \N}$ of $m$ where each $U_n$ is the [[Definition:Homeomorphism|homeomorphic]] [[Definition:Image|image]] of an [[Definition:Ope... | Locally Euclidean Space is Locally Path-Connected | https://proofwiki.org/wiki/Locally_Euclidean_Space_is_Locally_Path-Connected | https://proofwiki.org/wiki/Locally_Euclidean_Space_is_Locally_Path-Connected | [
"Locally Euclidean Spaces",
"Path-Connected Spaces"
] | [
"Definition:Locally Euclidean Space",
"Definition:Locally Path-Connected Space"
] | [
"Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls",
"Definition:Local Basis",
"Definition:Homeomorphism",
"Definition:Image",
"Definition:Open Ball",
"Definition:Open Ball",
"Definition:Homeomorphism",
"Open Ball in Normed Vector Space is Path-Connected",
"Definition:Path... |
proofwiki-20301 | Locally Euclidean Space is First-Countable | Let $M$ be a locally Euclidean space of some dimension $d$.
Then $M$ is first-countable. | Follows immediately from Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls.
{{qed}}
Category:Locally Euclidean Spaces
Category:First-Countable Spaces
39t7sdcwcgxewea5urhcbnimoh6crb4 | Let $M$ be a [[Definition:Locally Euclidean Space|locally Euclidean space]] of some dimension $d$.
Then $M$ is [[Definition:First-Countable Space|first-countable]]. | Follows immediately from [[Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls]].
{{qed}}
[[Category:Locally Euclidean Spaces]]
[[Category:First-Countable Spaces]]
39t7sdcwcgxewea5urhcbnimoh6crb4 | Locally Euclidean Space is First-Countable | https://proofwiki.org/wiki/Locally_Euclidean_Space_is_First-Countable | https://proofwiki.org/wiki/Locally_Euclidean_Space_is_First-Countable | [
"Locally Euclidean Spaces",
"First-Countable Spaces"
] | [
"Definition:Locally Euclidean Space",
"Definition:First-Countable Space"
] | [
"Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls",
"Category:Locally Euclidean Spaces",
"Category:First-Countable Spaces"
] |
proofwiki-20302 | Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints/Complex Plane | Let $\C$ denote the complex plane. | By definition of reparameterization, there exists a bijective differentiable strictly increasing real function $\phi: \closedint c d \closedint a b$ such that $\sigma = \gamma \circ \phi$.
As $\map {\phi^{-1} }{a} \in \closedint c d$:
: $c \le \map {\phi^{-1} }{a}$
As $\phi$ is strictly increasing:
: $\map \phi c \le \... | Let $\C$ denote the [[Definition:Complex Plane|complex plane]]. | By definition of [[Definition:Directed Smooth Curve/Parameterization/Complex Plane/Reparameterization|reparameterization]], there exists a [[Definition:Bijection|bijective]] [[Definition:Differentiable on Interval|differentiable]] [[Definition:Strictly Increasing Real Function|strictly increasing]] [[Definition:Real Fu... | Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints/Complex Plane | https://proofwiki.org/wiki/Reparameterization_of_Directed_Smooth_Curve_Maps_Endpoints_To_Endpoints/Complex_Plane | https://proofwiki.org/wiki/Reparameterization_of_Directed_Smooth_Curve_Maps_Endpoints_To_Endpoints/Complex_Plane | [
"Directed Smooth Curves (Complex Plane)"
] | [
"Definition:Complex Number/Complex Plane"
] | [
"Definition:Directed Smooth Curve/Parameterization/Complex Plane/Reparameterization",
"Definition:Bijection",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Strictly Increasing/Real Function",
"Definition:Real Function",
"Definition:Strictly Increasing/Real Function",
"Definitio... |
proofwiki-20303 | Arcsine in terms of Arctangent | :$\arcsin x = \map \arctan {\dfrac x {\sqrt {1 - x^2} } }$ | Let:
:$\theta = \arcsin x$
Then by the definition of arcsine:
:$x = \sin \theta$
and:
:$-\dfrac \pi 2 < \theta < \dfrac \pi 2$
Then:
{{begin-eqn}}
{{eqn | l = \map \arctan {\dfrac x {\sqrt {1 - x^2} } }
| r = \map \arctan {\dfrac {\sin \theta} {\sqrt {1 - \sin^2 \theta} } }
}}
{{eqn | r = \map \arctan {\dfrac ... | :$\arcsin x = \map \arctan {\dfrac x {\sqrt {1 - x^2} } }$ | Let:
:$\theta = \arcsin x$
Then by the definition of [[Definition:Real Arcsine|arcsine]]:
:$x = \sin \theta$
and:
:$-\dfrac \pi 2 < \theta < \dfrac \pi 2$
Then:
{{begin-eqn}}
{{eqn | l = \map \arctan {\dfrac x {\sqrt {1 - x^2} } }
| r = \map \arctan {\dfrac {\sin \theta} {\sqrt {1 - \sin^2 \theta} } }
}}
... | Arcsine in terms of Arctangent | https://proofwiki.org/wiki/Arcsine_in_terms_of_Arctangent | https://proofwiki.org/wiki/Arcsine_in_terms_of_Arctangent | [
"Arcsine Function",
"Arctangent Function"
] | [] | [
"Definition:Inverse Sine/Real/Arcsine",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-20304 | Arctangent in terms of Arcsine | :$\arctan x = \map \arcsin {\dfrac x {\sqrt {1 + x^2} } }$ | Let:
:$\theta = \arctan x$
Then by the definition of arctangent:
:$x = \tan \theta$
Then:
{{begin-eqn}}
{{eqn | l = \map \arcsin { \dfrac x {\sqrt {1 + x^2} } }
| r = \map \arcsin { \dfrac {\tan \theta} {\sqrt {1 + \tan^2 \theta} } }
}}
{{eqn | r = \map \arcsin { \dfrac {\tan \theta} {\sqrt {\sec^2 \theta} } ... | :$\arctan x = \map \arcsin {\dfrac x {\sqrt {1 + x^2} } }$ | Let:
:$\theta = \arctan x$
Then by the definition of [[Definition:Real Arctangent|arctangent]]:
:$x = \tan \theta$
Then:
{{begin-eqn}}
{{eqn | l = \map \arcsin { \dfrac x {\sqrt {1 + x^2} } }
| r = \map \arcsin { \dfrac {\tan \theta} {\sqrt {1 + \tan^2 \theta} } }
}}
{{eqn | r = \map \arcsin { \dfrac {\ta... | Arctangent in terms of Arcsine/Proof 1 | https://proofwiki.org/wiki/Arctangent_in_terms_of_Arcsine | https://proofwiki.org/wiki/Arctangent_in_terms_of_Arcsine/Proof_1 | [
"Arctangent in terms of Arcsine",
"Arcsine Function",
"Arctangent Function"
] | [] | [
"Definition:Inverse Tangent/Real/Arctangent",
"Sum of Squares of Sine and Cosine/Corollary 1"
] |
proofwiki-20305 | Arctangent in terms of Arcsine | :$\arctan x = \map \arcsin {\dfrac x {\sqrt {1 + x^2} } }$ | From Pfaff's Transformation:
:$\ds \map F {a, b; c; x} = \paren {1 - x}^{-a} \map F {a, c - b; c; \dfrac x {x - 1} }$
where $\map F {a, b; c; x}$ is the Gaussian hypergeometric function of $x$.
We have:
{{begin-eqn}}
{{eqn | l = \map \arctan x
| r = x \map F {\dfrac 1 2, 1; \dfrac 3 2; -x^2}
| c = Arctangen... | :$\arctan x = \map \arcsin {\dfrac x {\sqrt {1 + x^2} } }$ | From [[Pfaff's Transformation]]:
:$\ds \map F {a, b; c; x} = \paren {1 - x}^{-a} \map F {a, c - b; c; \dfrac x {x - 1} }$
where $\map F {a, b; c; x}$ is the [[Definition:Gaussian Hypergeometric Function|Gaussian hypergeometric function]] of $x$.
We have:
{{begin-eqn}}
{{eqn | l = \map \arctan x
| r = x \map ... | Arctangent in terms of Arcsine/Proof 2 | https://proofwiki.org/wiki/Arctangent_in_terms_of_Arcsine | https://proofwiki.org/wiki/Arctangent_in_terms_of_Arcsine/Proof_2 | [
"Arctangent in terms of Arcsine",
"Arcsine Function",
"Arctangent Function"
] | [] | [
"Pfaff's Transformation",
"Definition:Hypergeometric Function/Gaussian",
"Arctangent Function in terms of Gaussian Hypergeometric Function",
"Pfaff's Transformation",
"Arcsine Function in terms of Gaussian Hypergeometric Function"
] |
proofwiki-20306 | Derivative of Gaussian Hypergeometric Function | :$\map {\dfrac \d {\d x} } {\map F {a, b; c; x} } = \dfrac {a b} c \map F {a + 1, b + 1; c + 1; x} $ | {{begin-eqn}}
{{eqn | l = \map F {a, b; c; x}
| r = 1 + \dfrac {a b} c x + \dfrac {a \paren {a + 1} b \paren {b + 1} } {c \paren {c + 1} } \dfrac {x^2} {2!} + \cdots + \dfrac {a^{\overline k} b^{\overline k} } {c^{\overline k} } \dfrac {x^k} {k!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | l... | :$\map {\dfrac \d {\d x} } {\map F {a, b; c; x} } = \dfrac {a b} c \map F {a + 1, b + 1; c + 1; x} $ | {{begin-eqn}}
{{eqn | l = \map F {a, b; c; x}
| r = 1 + \dfrac {a b} c x + \dfrac {a \paren {a + 1} b \paren {b + 1} } {c \paren {c + 1} } \dfrac {x^2} {2!} + \cdots + \dfrac {a^{\overline k} b^{\overline k} } {c^{\overline k} } \dfrac {x^k} {k!}
| c = {{Defof|Gaussian Hypergeometric Function}}
}}
{{eqn | l... | Derivative of Gaussian Hypergeometric Function | https://proofwiki.org/wiki/Derivative_of_Gaussian_Hypergeometric_Function | https://proofwiki.org/wiki/Derivative_of_Gaussian_Hypergeometric_Function | [
"Gaussian Hypergeometric Function",
"Derivatives"
] | [] | [
"Derivative of Power of Function"
] |
proofwiki-20307 | Convex Real Function Composed with Increasing Convex Real Function is Convex | Let $I$ be real interval.
Let $f : I \to \R$ be a convex function.
Let $J$ be a real interval containing the image of $f$.
Let $g : J \to \R$ be a increasing convex function.
Then $g \circ f : I \to \R$ is a convex function. | Let $x, y \in I$ and $t \in \closedint 0 1$.
Since $f$ is convex, we have:
:$\map f {t x + \paren {1 - t} y} \le t \map f x + \paren {1 - t} \map f y$
Since $g$ is increasing, we then have:
:$\map {\paren {g \circ f} } {t x + \paren {1 - t} y} \le \map g {t \map f x + \paren {1 - t} \map f y}$
Since $g$ is convex, w... | Let $I$ be [[Definition:Real Interval|real interval]].
Let $f : I \to \R$ be a [[Definition:Convex Real Function|convex function]].
Let $J$ be a [[Definition:Real Interval|real interval]] containing the [[Definition:Image of Mapping|image]] of $f$.
Let $g : J \to \R$ be a [[Definition:Increasing Real Function|increa... | Let $x, y \in I$ and $t \in \closedint 0 1$.
Since $f$ is [[Definition:Convex Real Function|convex]], we have:
:$\map f {t x + \paren {1 - t} y} \le t \map f x + \paren {1 - t} \map f y$
Since $g$ is [[Definition:Increasing Real Function|increasing]], we then have:
:$\map {\paren {g \circ f} } {t x + \paren {1 -... | Convex Real Function Composed with Increasing Convex Real Function is Convex | https://proofwiki.org/wiki/Convex_Real_Function_Composed_with_Increasing_Convex_Real_Function_is_Convex | https://proofwiki.org/wiki/Convex_Real_Function_Composed_with_Increasing_Convex_Real_Function_is_Convex | [
"Convex Real Functions"
] | [
"Definition:Real Interval",
"Definition:Convex Real Function",
"Definition:Real Interval",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Increasing/Real Function",
"Definition:Convex Real Function",
"Definition:Convex Real Function"
] | [
"Definition:Convex Real Function",
"Definition:Increasing/Real Function",
"Definition:Convex Real Function",
"Definition:Convex Real Function",
"Category:Convex Real Functions"
] |
proofwiki-20308 | Power Function is Convex Real Function | Let $p \ge 1$ be a real number.
Define $f : \hointr 0 \infty \to \hointr 0 \infty$ by:
:$\map f x = x^p$
for each $x \in \hointr 0 \infty$.
Then $f$ is a convex function. | Applying Derivative of Power twice, we have that:
:$f$ is twice differentiable on $\openint 0 \infty$
with:
:$\map {f' '} x = p \paren {p - 1} x^{p - 2}$
for each $x \in \openint 0 \infty$.
Since $p \ge 1$, we have:
:$p \paren {p - 1} \ge 0$
and so:
:$\map {f' '} x \ge 0$
for each $x \in \openint 0 \infty$.
From Real... | Let $p \ge 1$ be a [[Definition:Real Number|real number]].
Define $f : \hointr 0 \infty \to \hointr 0 \infty$ by:
:$\map f x = x^p$
for each $x \in \hointr 0 \infty$.
Then $f$ is a [[Definition:Convex Real Function|convex function]]. | Applying [[Derivative of Power]] twice, we have that:
:$f$ is twice [[Definition:Differentiable Real Function|differentiable]] on $\openint 0 \infty$
with:
:$\map {f' '} x = p \paren {p - 1} x^{p - 2}$
for each $x \in \openint 0 \infty$.
Since $p \ge 1$, we have:
:$p \paren {p - 1} \ge 0$
and so:
:$\map {f' '... | Power Function is Convex Real Function | https://proofwiki.org/wiki/Power_Function_is_Convex_Real_Function | https://proofwiki.org/wiki/Power_Function_is_Convex_Real_Function | [
"Convex Real Functions"
] | [
"Definition:Real Number",
"Definition:Convex Real Function"
] | [
"Power Rule for Derivatives",
"Definition:Differentiable Mapping/Real Function",
"Real Function with Positive Derivative is Increasing",
"Definition:Increasing/Real Function",
"Real Function is Convex iff Derivative is Increasing",
"Definition:Convex Real Function",
"Definition:Convex Real Function",
... |
proofwiki-20309 | Power of Absolute Value is Convex Real Function | Let $p \ge 1$ be a real number.
Define $f : \R \to \R$ by:
:$\map f x = {\size x}^p$
for each $x \in \R$.
Then $f$ is a convex function. | From Absolute Value Function is Convex:
:$x \mapsto \size x$ is a convex function.
Note now that:
:$x \mapsto x^p$ is increasing on $\hointr 0 \infty$
From Power Function is Convex Real Function, we also have:
:$x \mapsto x^p$ is convex.
Since $f$ is the composition of the maps $x \mapsto \size x$ and $x \mapsto x^p... | Let $p \ge 1$ be a [[Definition:Real Number|real number]].
Define $f : \R \to \R$ by:
:$\map f x = {\size x}^p$
for each $x \in \R$.
Then $f$ is a [[Definition:Convex Real Function|convex function]]. | From [[Absolute Value Function is Convex]]:
:$x \mapsto \size x$ is a [[Definition:Convex Real Function|convex function]].
Note now that:
:$x \mapsto x^p$ is [[Definition:Increasing Real Function|increasing]] on $\hointr 0 \infty$
From [[Power Function is Convex Real Function]], we also have:
:$x \mapsto x^p$ i... | Power of Absolute Value is Convex Real Function | https://proofwiki.org/wiki/Power_of_Absolute_Value_is_Convex_Real_Function | https://proofwiki.org/wiki/Power_of_Absolute_Value_is_Convex_Real_Function | [
"Convex Real Functions"
] | [
"Definition:Real Number",
"Definition:Convex Real Function"
] | [
"Absolute Value Function is Convex",
"Definition:Convex Real Function",
"Definition:Increasing/Real Function",
"Power Function is Convex Real Function",
"Definition:Convex Real Function",
"Definition:Convex Real Function",
"Convex Real Function Composed with Increasing Convex Real Function is Convex",
... |
proofwiki-20310 | Components of Vector between two Points | Let $A, B$ be points in the Euclidean space $\R^n$.
Let their Cartesian coordinates be given by:
{{begin-eqn}}
{{eqn | l = A
| r = \tuple {a_1 , a_2, \ldots, a_n }
}}
{{eqn | l = B
| r = \tuple {b_1 , b_2, \ldots, b_n }
}}
{{end-eqn}}
Let $\vec {AB}$ be the vector quantity that represents the directed line ... | Let $O$ denote the origin of $\R^n$.
Let $\vec {OA}$ and $\vec {OB}$ be the positions vectors of $A$ and $B$.
By definition of positions vectors, their components are:
{{begin-eqn}}
{{eqn | l = \vec {OA}
| r = \tuple {a_1 , a_2, \ldots, a_n }
}}
{{eqn | l = \vec {OB}
| r = \tuple {b_1 , b_2, \ldots, b_n }
}... | Let $A, B$ be [[Definition:Point|points]] in the [[Definition:Real Euclidean Space|Euclidean space]] $\R^n$.
Let their [[Definition:Cartesian Coordinate System|Cartesian coordinates]] be given by:
{{begin-eqn}}
{{eqn | l = A
| r = \tuple {a_1 , a_2, \ldots, a_n }
}}
{{eqn | l = B
| r = \tuple {b_1 , b_2, ... | Let $O$ denote the [[Definition:Origin|origin]] of $\R^n$.
Let $\vec {OA}$ and $\vec {OB}$ be the [[Definition:Position Vector|positions vectors]] of $A$ and $B$.
By definition of [[Definition:Position Vector|positions vectors]], their [[Definition:Vector Component|components]] are:
{{begin-eqn}}
{{eqn | l = \vec {O... | Components of Vector between two Points | https://proofwiki.org/wiki/Components_of_Vector_between_two_Points | https://proofwiki.org/wiki/Components_of_Vector_between_two_Points | [
"Vectors"
] | [
"Definition:Point",
"Definition:Euclidean Space/Real",
"Definition:Cartesian Coordinate System",
"Definition:Vector Quantity",
"Definition:Directed Line Segment",
"Definition:Line/Endpoint",
"Definition:Line/Endpoint",
"Definition:Vector Quantity/Component"
] | [
"Definition:Coordinate System/Origin",
"Definition:Position Vector",
"Definition:Position Vector",
"Definition:Vector Quantity/Component",
"Definition:Vector Quantity/Component",
"Definition:Vector Sum/Triangle Law",
"Definition:Vector Quantity/Arrow Representation",
"Equivalence of Definitions of Vec... |
proofwiki-20311 | Banach Algebra with Unity is Unital Banach Algebra | Let $\mathbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a non-trivial Banach algebra over $\mathbb F$.
Suppose that $X$ has an identity element $\mathbf 1_X$.
Then there exists a norm $\norm \cdot '$ on $X$ equivalent to $\norm \cdot$ such that $\struct {X, \norm \cdot '}$ is a unital Banach algebra.
That ... | Define $\norm \cdot' : X \to \closedint 0 \infty$ by:
:$\norm a' = \sup \set {\norm {a b} : \norm b \le 1}$
for each $a \in X$.
Note that for each $a, b \in X$ with $\norm b \le 1$, we have:
:$\norm {a b} \le \norm a \norm b \le \norm a$
so that:
:$\norm a' \in \hointr 0 \infty$
for each $a \in X$.
We now verify... | Let $\mathbb F \in \set {\R, \C}$.
Let $\struct {X, \norm \cdot}$ be a non-[[Definition:Trivial Vector Space|trivial]] [[Definition:Banach Algebra|Banach algebra]] over $\mathbb F$.
Suppose that $X$ has an [[Definition:Identity Element|identity element]] $\mathbf 1_X$.
Then there exists a [[Definition:Norm on Vecto... | Define $\norm \cdot' : X \to \closedint 0 \infty$ by:
:$\norm a' = \sup \set {\norm {a b} : \norm b \le 1}$
for each $a \in X$.
Note that for each $a, b \in X$ with $\norm b \le 1$, we have:
:$\norm {a b} \le \norm a \norm b \le \norm a$
so that:
:$\norm a' \in \hointr 0 \infty$
for each $a \in X$.
We n... | Banach Algebra with Unity is Unital Banach Algebra | https://proofwiki.org/wiki/Banach_Algebra_with_Unity_is_Unital_Banach_Algebra | https://proofwiki.org/wiki/Banach_Algebra_with_Unity_is_Unital_Banach_Algebra | [
"Unital Banach Algebras",
"Banach Algebras",
"Unital Banach Algebras"
] | [
"Definition:Trivial Vector Space",
"Definition:Banach Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Norm/Vector Space",
"Definition:Equivalence of Norms",
"Definition:Unital Banach Algebra",
"Definition:Equivalence of Norms",
"Definition:Banach Algebra"
] | [
"Definition:Norm/Vector Space",
"Multiple of Supremum",
"Definition:Supremum of Set/Real Numbers",
"Definition:Norm/Vector Space",
"Definition:Equivalence of Norms",
"Definition:Equivalence of Norms",
"Norm Equivalence Preserves Completeness",
"Definition:Banach Space",
"Definition:Banach Algebra",
... |
proofwiki-20312 | Regular Polygon is composed of Isosceles Triangles | Let $P$ be a regular $n$-gon.
Let $O$ be the center of $P$.
Then there exists a triangulation of $P$ into $n$ congruent isosceles triangles.
The three vertices of each triangle are $O$ and two adjacent vertices of $P$. | From Regular Polygon is Cyclic, it follows that $O$ is the center of the circumcircle of $P$.
Let $V_1 , V_2$ be adjacent vertices of $P$.
By definition of circumcircle, $V_1 O$ and $V_2 O$ are radii of the circumcircle.
As $V_1 O$ and $V_2 O$ have the same length, it follows that $\triangle V_1 V_2 O$ is an isosceles ... | Let $P$ be a [[Definition:Regular Polygon|regular]] [[Definition:N-Gon|$n$-gon]].
Let $O$ be the [[Definition:Center of Regular Polygon|center]] of $P$.
Then there exists a [[Definition:Triangulation of Polygon|triangulation]] of $P$ into $n$ [[Definition:Congruence (Geometry)|congruent]] [[Definition:Isosceles Tria... | From [[Regular Polygon is Cyclic]], it follows that $O$ is the [[Definition:Center of Circle|center]] of the [[Definition:Circumcircle|circumcircle]] of $P$.
Let $V_1 , V_2$ be [[Definition:Adjacent Vertices of Polygon|adjacent vertices]] of $P$.
By definition of [[Definition:Circumcircle|circumcircle]], $V_1 O$ and ... | Regular Polygon is composed of Isosceles Triangles | https://proofwiki.org/wiki/Regular_Polygon_is_composed_of_Isosceles_Triangles | https://proofwiki.org/wiki/Regular_Polygon_is_composed_of_Isosceles_Triangles | [
"Regular Polygons",
"Isosceles Triangles"
] | [
"Definition:Polygon/Regular",
"Definition:Polygon/Multilateral",
"Definition:Polygon/Regular/Center",
"Definition:Triangulation of Polygon",
"Definition:Congruence (Geometry)",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Polygon/Vertex",
"Definition:Triangle (Geometry)",
"Definition:Poly... | [
"Regular Polygon is Cyclic",
"Definition:Circle/Center",
"Definition:Circumcircle",
"Definition:Polygon/Adjacent/Vertices",
"Definition:Circumcircle",
"Definition:Circle/Radius",
"Definition:Circumcircle",
"Definition:Linear Measure/Length",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:... |
proofwiki-20313 | Regular Hexagon is composed of Equilateral Triangles | Let $P$ be a regular hexagon.
Let $O$ be the center of $P$.
Then there exists a triangulation of $P$ into six congruent equilateral triangles.
The three vertices of each triangle are $O$ and two adjacent vertices of $P$. | Let $V_2$ be a vertex of $P$.
Let $V_1, V_3$ be the two adjacent vertices of $V_1$.
From Regular Polygon is composed of Isosceles Triangles, it follows that there exists a triangulation of $P$ into six congruent isosceles triangles of the type $V_1 V_2 O$.
As $\angle V_1 V_2 O$ and $\angle V_3 V_2 O$ are corresponding ... | Let $P$ be a [[Definition:Regular Hexagon|regular hexagon]].
Let $O$ be the [[Definition:Center of Regular Polygon|center]] of $P$.
Then there exists a [[Definition:Triangulation of Polygon|triangulation]] of $P$ into six [[Definition:Congruence (Geometry)|congruent]] [[Definition:Equilateral Triangle|equilateral tr... | Let $V_2$ be a [[Definition:Vertex of Polygon|vertex]] of $P$.
Let $V_1, V_3$ be the two [[Definition:Adjacent Vertices of Polygon|adjacent vertices]] of $V_1$.
From [[Regular Polygon is composed of Isosceles Triangles]], it follows that there exists a [[Definition:Triangulation of Polygon|triangulation]] of $P$ into... | Regular Hexagon is composed of Equilateral Triangles | https://proofwiki.org/wiki/Regular_Hexagon_is_composed_of_Equilateral_Triangles | https://proofwiki.org/wiki/Regular_Hexagon_is_composed_of_Equilateral_Triangles | [
"Hexagons",
"Equilateral Triangles"
] | [
"Definition:Hexagon/Regular",
"Definition:Polygon/Regular/Center",
"Definition:Triangulation of Polygon",
"Definition:Congruence (Geometry)",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Polygon/Vertex",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Adjacent/Vertices"
] | [
"Definition:Polygon/Vertex",
"Definition:Polygon/Adjacent/Vertices",
"Regular Polygon is composed of Isosceles Triangles",
"Definition:Triangulation of Polygon",
"Definition:Congruence (Geometry)",
"Definition:Triangle (Geometry)/Isosceles",
"Definition:Polygon/Internal Angle",
"Definition:Congruence ... |
proofwiki-20314 | Morley's Formula | Let $n \in \C$ be a complex number.
Let $\map \Re n < \dfrac 2 3$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^\infty \paren {\dfrac {n^{\overline k} } {k!} }^3
| r = \dfrac {\map \Gamma {1 - \dfrac {3 n} 2} } {\map {\Gamma^3} {1 - \dfrac n 2} } \map \cos {\dfrac {\pi n} 2}
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^\infty \paren {\dfrac {n^{\overline k} } {k!} }^3
| r = \dfrac {6 \map \sin {\dfrac {\pi n} 2} \map \sin {\pi n} } {\pi^2 n^2 \paren {1 + 2 \map \cos {\pi n} } } \times \dfrac {\map {\Gamma^3} {\dfrac n 2 + 1} } {\map \Gamma {\dfrac {3 n} 2 + 1} }
| c = {{Corol... | Let $n \in \C$ be a [[Definition:Complex Number|complex number]].
Let $\map \Re n < \dfrac 2 3$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^\infty \paren {\dfrac {n^{\overline k} } {k!} }^3
| r = \dfrac {\map \Gamma {1 - \dfrac {3 n} 2} } {\map {\Gamma^3} {1 - \dfrac n 2} } \map \cos {\dfrac {\pi n} ... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^\infty \paren {\dfrac {n^{\overline k} } {k!} }^3
| r = \dfrac {6 \map \sin {\dfrac {\pi n} 2} \map \sin {\pi n} } {\pi^2 n^2 \paren {1 + 2 \map \cos {\pi n} } } \times \dfrac {\map {\Gamma^3} {\dfrac n 2 + 1} } {\map \Gamma {\dfrac {3 n} 2 + 1} }
| c = {{Corol... | Morley's Formula | https://proofwiki.org/wiki/Morley's_Formula | https://proofwiki.org/wiki/Morley's_Formula | [
"Hypergeometric Functions",
"Gamma Function"
] | [
"Definition:Complex Number"
] | [
"Euler's Reflection Formula",
"Sine of Integer Multiple of Argument/Formulation 6",
"Double Angle Formulas/Sine"
] |
proofwiki-20315 | Equivalence of Formulations of Axiom of Choice/Formulation 1 implies Formulation 2 | The following formulation of the Axiom of Choice: | Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of sets all of which are non-empty, indexed by $I$ which is also non-empty.
By hypothesis, Formulation 1 holds.
That is, there exists a choice function on every set of non-empty sets.
Let $f$ be a choice function on $\set{X_i}$.
Let $x_i = \map f {X_i}$.
By def... | The following formulation of the [[Axiom:Axiom of Choice|Axiom of Choice]]: | Let $\family {X_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family of Sets|indexed family of sets]] all of which are [[Definition:Non-Empty Set|non-empty]], indexed by $I$ which is also [[Definition:Non-Empty Set|non-empty]].
By hypothesis, [[Axiom:Axiom of Choice/Formulation 1|Formulation 1]] holds.
That is, th... | Equivalence of Formulations of Axiom of Choice/Formulation 1 implies Formulation 2 | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Choice/Formulation_1_implies_Formulation_2 | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Choice/Formulation_1_implies_Formulation_2 | [
"Equivalence of Formulations of Axiom of Choice"
] | [
"Axiom:Axiom of Choice",
"Axiom:Axiom of Choice"
] | [
"Definition:Indexing Set/Family of Sets",
"Definition:Non-Empty Set",
"Definition:Non-Empty Set",
"Axiom:Axiom of Choice/Formulation 1",
"Definition:Choice Function",
"Definition:Set of Sets",
"Definition:Non-Empty Set",
"Definition:Choice Function",
"Definition:Choice Function",
"Axiom:Axiom of C... |
proofwiki-20316 | Kummer's Quadratic Transformation | :$\ds \map F {a, b; 1 + a - b; z} = \paren {1 - z}^{-a} \map F {\dfrac a 2, \dfrac {1 + a} 2 - b; 1 + a - b; \dfrac {-4 z} {\paren {1 - z}^2} }$ | On the {{RHS}}, our $z$ variable transforms to $\dfrac {-4 z} {\paren {1 - z}^2}$, therefore:
{{begin-eqn}}
{{eqn | l = \size {\dfrac {-4 z} {\paren {1 - z}^2} }
| o = <
| r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = 4 z
| o = <
| r = \paren {1 - z}^2
| c =
}}
{{eqn | ll= \leads... | :$\ds \map F {a, b; 1 + a - b; z} = \paren {1 - z}^{-a} \map F {\dfrac a 2, \dfrac {1 + a} 2 - b; 1 + a - b; \dfrac {-4 z} {\paren {1 - z}^2} }$ | On the {{RHS}}, our $z$ variable transforms to $\dfrac {-4 z} {\paren {1 - z}^2}$, therefore:
{{begin-eqn}}
{{eqn | l = \size {\dfrac {-4 z} {\paren {1 - z}^2} }
| o = <
| r = 1
| c =
}}
{{eqn | ll= \leadsto
| l = 4 z
| o = <
| r = \paren {1 - z}^2
| c =
}}
{{eqn | ll= \leads... | Kummer's Quadratic Transformation | https://proofwiki.org/wiki/Kummer's_Quadratic_Transformation | https://proofwiki.org/wiki/Kummer's_Quadratic_Transformation | [
"Kummer's Quadratic Transformation",
"Gaussian Hypergeometric Function",
"Gamma Function"
] | [] | [
"Solution to Quadratic Equation",
"Definition:Analytic Function/Complex Plane",
"Definition:Hypergeometric Function/Gaussian",
"Binomial Theorem/Extended",
"Negated Upper Index of Binomial Coefficient",
"Definition:Coefficient",
"Rising Factorial as Quotient of Factorials",
"Legendre's Duplication For... |
proofwiki-20317 | Element of Unital Banach Algebra Close to Identity is Invertible | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {A, \norm \cdot}$ be a unital Banach algebra over $\Bbb F$ with identity element $\mathbf 1_A$.
Let $a \in A$ be such that:
:$\norm {\mathbf 1_A - a} < 1$
Then $a$ is invertible with inverse element $a^{-1}$ satisfying:
:$\ds \norm {a^{-1} } \le \frac 1 {1 - \norm {\mat... | Let:
:$x = \mathbf 1_A - a$
From Bound on Norm of Power of Element in Normed Algebra, we have:
:$\norm {x^n} \le \norm x^n$
for each $n \in \Z_{\ge 0}$.
Then we have:
:$\ds \sum_{n \mathop = 0}^\infty \norm {x^n} \le \sum_{n \mathop = 0}^\infty \norm x^n$
Since $\norm x < 1$, we have:
:$\ds \sum_{n \mathop = 0}^\in... | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {A, \norm \cdot}$ be a [[Definition:Unital Banach Algebra|unital Banach algebra]] over $\Bbb F$ with [[Definition:Identity Element|identity element]] $\mathbf 1_A$.
Let $a \in A$ be such that:
:$\norm {\mathbf 1_A - a} < 1$
Then $a$ is [[Definition:Invertible Elemen... | Let:
:$x = \mathbf 1_A - a$
From [[Bound on Norm of Power of Element in Normed Algebra]], we have:
:$\norm {x^n} \le \norm x^n$
for each $n \in \Z_{\ge 0}$.
Then we have:
:$\ds \sum_{n \mathop = 0}^\infty \norm {x^n} \le \sum_{n \mathop = 0}^\infty \norm x^n$
Since $\norm x < 1$, we have:
:$\ds \sum_{n \mat... | Element of Unital Banach Algebra Close to Identity is Invertible | https://proofwiki.org/wiki/Element_of_Unital_Banach_Algebra_Close_to_Identity_is_Invertible | https://proofwiki.org/wiki/Element_of_Unital_Banach_Algebra_Close_to_Identity_is_Invertible | [
"Unital Banach Algebras"
] | [
"Definition:Unital Banach Algebra",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Invertible Element",
"Definition:Inverse (Abstract Algebra)/Inverse"
] | [
"Bound on Norm of Power of Element in Normed Algebra",
"Definition:Convergent Series",
"Sum of Infinite Geometric Sequence",
"Definition:Convergent Series",
"Definition:Banach Space",
"Definition:Convergent Series",
"Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach",... |
proofwiki-20318 | Real Cosine Function has Zeroes | :$\exists \theta \in \R: \map \cos \theta = 0 \text { and } \map \cos {-\theta} = 0$ | {{AimForCont}} $\cos x$ is positive everywhere on $\R$.
From Derivative of Cosine Function and Derivative of Sine Function:
:$\map {\dfrac {\d^2} {\d x^2} } {\cos x} = \map {\dfrac \d {\d x} } {-\sin x} = -\cos x$
Thus $\map {\dfrac {\d^2} {\d x^2} } {\cos x} = -\cos x$ would always be negative.
Thus from Second Deriva... | :$\exists \theta \in \R: \map \cos \theta = 0 \text { and } \map \cos {-\theta} = 0$ | {{AimForCont}} $\cos x$ is [[Definition:Positive Real Function|positive]] everywhere on $\R$.
From [[Derivative of Cosine Function]] and [[Derivative of Sine Function]]:
:$\map {\dfrac {\d^2} {\d x^2} } {\cos x} = \map {\dfrac \d {\d x} } {-\sin x} = -\cos x$
Thus $\map {\dfrac {\d^2} {\d x^2} } {\cos x} = -\cos x$ w... | Real Cosine Function has Zeroes | https://proofwiki.org/wiki/Real_Cosine_Function_has_Zeroes | https://proofwiki.org/wiki/Real_Cosine_Function_has_Zeroes | [
"Cosine Function"
] | [] | [
"Definition:Positive Real Function",
"Derivative of Cosine Function",
"Derivative of Sine Function",
"Definition:Negative Real Function",
"Second Derivative of Concave Real Function is Non-Positive",
"Definition:Concave Real Function",
"Real Cosine Function is Bounded",
"Definition:Bounded Mapping/Rea... |
proofwiki-20319 | Sine and Cosine are Periodic on Reals/Sine/Proof 1 | The sine function is periodic with the same period as the cosine function.
:820px | Since Real Cosine Function is Periodic, let $K$ be its period.
Then:
:$\cos K = \map \cos {0 + K} = \cos 0$
Because Cosine of Zero is One:
:$\cos K = 1$
Furthermore:
{{begin-eqn}}
{{eqn | l = \cos^2 K + \sin^2 K
| r = 1
| c = Sum of Squares of Sine and Cosine
}}
{{eqn | l = \sin^2 K
| r = 0
| c ... | The [[Definition:Real Sine Function|sine]] function is [[Definition:Periodic Real Function|periodic]] with the same [[Definition:Period of Periodic Real Function|period]] as the [[Definition:Real Cosine Function|cosine]] function.
:[[File:SineCos.png|820px]] | Since [[Real Cosine Function is Periodic]], let $K$ be its [[Definition:Period of Periodic Real Function|period]].
Then:
:$\cos K = \map \cos {0 + K} = \cos 0$
Because [[Cosine of Zero is One]]:
:$\cos K = 1$
Furthermore:
{{begin-eqn}}
{{eqn | l = \cos^2 K + \sin^2 K
| r = 1
| c = [[Sum of Squares of Si... | Sine and Cosine are Periodic on Reals/Sine/Proof 1 | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Sine/Proof_1 | https://proofwiki.org/wiki/Sine_and_Cosine_are_Periodic_on_Reals/Sine/Proof_1 | [
"Sine and Cosine are Periodic on Reals"
] | [
"Definition:Sine/Real Function",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Cosine/Real Function",
"File:SineCos.png"
] | [
"Sine and Cosine are Periodic on Reals/Cosine",
"Definition:Periodic Real Function/Period",
"Cosine of Zero is One",
"Sum of Squares of Sine and Cosine",
"Sine of Sum",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"Definition:Periodic Real Function/Period",
"Sine ... |
proofwiki-20320 | Constant Loop is Loop | Let $\struct {T, \tau}$ be a topological space.
Let $p \in T$.
Let $c_p : \closedint 0 1 \to T$ be the constant mapping defined by:
:$\forall t \in \closedint 0 1 : \map {c_p} t = p$
Then $c_p$ is a loop in $T$. | From Constant Mapping is Continuous, it follows that $c_p$ is continuous.
By definition of path, it follows that $c_p$ is a path in $T$.
We have:
:$\map {c_p} 0 = \map {c_p} 1 = p$
Hence, $c_p$ is a loop in $T$.
{{qed}}
Category:Loops (Topology)
cmezlz5elde4qtdq5udjhkpwekca3ug | Let $\struct {T, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $p \in T$.
Let $c_p : \closedint 0 1 \to T$ be the [[Definition:Constant Mapping|constant mapping]] defined by:
:$\forall t \in \closedint 0 1 : \map {c_p} t = p$
Then $c_p$ is a [[Definition:Loop (Topology)|loop]] in $T$. | From [[Constant Mapping is Continuous]], it follows that $c_p$ is [[Definition:Continuous Mapping (Topology)|continuous]].
By definition of [[Definition:Path (Topology)|path]], it follows that $c_p$ is a [[Definition:Path (Topology)|path]] in $T$.
We have:
:$\map {c_p} 0 = \map {c_p} 1 = p$
Hence, $c_p$ is a [[Def... | Constant Loop is Loop | https://proofwiki.org/wiki/Constant_Loop_is_Loop | https://proofwiki.org/wiki/Constant_Loop_is_Loop | [
"Loops (Topology)"
] | [
"Definition:Topological Space",
"Definition:Constant Mapping",
"Definition:Loop (Topology)"
] | [
"Constant Mapping is Continuous",
"Definition:Continuous Mapping (Topology)",
"Definition:Path (Topology)",
"Definition:Path (Topology)",
"Definition:Loop (Topology)",
"Category:Loops (Topology)"
] |
proofwiki-20321 | Basic Inequality/One-Sided Shift Space of Finite Type | Let $\struct {X_\mathbf A ^+, \sigma}$ be a one-sided shift of finite type.
Let $F_\theta ^+$ be the space of Lipschitz functions on $X_\mathbf A ^+$.
Let $\size \cdot_\theta$ be the Lipschitz seminorm on $F_\theta ^+$.
Let $\norm \cdot_\infty$ be the supremum norm on $F_\theta ^+$.
Let $f \in F_\theta ^+$.
Let $u := \... | Let $x, y \in X_\mathbf A ^+$.
If $x_0 = y_0$, then:
{{begin-eqn}}
{{eqn | l = \cmod {\map {\LL_f w} x - \map {\LL_f w} y}
| r = \cmod {\sum_{\map {\mathbf A} {k, x_0} = 1} \paren {e^{\map f {k x} } \map w {k x} - e^{\map f {k y} } \map w {k y} } }
}}
{{eqn | o = \le
| r = \sum_{\map {\mathbf A} {k, x_0} = ... | Let $\struct {X_\mathbf A ^+, \sigma}$ be a [[Definition:One-Sided Shift of Finite Type|one-sided shift of finite type]].
Let $F_\theta ^+$ be the [[Definition:Space of Lipschitz Functions/One-Sided Shift of Finite Type|space of Lipschitz functions]] on $X_\mathbf A ^+$.
Let $\size \cdot_\theta$ be the [[Definition:L... | Let $x, y \in X_\mathbf A ^+$.
If $x_0 = y_0$, then:
{{begin-eqn}}
{{eqn | l = \cmod {\map {\LL_f w} x - \map {\LL_f w} y}
| r = \cmod {\sum_{\map {\mathbf A} {k, x_0} = 1} \paren {e^{\map f {k x} } \map w {k x} - e^{\map f {k y} } \map w {k y} } }
}}
{{eqn | o = \le
| r = \sum_{\map {\mathbf A} {k, x_0} =... | Basic Inequality/One-Sided Shift Space of Finite Type | https://proofwiki.org/wiki/Basic_Inequality/One-Sided_Shift_Space_of_Finite_Type | https://proofwiki.org/wiki/Basic_Inequality/One-Sided_Shift_Space_of_Finite_Type | [
"Ergodic Theory",
"Functional Analysis"
] | [
"Definition:One-Sided Shift of Finite Type",
"Definition:Space of Lipschitz Functions/One-Sided Shift of Finite Type",
"Definition:Lipschitz Seminorm",
"Definition:Supremum Norm",
"Definition:Complex Number/Real Part",
"Definition:Ruelle-Perron-Frobenius Operator/One-Sided Shift Space of Finite Type",
"... | [] |
proofwiki-20322 | Lasota-Yorke Inequality/One-Sided Shift Space of Finite Type | Let $\struct {X_\mathbf A ^+, \sigma}$ be a one-sided shift of finite type.
Let $F_\theta ^+$ be the space of Lipschitz functions on $X_\mathbf A ^+$.
Let $\norm \cdot_\theta$ be the Lipschitz norm on $F_\theta ^+$.
Let $\norm \cdot_\infty$ be the supremum norm on $F_\theta ^+$.
Let $f \in F_\theta ^+$.
Let $u := \map ... | Recall the basic inequality:
:There exists a $C_0 > 0$ so that we have:
{{begin-eqn}}
{{eqn | n = 1
| l = \size {\LL_f ^n w}_\theta
| o = \le
| r = C_0 \norm w_\infty + \theta ^n \size w_\theta
}}
{{end-eqn}}
:for all $w \in F_\theta ^+$ and $n \in \N$.
On the other hand, we have:
{{begin-eqn}}
{{eqn ... | Let $\struct {X_\mathbf A ^+, \sigma}$ be a [[Definition:One-Sided Shift of Finite Type|one-sided shift of finite type]].
Let $F_\theta ^+$ be the [[Definition:Space of Lipschitz Functions/One-Sided Shift of Finite Type|space of Lipschitz functions]] on $X_\mathbf A ^+$.
Let $\norm \cdot_\theta$ be the [[Definition:L... | Recall the [[Basic Inequality/One-Sided Shift Space of Finite Type|basic inequality]]:
:There exists a $C_0 > 0$ so that we have:
{{begin-eqn}}
{{eqn | n = 1
| l = \size {\LL_f ^n w}_\theta
| o = \le
| r = C_0 \norm w_\infty + \theta ^n \size w_\theta
}}
{{end-eqn}}
:for all $w \in F_\theta ^+$ and $n... | Lasota-Yorke Inequality/One-Sided Shift Space of Finite Type | https://proofwiki.org/wiki/Lasota-Yorke_Inequality/One-Sided_Shift_Space_of_Finite_Type | https://proofwiki.org/wiki/Lasota-Yorke_Inequality/One-Sided_Shift_Space_of_Finite_Type | [
"Ergodic Theory",
"Functional Analysis"
] | [
"Definition:One-Sided Shift of Finite Type",
"Definition:Space of Lipschitz Functions/One-Sided Shift of Finite Type",
"Definition:Lipschitz Norm",
"Definition:Supremum Norm",
"Definition:Complex Number/Real Part",
"Definition:Ruelle-Perron-Frobenius Operator/One-Sided Shift Space of Finite Type",
"Defi... | [
"Basic Inequality/One-Sided Shift Space of Finite Type",
"Category:Ergodic Theory",
"Category:Functional Analysis"
] |
proofwiki-20323 | Thomae's Transformation | :$\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1} = \dfrac {\map \Gamma e \map \Gamma f \map \Gamma s } {\map \Gamma a \map \Gamma {s + b} \map \Gamma {s + c} } \map { {}_3 \operatorname F_2} { { {e - a, f - a, s} \atop {s + b, s + c} } \, \middle \vert \, 1} $ | First, we observe that:
{{begin-eqn}}
{{eqn | l = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma e \map \Gamma f } \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1}
| r = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma e \map \Gamma f } \sum_{n \... | :$\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1} = \dfrac {\map \Gamma e \map \Gamma f \map \Gamma s } {\map \Gamma a \map \Gamma {s + b} \map \Gamma {s + c} } \map { {}_3 \operatorname F_2} { { {e - a, f - a, s} \atop {s + b, s + c} } \, \middle \vert \, 1} $ | First, we observe that:
{{begin-eqn}}
{{eqn | l = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma e \map \Gamma f } \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1}
| r = \dfrac {\map \Gamma a \map \Gamma b \map \Gamma c } {\map \Gamma e \map \Gamma f } \sum_{n \... | Thomae's Transformation | https://proofwiki.org/wiki/Thomae's_Transformation | https://proofwiki.org/wiki/Thomae's_Transformation | [
"Thomae's Transformation",
"Gamma Function",
"Hypergeometric Functions"
] | [] | [
"Rising Factorial as Quotient of Factorials",
"Gauss's Hypergeometric Theorem",
"Rising Factorial as Quotient of Factorials",
"Rising Factorial as Quotient of Factorials",
"Gauss's Hypergeometric Theorem",
"Rising Factorial as Quotient of Factorials"
] |
proofwiki-20324 | Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls | Let $M$ be a locally Euclidean space of some dimension $d$.
Let $m \in M$.
Then:
:there exists a countable local basis $\family{U_n}_{n \in \N}$ of $m$ where each $U_n$ is the homeomorphic image of an open ball of $\R^d$ | By definition of a locally Euclidean space:
:there exists an open neighbourhood $U$ of $m$ which is homeomorphic to an open subset $V$ of Euclidean space $\R^d$.
Let $\phi: U \to V$ be a homeomorphism.
By definition of the Euclidean space $\R^d$ the topology on $\R^d$ is the topology induced by the metric:
:$\ds \map {... | Let $M$ be a [[Definition:Locally Euclidean Space|locally Euclidean space]] of some dimension $d$.
Let $m \in M$.
Then:
:there exists a [[Definition:Countable|countable]] [[Definition:Local Basis|local basis]] $\family{U_n}_{n \in \N}$ of $m$ where each $U_n$ is the [[Definition:Homeomorphism|homeomorphic]] [[Defin... | By definition of a [[Definition:Locally Euclidean Space|locally Euclidean space]]:
:there exists an [[Definition:Open Set (Topology)|open neighbourhood]] $U$ of $m$ which is [[Definition:Homeomorphic Topological Spaces|homeomorphic]] to an [[Definition:Open Set (Topology)|open subset]] $V$ of [[Definition:Euclidean Spa... | Locally Euclidean Space has Countable Local Basis Homeomorphic to Open Balls | https://proofwiki.org/wiki/Locally_Euclidean_Space_has_Countable_Local_Basis_Homeomorphic_to_Open_Balls | https://proofwiki.org/wiki/Locally_Euclidean_Space_has_Countable_Local_Basis_Homeomorphic_to_Open_Balls | [
"Locally Euclidean Spaces"
] | [
"Definition:Locally Euclidean Space",
"Definition:Countable Set",
"Definition:Local Basis",
"Definition:Homeomorphism",
"Definition:Image",
"Definition:Open Ball"
] | [
"Definition:Locally Euclidean Space",
"Definition:Open Set/Topology",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Open Set/Topology",
"Definition:Euclidean Space",
"Definition:Homeomorphism",
"Definition:Euclidean Space",
"Definition:Euclidean Space/Euclidean Topology",
"Definition:To... |
proofwiki-20325 | Homeomorphic Image of Local Basis is Local Basis | Let $T_\alpha = \struct{S_\alpha, \tau_\alpha}$ and $T_\beta = \struct{S_\beta, \tau_\beta}$ be topological spaces.
Let $\phi: T_\alpha \to T_\beta$ be a homeomorphism.
Let $s \in S_\alpha$.
Let $\BB$ be a local basis of $s$ in $T_\alpha$.
Then:
:$\BB' = \set{ \phi \sqbrk B : B \in \BB}$ is a local basis of $\map \phi ... | By definition of homeomorphism:
:$\forall U \in \tau_\alpha : \phi \sqbrk U \in \tau_\beta$
Hence:
:$\BB'$ is a set of open sets in $T_\beta$ containing $\map \phi s$
Let $U \in \tau_\beta$ containing $\map \phi s$.
By definition of homeomorphism:
:$\phi^{-1} \sqbrk U \in \tau_\alpha$ containing $s$
By definition of lo... | Let $T_\alpha = \struct{S_\alpha, \tau_\alpha}$ and $T_\beta = \struct{S_\beta, \tau_\beta}$ be [[Definition:Topological Space|topological spaces]].
Let $\phi: T_\alpha \to T_\beta$ be a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]].
Let $s \in S_\alpha$.
Let $\BB$ be a [[Definition:Local Basis|loc... | By definition of [[Definition:Homeomorphism|homeomorphism]]:
:$\forall U \in \tau_\alpha : \phi \sqbrk U \in \tau_\beta$
Hence:
:$\BB'$ is a [[Definition:Set|set]] of [[Definition:Open Set (Topology)|open sets]] in $T_\beta$ [[Definition:Element|containing]] $\map \phi s$
Let $U \in \tau_\beta$ [[Definition:Element|... | Homeomorphic Image of Local Basis is Local Basis | https://proofwiki.org/wiki/Homeomorphic_Image_of_Local_Basis_is_Local_Basis | https://proofwiki.org/wiki/Homeomorphic_Image_of_Local_Basis_is_Local_Basis | [
"Homeomorphisms (Topological Spaces)",
"Local Bases"
] | [
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Local Basis",
"Definition:Local Basis"
] | [
"Definition:Homeomorphism",
"Definition:Set",
"Definition:Open Set/Topology",
"Definition:Element",
"Definition:Element",
"Definition:Homeomorphism",
"Definition:Element",
"Definition:Local Basis",
"Definition:Image (Set Theory)/Mapping/Subset",
"Image of Subset under Mapping is Subset of Image",
... |
proofwiki-20326 | Local Basis of Open Subspace iff Local Basis | Let $T = \struct {S, \tau}$ be a topological space.
Let $U \subseteq S$ be an open subset
Let $\tau_U$ denote the subspace topology on $U$.
Let $s \in U$.
Let $\BB \subseteq \powerset U$.
Then:
:$\BB$ is a local basis of $s$ in $\struct {U, \tau_U}$
{{iff}}:
:$\BB$ is a local basis of $s$ in $\struct {S, \tau}$. | Let $\map \BB s$ denote the set of open sets containing $s$ in $\struct {S, \tau}$
Let $\map \CC s$ denote the set of open sets containing $s$ in $\struct {U, \tau_U}$
From Open Set in Open Subspace:
:$\BB \subseteq \map \BB s$ {{iff}} $\BB \subseteq \map \CC s$ | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $U \subseteq S$ be an [[Definition:Open Set (Topology)|open subset]]
Let $\tau_U$ denote the [[Definition:Subspace Topology|subspace topology]] on $U$.
Let $s \in U$.
Let $\BB \subseteq \powerset U$.
Then:
:$\BB$ is a [[Def... | Let $\map \BB s$ denote the [[Definition:Set|set]] of [[Definition:Open Set (Topology)|open sets]] [[Definition:Element|containing]] $s$ in $\struct {S, \tau}$
Let $\map \CC s$ denote the [[Definition:Set|set]] of [[Definition:Open Set (Topology)|open sets]] [[Definition:Element|containing]] $s$ in $\struct {U, \tau_U... | Local Basis of Open Subspace iff Local Basis | https://proofwiki.org/wiki/Local_Basis_of_Open_Subspace_iff_Local_Basis | https://proofwiki.org/wiki/Local_Basis_of_Open_Subspace_iff_Local_Basis | [
"Topological Subspaces",
"Local Bases"
] | [
"Definition:Topological Space",
"Definition:Open Set/Topology",
"Definition:Topological Subspace",
"Definition:Local Basis",
"Definition:Local Basis"
] | [
"Definition:Set",
"Definition:Open Set/Topology",
"Definition:Element",
"Definition:Set",
"Definition:Open Set/Topology",
"Definition:Element",
"Open Set in Open Subspace",
"Open Set in Open Subspace"
] |
proofwiki-20327 | Distance between Points in Regular Hexagon | Let $H$ be a regular hexagon embedded in the Euclidean plane $\R^2$.
Let $s \in \R_{>0}$ be the side length of $H$.
Let $\mathbf x, \mathbf y \in \R^2$ such that $\mathbf x$ and $\mathbf y$ lie in the interior of $H$, or on the circumference of $H$.
Then:
:$\map d {\mathbf x, \mathbf y} \le 2s$
where $\map d {\mathbf x... | From Regular Polygon is Cyclic, it follows that $H$ can be inscribed in a circle with center $\mathbf c$.
The circumcircle intersects all vertices of $H$.
From Regular Hexagon is composed of Equilateral Triangles, it follows that the side length $s$ is equal to the distance from $\mathbf c$ to any vertex of $H$.
It fol... | Let $H$ be a [[Definition:Regular Hexagon|regular hexagon]] [[Definition:Embedding (Topology)|embedded]] in the [[Definition:Euclidean Plane|Euclidean plane]] $\R^2$.
Let $s \in \R_{>0}$ be the [[Definition:Side of Polygon|side]] [[Definition:Length of Line|length]] of $H$.
Let $\mathbf x, \mathbf y \in \R^2$ such th... | From [[Regular Polygon is Cyclic]], it follows that $H$ can be [[Definition:Polygon Inscribed in Circle|inscribed]] in a [[Definition:Circle|circle]] with [[Definition:Center of Circle|center]] $\mathbf c$.
The [[Definition:Circumcircle|circumcircle]] intersects all [[Definition:Vertex of Polygon|vertices]] of $H$.
F... | Distance between Points in Regular Hexagon | https://proofwiki.org/wiki/Distance_between_Points_in_Regular_Hexagon | https://proofwiki.org/wiki/Distance_between_Points_in_Regular_Hexagon | [
"Hexagons"
] | [
"Definition:Hexagon/Regular",
"Definition:Embedding (Topology)",
"Definition:Euclidean Plane",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Region",
"Definition:Circumference of Geometric Figure",
"Definition:Euclidean Metric/Real Number Plane"
] | [
"Regular Polygon is Cyclic",
"Definition:Inscribe/Polygon in Circle",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circumcircle",
"Definition:Polygon/Vertex",
"Regular Hexagon is composed of Equilateral Triangles",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Def... |
proofwiki-20328 | Homeomorphic Image of Neighborhood Basis is Neighborhood Basis | Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be topological spaces.
Let $\phi: T_\alpha \to T_\beta$ be a homeomorphism.
Let $s \in S_\alpha$.
Let $\NN$ be a neighborhood basis of $s$ in $T_\alpha$.
Then:
:$\NN' = \set {\phi \sqbrk N : N \in \NN}$ is a neighborhood basi... | Let $N$ be a neighborhood of $s$ in $T_\alpha$.
By definition of neighborhood:
:$\exists V \in \tau_\alpha : s \in V \subseteq N$
By definition of image of subset:
:$\map \phi s \in \phi \sqbrk V$
From Subset Maps to Subset:
:$\phi \sqbrk V \subseteq \phi \sqbrk N$
By definition of homeomorphism:
:$\phi \sqbrk V \in \t... | Let $T_\alpha = \struct {S_\alpha, \tau_\alpha}$ and $T_\beta = \struct {S_\beta, \tau_\beta}$ be [[Definition:Topological Space|topological spaces]].
Let $\phi: T_\alpha \to T_\beta$ be a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]].
Let $s \in S_\alpha$.
Let $\NN$ be a [[Definition:Neighborhood ... | Let $N$ be a [[Definition:Neighborhood (Topology)|neighborhood]] of $s$ in $T_\alpha$.
By definition of [[Definition:Neighborhood (Topology)|neighborhood]]:
:$\exists V \in \tau_\alpha : s \in V \subseteq N$
By definition of [[Definition:Image of Subset under Mapping|image of subset]]:
:$\map \phi s \in \phi \sqbrk V... | Homeomorphic Image of Neighborhood Basis is Neighborhood Basis | https://proofwiki.org/wiki/Homeomorphic_Image_of_Neighborhood_Basis_is_Neighborhood_Basis | https://proofwiki.org/wiki/Homeomorphic_Image_of_Neighborhood_Basis_is_Neighborhood_Basis | [
"Homeomorphisms (Topological Spaces)",
"Neighborhood Bases"
] | [
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Neighborhood Basis",
"Definition:Neighborhood Basis"
] | [
"Definition:Neighborhood (Topology)",
"Definition:Neighborhood (Topology)",
"Definition:Image (Set Theory)/Mapping/Subset",
"Image of Subset under Mapping is Subset of Image",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Neighborhood (Topology)",
"Definition:Set",
"Definition:Neighborhoo... |
proofwiki-20329 | Neighborhood Basis of Open Subspace iff Neighborhood Basis | Let $T = \struct{S, \tau}$ be a topological space.
Let $U \subseteq S$ be an open subset
Let $\tau_U$ denote the subspace topology on $U$.
Let $s \in U$.
Let $\NN \subseteq \powerset U$.
Then:
:$\NN$ is a neighborhood basis of $s$ in $\struct{U, \tau_U}$ {{iff}} $\NN$ is a neighborhood basis of $s$ in $\struct{S, \tau}... | Let $\map \NN s$ denote the set of neighborhoods of $s$ in $\struct{S, \tau}$
Let $\map \MM s$ denote the set of neighborhoods of $s$ in $\struct{U, \tau_U}$
From Neighborhood in Open Subspace:
:$\NN \subseteq \map \NN s$ {{iff}} $\NN \subseteq \map \MM s$ | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $U \subseteq S$ be an [[Definition:Open Set (Topology)|open subset]]
Let $\tau_U$ denote the [[Definition:Subspace Topology|subspace topology]] on $U$.
Let $s \in U$.
Let $\NN \subseteq \powerset U$.
Then:
:$\NN$ is a [[Defi... | Let $\map \NN s$ denote the [[Definition:Set|set]] of [[Definition:Neighborhood (Topology)|neighborhoods]] of $s$ in $\struct{S, \tau}$
Let $\map \MM s$ denote the [[Definition:Set|set]] of [[Definition:Neighborhood (Topology)|neighborhoods]] of $s$ in $\struct{U, \tau_U}$
From [[Neighborhood in Open Subspace]]:
:$\N... | Neighborhood Basis of Open Subspace iff Neighborhood Basis | https://proofwiki.org/wiki/Neighborhood_Basis_of_Open_Subspace_iff_Neighborhood_Basis | https://proofwiki.org/wiki/Neighborhood_Basis_of_Open_Subspace_iff_Neighborhood_Basis | [
"Topological Subspaces",
"Neighborhood Bases"
] | [
"Definition:Topological Space",
"Definition:Open Set/Topology",
"Definition:Topological Subspace",
"Definition:Neighborhood Basis",
"Definition:Neighborhood Basis"
] | [
"Definition:Set",
"Definition:Neighborhood (Topology)",
"Definition:Set",
"Definition:Neighborhood (Topology)",
"Neighborhood in Open Subspace",
"Definition:Neighborhood (Topology)",
"Neighborhood in Open Subspace"
] |
proofwiki-20330 | Locally Euclidean Space has Countable Neighborhood Basis Homeomorphic to Closed Balls | Let $M$ be a locally Euclidean space of some dimension $d$.
Let $m \in M$.
Then:
:there exists a countable neighborhood basis $\family{N_n}_{n \in \N}$ of $m$ where each $N_n$ is the homeomorphic image of a closed ball of $\R^d$ | By definition of a locally Euclidean space:
:there exists an open neighbourhood $U$ of $m$ which is homeomorphic to an open subset $V$ of Euclidean space $\R^d$.
Let $\phi: U \to V$ be a homeomorphism.
By definition of the Euclidean space $\R^d$ the topology on $\R^d$ is the topology induced by the metric:
:$\ds \map {... | Let $M$ be a [[Definition:Locally Euclidean Space|locally Euclidean space]] of some dimension $d$.
Let $m \in M$.
Then:
:there exists a [[Definition:Countable|countable]] [[Definition:Neighborhood Basis|neighborhood basis]] $\family{N_n}_{n \in \N}$ of $m$ where each $N_n$ is the [[Definition:Homeomorphism|homeomor... | By definition of a [[Definition:Locally Euclidean Space|locally Euclidean space]]:
:there exists an [[Definition:Open Set (Topology)|open neighbourhood]] $U$ of $m$ which is [[Definition:Homeomorphic Topological Spaces|homeomorphic]] to an [[Definition:Open Set (Topology)|open subset]] $V$ of [[Definition:Euclidean Spa... | Locally Euclidean Space has Countable Neighborhood Basis Homeomorphic to Closed Balls | https://proofwiki.org/wiki/Locally_Euclidean_Space_has_Countable_Neighborhood_Basis_Homeomorphic_to_Closed_Balls | https://proofwiki.org/wiki/Locally_Euclidean_Space_has_Countable_Neighborhood_Basis_Homeomorphic_to_Closed_Balls | [
"Locally Euclidean Spaces"
] | [
"Definition:Locally Euclidean Space",
"Definition:Countable Set",
"Definition:Neighborhood Basis",
"Definition:Homeomorphism",
"Definition:Image",
"Definition:Closed Ball/Metric Space"
] | [
"Definition:Locally Euclidean Space",
"Definition:Open Set/Topology",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Open Set/Topology",
"Definition:Euclidean Space",
"Definition:Homeomorphism",
"Definition:Euclidean Space",
"Definition:Euclidean Space/Euclidean Topology",
"Definition:To... |
proofwiki-20331 | Watson's Hypergeometric Theorem | :$\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {\dfrac 1 2 \paren {a + b + 1}, 2 c } } \, \middle \vert \, 1} = \dfrac {\map \Gamma {\dfrac 1 2} \map \Gamma {\dfrac 1 2 + c} \map \Gamma {\dfrac 1 2 \paren {1 + a + b} } \map \Gamma {\dfrac 1 2 \paren {1 - a - b} + c } } {\map \Gamma {\dfrac 1 2 \paren {1 + a} ... | From Thomae's Transformation, we have:
{{begin-eqn}}
{{eqn | l = \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1}
| r = \dfrac {\map \Gamma e \map \Gamma f \map \Gamma s } {\map \Gamma a \map \Gamma {s + b} \map \Gamma {s + c} } \map { {}_3 \operatorname F_2} { { {e - a, f - a, s... | :$\ds \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {\dfrac 1 2 \paren {a + b + 1}, 2 c } } \, \middle \vert \, 1} = \dfrac {\map \Gamma {\dfrac 1 2} \map \Gamma {\dfrac 1 2 + c} \map \Gamma {\dfrac 1 2 \paren {1 + a + b} } \map \Gamma {\dfrac 1 2 \paren {1 - a - b} + c } } {\map \Gamma {\dfrac 1 2 \paren {1 + a} ... | From [[Thomae's Transformation]], we have:
{{begin-eqn}}
{{eqn | l = \map { {}_3 \operatorname F_2} { { {a, b, c} \atop {e, f} } \, \middle \vert \, 1}
| r = \dfrac {\map \Gamma e \map \Gamma f \map \Gamma s } {\map \Gamma a \map \Gamma {s + b} \map \Gamma {s + c} } \map { {}_3 \operatorname F_2} { { {e - a, f - ... | Watson's Hypergeometric Theorem | https://proofwiki.org/wiki/Watson's_Hypergeometric_Theorem | https://proofwiki.org/wiki/Watson's_Hypergeometric_Theorem | [
"Watson's Hypergeometric Theorem",
"Hypergeometric Functions"
] | [] | [
"Thomae's Transformation",
"Thomae's Transformation",
"Definition:Hypergeometric Function/Generalized",
"Dixon's Hypergeometric Theorem",
"Dixon's Hypergeometric Theorem",
"Definition:Hypergeometric Function/Generalized",
"Dixon's Hypergeometric Theorem",
"Dixon's Hypergeometric Theorem",
"Legendre'... |
proofwiki-20332 | Generated Submodule may not equal Set of Linear Combinations | Let $\struct { R, +_R, \times_R }$ be a ring.
Let $\struct { M, +_M, \circ }_R$ be an $R$-module.
Let $S$ be a subset of $M$.
Let $H_1$ be the submodule generated by $S$.
Let $H_2$ be the set of all linear combinations of elements of $S$.
Then it is possible to select $\struct { R, +_R, \times_R }$, $\struct { M, +_M,... | Let $R = \set {2 k : k \in \Z}$ be the set of all even integers.
From Integers form Commutative Ring, it follows that $\Z$ is a ring.
From Ideal of Ring/Examples/Set of Even Integers, it follows that $R$ is an ideal of $\Z$.
From Ideal is Subring, it follows that $\struct {R, +, \times}$ is a ring, where $+$ denotes in... | Let $\struct { R, +_R, \times_R }$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct { M, +_M, \circ }_R$ be an [[Definition:Module over Ring|$R$-module]].
Let $S$ be a [[Definition:Subset|subset]] of $M$.
Let $H_1$ be the [[Definition:Generated Submodule|submodule generated by $S$]].
Let $H_2$ be the ... | Let $R = \set {2 k : k \in \Z}$ be the [[Definition:Set|set]] of all [[Definition:Even Integer|even integers]].
From [[Integers form Commutative Ring]], it follows that $\Z$ is a [[Definition:Ring (Abstract Algebra)|ring]].
From [[Ideal of Ring/Examples/Set of Even Integers]], it follows that $R$ is an [[Definition:I... | Generated Submodule may not equal Set of Linear Combinations | https://proofwiki.org/wiki/Generated_Submodule_may_not_equal_Set_of_Linear_Combinations | https://proofwiki.org/wiki/Generated_Submodule_may_not_equal_Set_of_Linear_Combinations | [
"Generators of Modules",
"Linear Combinations"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Module over Ring",
"Definition:Subset",
"Definition:Generated Submodule",
"Definition:Set",
"Definition:Linear Combination",
"Definition:Element"
] | [
"Definition:Set",
"Definition:Even Integer",
"Integers form Commutative Ring",
"Definition:Ring (Abstract Algebra)",
"Ideal of Ring/Examples/Set of Even Integers",
"Definition:Ideal of Ring",
"Ideal is Subring",
"Definition:Ring (Abstract Algebra)",
"Definition:Addition/Integers",
"Definition:Mult... |
proofwiki-20333 | Relation Between First and Second Form of Binet Form | Let $m \in \R$.
Define:
{{begin-eqn}}
{{eqn | l = \Delta
| r = \sqrt {m^2 + 4}
}}
{{eqn | l = \alpha
| r = \frac {m + \Delta} 2
}}
{{eqn | l = \beta
| r = \frac {m - \Delta} 2
}}
{{end-eqn}} | Proof by induction:
Let $\map P n$ be the proposition:
:$U_{n - 1} + U_{n + 1} = V_n$ | Let $m \in \R$.
Define:
{{begin-eqn}}
{{eqn | l = \Delta
| r = \sqrt {m^2 + 4}
}}
{{eqn | l = \alpha
| r = \frac {m + \Delta} 2
}}
{{eqn | l = \beta
| r = \frac {m - \Delta} 2
}}
{{end-eqn}} | Proof by [[Principle of Mathematical Induction|induction]]:
Let $\map P n$ be the proposition:
:$U_{n - 1} + U_{n + 1} = V_n$ | Relation Between First and Second Form of Binet Form | https://proofwiki.org/wiki/Relation_Between_First_and_Second_Form_of_Binet_Form | https://proofwiki.org/wiki/Relation_Between_First_and_Second_Form_of_Binet_Form | [
"Binet Form"
] | [] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-20334 | Iteration of Ruelle-Perron-Frobenius Operator/One-Sided Shift Space of Finite Type | Let $\struct {X_\mathbf A ^+, \sigma}$ be a one-sided shift of finite type.
Let $\map C {X _\mathbf A ^+}$ be the complex-valued continuous mapping space.
Let $f \in \map C {X _\mathbf A ^+}$.
Let $\LL_f$ be the Ruelle-Perron-Frobenius operator.
For all $w \in \map C {X _\mathbf A ^+}$, $n \in \N$ and $x \in X _\mathbf... | {{ProofWanted}}
Category:Ergodic Theory
Category:Functional Analysis
9bc7b9n5md2vlapnejy9oubspavkczu | Let $\struct {X_\mathbf A ^+, \sigma}$ be a [[Definition:One-Sided Shift of Finite Type|one-sided shift of finite type]].
Let $\map C {X _\mathbf A ^+}$ be the [[Definition:Complex-Valued Function|complex-valued]] [[Definition:Continuous Mapping Space|continuous mapping space]].
Let $f \in \map C {X _\mathbf A ^+}$.
... | {{ProofWanted}}
[[Category:Ergodic Theory]]
[[Category:Functional Analysis]]
9bc7b9n5md2vlapnejy9oubspavkczu | Iteration of Ruelle-Perron-Frobenius Operator/One-Sided Shift Space of Finite Type | https://proofwiki.org/wiki/Iteration_of_Ruelle-Perron-Frobenius_Operator/One-Sided_Shift_Space_of_Finite_Type | https://proofwiki.org/wiki/Iteration_of_Ruelle-Perron-Frobenius_Operator/One-Sided_Shift_Space_of_Finite_Type | [
"Ergodic Theory",
"Functional Analysis"
] | [
"Definition:One-Sided Shift of Finite Type",
"Definition:Complex-Valued Function",
"Definition:Continuous Mapping Space",
"Definition:Ruelle-Perron-Frobenius Operator/One-Sided Shift Space of Finite Type",
"Definition:Preimage/Mapping/Element"
] | [
"Category:Ergodic Theory",
"Category:Functional Analysis"
] |
proofwiki-20335 | Neighborhood in Open Subspace | Let $T = \struct{S, \tau}$ be a topological space.
Let $U \subseteq S$ be an open subset.
Let $\tau_U$ denote the subspace topology on $U$.
Let $s \in U$
Let $N \subseteq U$ be a subset.
Then:
:$N$ is a neighborhood of $s$ in $\struct{U, \tau_U}$
{{iff}}
:$N$ is a neighborhood of $s$ in $\struct{S, \tau}$ | === Necessary Condition ===
Let $N$ be a neighborhood of $s$ in $\struct{U, \tau_U}$.
By definition of neighborhood:
:$\exists V \in \tau_U : x \in V \subseteq N$
From Open Set in Open Subspace:
:$V \in \tau$
Hence:
:$\exists V \in \tau : x \in V \subseteq N$
It follows that $N$ is a neighborhood of $s$ in $\struct{S, ... | Let $T = \struct{S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $U \subseteq S$ be an [[Definition:Open Set (Topology)|open]] [[Definition:Subset|subset]].
Let $\tau_U$ denote the [[Definition:Topological Subspace|subspace topology]] on $U$.
Let $s \in U$
Let $N \subseteq U$ be a [[Definit... | === Necessary Condition ===
Let $N$ be a [[Definition:Neighborhood (Topology)|neighborhood]] of $s$ in $\struct{U, \tau_U}$.
By definition of [[Definition:Neighborhood (Topology)|neighborhood]]:
:$\exists V \in \tau_U : x \in V \subseteq N$
From [[Open Set in Open Subspace]]:
:$V \in \tau$
Hence:
:$\exists V \in \t... | Neighborhood in Open Subspace | https://proofwiki.org/wiki/Neighborhood_in_Open_Subspace | https://proofwiki.org/wiki/Neighborhood_in_Open_Subspace | [
"Neighborhoods",
"Topological Subspaces"
] | [
"Definition:Topological Space",
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Topological Subspace",
"Definition:Subset",
"Definition:Neighborhood (Topology)",
"Definition:Neighborhood (Topology)"
] | [
"Definition:Neighborhood (Topology)",
"Definition:Neighborhood (Topology)",
"Open Set in Open Subspace",
"Definition:Neighborhood (Topology)",
"Definition:Neighborhood (Topology)",
"Definition:Neighborhood (Topology)",
"Open Set in Open Subspace",
"Definition:Neighborhood (Topology)"
] |
proofwiki-20336 | Derivative of Matrix Exponential | :$\dfrac \d {\d t} e^{\mathbf A t} = \mathbf A e^{\mathbf A t}$ | From the definition of the matrix exponential, $e^{\mathbf A t}$ is defined as being the square matrix $X$ with the properties:
:$(1): \quad \map {\dfrac \d {\d t} } X = \mathbf A X$
:$(2): \quad \map X {\mathbf 0_n} = \mathbf I_n$
The result follows directly;
{{qed}}
Category:Matrix Exponential
Category:Derivatives
7f... | :$\dfrac \d {\d t} e^{\mathbf A t} = \mathbf A e^{\mathbf A t}$ | From the definition of the [[Definition:Matrix Exponential|matrix exponential]], $e^{\mathbf A t}$ is defined as being the [[Definition:Square Matrix|square matrix]] $X$ with the properties:
:$(1): \quad \map {\dfrac \d {\d t} } X = \mathbf A X$
:$(2): \quad \map X {\mathbf 0_n} = \mathbf I_n$
The result follows dire... | Derivative of Matrix Exponential | https://proofwiki.org/wiki/Derivative_of_Matrix_Exponential | https://proofwiki.org/wiki/Derivative_of_Matrix_Exponential | [
"Matrix Exponential",
"Derivatives"
] | [] | [
"Definition:Matrix Exponential",
"Definition:Matrix/Square Matrix",
"Category:Matrix Exponential",
"Category:Derivatives"
] |
proofwiki-20337 | Determinant of Matrix Exponential is Non-Zero | :$\det e^{\mathbf A t} \ne 0$
where $\det$ denotes the determinant. | {{ProofWanted|The below outlines an approach. Details need to be completed.}}
The linear system $x' = \mathbf A x$ has $n$ linearly independent solutions.
Putting together these solutions as columns in a matrix creates a matrix solution to the differential equation, considering the initial conditions for the matrix exp... | :$\det e^{\mathbf A t} \ne 0$
where $\det$ denotes the [[Definition:Determinant of Matrix|determinant]]. | {{ProofWanted|The below outlines an approach. Details need to be completed.}}
The linear system $x' = \mathbf A x$ has $n$ [[Linearly Independent Solutions to 1st Order Systems|linearly independent solutions]].
Putting together these solutions as columns in a matrix creates a matrix solution to the differential equat... | Determinant of Matrix Exponential is Non-Zero | https://proofwiki.org/wiki/Determinant_of_Matrix_Exponential_is_Non-Zero | https://proofwiki.org/wiki/Determinant_of_Matrix_Exponential_is_Non-Zero | [
"Matrix Exponential",
"Determinants"
] | [
"Definition:Determinant/Matrix"
] | [
"Linearly Independent Solutions to 1st Order Systems",
"Existence and Uniqueness Theorem for 1st Order IVPs",
"Definition:Unique",
"Liouville's Theorem (Differential Equations)",
"Category:Matrix Exponential",
"Category:Determinants"
] |
proofwiki-20338 | Same-Matrix Product of Matrix Exponentials | :$e^{\mathbf A t} e^{\mathbf A s} = e^{\mathbf A \paren {t + s} }$ | Let
:$\map \Phi t = e^{\mathbf A t} e^{\mathbf A s} - e^{\mathbf A \paren {t + s} }$
for some fixed $s \in \R$.
From Derivative of Matrix Exponential:
{{begin-eqn}}
{{eqn | l = \map {\Phi'} t
| r = \mathbf A e^{\mathbf A t} e^{\mathbf A s} - \mathbf A e^{\mathbf A \paren {t + s} }
| c =
}}
{{eqn | r = \mat... | :$e^{\mathbf A t} e^{\mathbf A s} = e^{\mathbf A \paren {t + s} }$ | Let
:$\map \Phi t = e^{\mathbf A t} e^{\mathbf A s} - e^{\mathbf A \paren {t + s} }$
for some fixed $s \in \R$.
From [[Derivative of Matrix Exponential]]:
{{begin-eqn}}
{{eqn | l = \map {\Phi'} t
| r = \mathbf A e^{\mathbf A t} e^{\mathbf A s} - \mathbf A e^{\mathbf A \paren {t + s} }
| c =
}}
{{eqn | r =... | Same-Matrix Product of Matrix Exponentials | https://proofwiki.org/wiki/Same-Matrix_Product_of_Matrix_Exponentials | https://proofwiki.org/wiki/Same-Matrix_Product_of_Matrix_Exponentials | [
"Matrix Exponential"
] | [] | [
"Derivative of Matrix Exponential",
"Category:Matrix Exponential"
] |
proofwiki-20339 | Inverse of Matrix Exponential | :$\paren {e^{\mathbf A t} }^{-1} = e^{-\mathbf A t}$
where $\paren {e^{\mathbf A t} }^{-1}$ denotes the inverse of $e^{\mathbf A t}$. | {{begin-eqn}}
{{eqn | l = e^{\mathbf A t} e^{-\mathbf A t}
| r = e^{\mathbf A \paren {t - t} }
| c = Same-Matrix Product of Matrix Exponentials
}}
{{eqn | r = e^{\mathbf 0}
| c = {{Defof|Matrix Scalar Product}}: $\mathbf A 0 = \mathbf 0$
}}
{{eqn | r = \mathbf I
| c = Matrix Exponential of Zero ... | :$\paren {e^{\mathbf A t} }^{-1} = e^{-\mathbf A t}$
where $\paren {e^{\mathbf A t} }^{-1}$ denotes the [[Definition:Inverse Matrix|inverse]] of $e^{\mathbf A t}$. | {{begin-eqn}}
{{eqn | l = e^{\mathbf A t} e^{-\mathbf A t}
| r = e^{\mathbf A \paren {t - t} }
| c = [[Same-Matrix Product of Matrix Exponentials]]
}}
{{eqn | r = e^{\mathbf 0}
| c = {{Defof|Matrix Scalar Product}}: $\mathbf A 0 = \mathbf 0$
}}
{{eqn | r = \mathbf I
| c = [[Matrix Exponential of... | Inverse of Matrix Exponential | https://proofwiki.org/wiki/Inverse_of_Matrix_Exponential | https://proofwiki.org/wiki/Inverse_of_Matrix_Exponential | [
"Matrix Exponential",
"Inverse Matrices"
] | [
"Definition:Inverse Matrix"
] | [
"Same-Matrix Product of Matrix Exponentials",
"Matrix Exponential of Zero Matrix",
"Definition:Zero Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Unit Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Inverse Matrix",
"Category:Matrix Exponential",
"Category:Inverse Matri... |
proofwiki-20340 | Product with Matrix Exponential of Commutative Matrices | Let $\mathbf A \mathbf B = \mathbf B \mathbf A$.
Then:
:$e^{\mathbf A t} \mathbf B = \mathbf B e^{\mathbf A t}$ | Let:
:$\map \Phi t = e^{\mathbf A t} \mathbf B - \mathbf B e^{\mathbf A t}$
{{ProofWanted|and then follow the same program outlined in the Same-Matrix Product of Matrix Exponentials proof.}}
Category:Matrix Exponential
Category:Commutativity
r8p9y8xiyh8n22f197qlkg6cgyx4ly3 | Let $\mathbf A \mathbf B = \mathbf B \mathbf A$.
Then:
:$e^{\mathbf A t} \mathbf B = \mathbf B e^{\mathbf A t}$ | Let:
:$\map \Phi t = e^{\mathbf A t} \mathbf B - \mathbf B e^{\mathbf A t}$
{{ProofWanted|and then follow the same program outlined in the [[Same-Matrix Product of Matrix Exponentials]] proof.}}
[[Category:Matrix Exponential]]
[[Category:Commutativity]]
r8p9y8xiyh8n22f197qlkg6cgyx4ly3 | Product with Matrix Exponential of Commutative Matrices | https://proofwiki.org/wiki/Product_with_Matrix_Exponential_of_Commutative_Matrices | https://proofwiki.org/wiki/Product_with_Matrix_Exponential_of_Commutative_Matrices | [
"Matrix Exponential",
"Commutativity"
] | [] | [
"Same-Matrix Product of Matrix Exponentials",
"Category:Matrix Exponential",
"Category:Commutativity"
] |
proofwiki-20341 | Matrix Exponential of Sum of Commutative Matrices | Let $\mathbf A \mathbf B = \mathbf B \mathbf A$.
Then:
:$e^{\mathbf A t} e^{\mathbf B t} = e^{\paren {\mathbf A + \mathbf B} t}$ | Let:
:$\map \Phi t = e^{\mathbf A t} e^{\mathbf B t} - e^{\paren {\mathbf A + \mathbf B} t}$
{{ProofWanted|and then follow the same program outlined in the Same-Matrix Product of Matrix Exponentials proof.}} | Let $\mathbf A \mathbf B = \mathbf B \mathbf A$.
Then:
:$e^{\mathbf A t} e^{\mathbf B t} = e^{\paren {\mathbf A + \mathbf B} t}$ | Let:
:$\map \Phi t = e^{\mathbf A t} e^{\mathbf B t} - e^{\paren {\mathbf A + \mathbf B} t}$
{{ProofWanted|and then follow the same program outlined in the [[Same-Matrix Product of Matrix Exponentials]] proof.}} | Matrix Exponential of Sum of Commutative Matrices | https://proofwiki.org/wiki/Matrix_Exponential_of_Sum_of_Commutative_Matrices | https://proofwiki.org/wiki/Matrix_Exponential_of_Sum_of_Commutative_Matrices | [
"Matrix Exponential",
"Commutativity"
] | [] | [
"Same-Matrix Product of Matrix Exponentials"
] |
proofwiki-20342 | Series Expansion of Matrix Exponential | :$\ds e^{\mathbf A t} = \sum_{n \mathop = 0}^\infty \frac {t^n} {n!} \mathbf A^n$ | {{ProofWanted|Differentiating the series term-by-term and evaluating at $t {{=}} 0$ proves the series satisfies the same definition as the matrix exponential, and hence by uniqueness is equal.}}
{{finish|It needs to be shown that it converges and can be differentiated termwise}} | :$\ds e^{\mathbf A t} = \sum_{n \mathop = 0}^\infty \frac {t^n} {n!} \mathbf A^n$ | {{ProofWanted|Differentiating the series term-by-term and evaluating at $t {{=}} 0$ proves the series satisfies the same definition as the matrix exponential, and hence by uniqueness is equal.}}
{{finish|It needs to be shown that it converges and can be differentiated termwise}} | Series Expansion of Matrix Exponential | https://proofwiki.org/wiki/Series_Expansion_of_Matrix_Exponential | https://proofwiki.org/wiki/Series_Expansion_of_Matrix_Exponential | [
"Matrix Exponential"
] | [] | [] |
proofwiki-20343 | Algebra of Sets is Closed under Intersection | Let $S$ be a set.
Let $\RR$ be an algebra of sets on $S$.
Then:
:$\forall A, B \in S: A \cap B \in \RR$ | By definition $2$ of Algebra of Sets:
:$\RR$ is a ring of sets with a unit.
By definition $1$ of Ring of Sets:
{{begin-axiom}}
{{axiom | n = \text {RS} 2_1
| lc= Closure under Intersection:
| q = \forall A, B \in \RR
| m = A \cap B \in \RR
}}
{{end-axiom}}
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\RR$ be an [[Definition:Algebra of Sets|algebra of sets]] on $S$.
Then:
:$\forall A, B \in S: A \cap B \in \RR$ | By [[Definition:Algebra of Sets/Definition 2|definition $2$ of Algebra of Sets]]:
:$\RR$ is a [[Definition:Ring of Sets|ring of sets]] with a [[Definition:Unit of System of Sets|unit]].
By [[Definition:Ring of Sets/Definition 1|definition $1$ of Ring of Sets]]:
{{begin-axiom}}
{{axiom | n = \text {RS} 2_1
| ... | Algebra of Sets is Closed under Intersection/Proof 1 | https://proofwiki.org/wiki/Algebra_of_Sets_is_Closed_under_Intersection | https://proofwiki.org/wiki/Algebra_of_Sets_is_Closed_under_Intersection/Proof_1 | [
"Algebra of Sets is Closed under Intersection",
"Algebras of Sets",
"Set Intersection"
] | [
"Definition:Set",
"Definition:Algebra of Sets"
] | [
"Definition:Algebra of Sets/Definition 2",
"Definition:Ring of Sets",
"Definition:Unit of System of Sets",
"Definition:Ring of Sets/Definition 1",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Intersection"
] |
proofwiki-20344 | Algebra of Sets is Closed under Intersection | Let $S$ be a set.
Let $\RR$ be an algebra of sets on $S$.
Then:
:$\forall A, B \in S: A \cap B \in \RR$ | By definition $1$ of algebra of sets, we have that:
{{begin-axiom}}
{{axiom | n = \text {AS} 2
| lc= Closure under Union:
| q = \forall A, B \in \RR
| m = A \cup B \in \RR
}}
{{axiom | n = \text {AS} 3
| lc= Closure under Complement Relative to $S$:
| q = \forall A \in \RR
... | Let $S$ be a [[Definition:Set|set]].
Let $\RR$ be an [[Definition:Algebra of Sets|algebra of sets]] on $S$.
Then:
:$\forall A, B \in S: A \cap B \in \RR$ | By [[Definition:Algebra of Sets/Definition 1|definition $1$ of algebra of sets]], we have that:
{{begin-axiom}}
{{axiom | n = \text {AS} 2
| lc= [[Definition:Closed Algebraic Structure|Closure]] under [[Definition:Set Union|Union]]:
| q = \forall A, B \in \RR
| m = A \cup B \in \RR
}}
{{axiom |... | Algebra of Sets is Closed under Intersection/Proof 2 | https://proofwiki.org/wiki/Algebra_of_Sets_is_Closed_under_Intersection | https://proofwiki.org/wiki/Algebra_of_Sets_is_Closed_under_Intersection/Proof_2 | [
"Algebra of Sets is Closed under Intersection",
"Algebras of Sets",
"Set Intersection"
] | [
"Definition:Set",
"Definition:Algebra of Sets"
] | [
"Definition:Algebra of Sets/Definition 1",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Union",
"Definition:Closed under Mapping",
"Definition:Relative Complement",
"De Morgan's Laws (Set Theory)/Relative Complement/Complement of Intersection"
] |
proofwiki-20345 | Algebra of Sets is Closed under Set Difference | Let $S$ be a set.
Let $\RR$ be an algebra of sets on $S$.
Then:
:$\forall A, B \in S: A \setminus B \in \RR$ | By definition $1$ of algebra of sets, we have that:
{{begin-axiom}}
{{axiom | n = \text {AS} 3
| lc= Closure under Complement Relative to $S$:
| q = \forall A \in \RR
| m = \relcomp S A \in \RR
}}
{{end-axiom}}
Thus:
{{begin-eqn}}
{{eqn | l = A, B
| o = \in
| r = \RR
| c =
}}
... | Let $S$ be a [[Definition:Set|set]].
Let $\RR$ be an [[Definition:Algebra of Sets|algebra of sets]] on $S$.
Then:
:$\forall A, B \in S: A \setminus B \in \RR$ | By [[Definition:Algebra of Sets/Definition 1|definition $1$ of algebra of sets]], we have that:
{{begin-axiom}}
{{axiom | n = \text {AS} 3
| lc= [[Definition:Closed under Mapping|Closure]] under [[Definition:Relative Complement|Complement Relative to $S$]]:
| q = \forall A \in \RR
| m = \relcom... | Algebra of Sets is Closed under Set Difference/Proof 2 | https://proofwiki.org/wiki/Algebra_of_Sets_is_Closed_under_Set_Difference | https://proofwiki.org/wiki/Algebra_of_Sets_is_Closed_under_Set_Difference/Proof_2 | [
"Algebra of Sets is Closed under Set Difference",
"Algebras of Sets",
"Set Difference"
] | [
"Definition:Set",
"Definition:Algebra of Sets"
] | [
"Definition:Algebra of Sets/Definition 1",
"Definition:Closed under Mapping",
"Definition:Relative Complement",
"Algebra of Sets is Closed under Intersection",
"Set Difference as Intersection with Relative Complement"
] |
proofwiki-20346 | Algebra of Sets contains Underlying Set | Let $S$ be a set.
Let $\RR$ be an algebra of sets on $S$.
Then:
:$S \in \RR$ | By definition $1$ of algebra of sets, we have that:
{{begin-axiom}}
{{axiom | n = \text {AS} 1
| lc= Unit:
| m = S \in \RR
}}
{{end-axiom}}
The result is hence immediate.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\RR$ be an [[Definition:Algebra of Sets|algebra of sets]] on $S$.
Then:
:$S \in \RR$ | By [[Definition:Algebra of Sets/Definition 1|definition $1$ of algebra of sets]], we have that:
{{begin-axiom}}
{{axiom | n = \text {AS} 1
| lc= [[Definition:Unit of System of Sets|Unit]]:
| m = S \in \RR
}}
{{end-axiom}}
The result is hence immediate.
{{qed}} | Algebra of Sets contains Underlying Set/Proof 1 | https://proofwiki.org/wiki/Algebra_of_Sets_contains_Underlying_Set | https://proofwiki.org/wiki/Algebra_of_Sets_contains_Underlying_Set/Proof_1 | [
"Algebra of Sets contains Underlying Set",
"Algebras of Sets"
] | [
"Definition:Set",
"Definition:Algebra of Sets"
] | [
"Definition:Algebra of Sets/Definition 1",
"Definition:Unit of System of Sets"
] |
proofwiki-20347 | Algebra of Sets contains Underlying Set | Let $S$ be a set.
Let $\RR$ be an algebra of sets on $S$.
Then:
:$S \in \RR$ | By definition $1$ of algebra of sets, we have that:
{{begin-axiom}}
{{axiom | n = \text {AS} 2
| lc= Closure under Union:
| q = \forall A, B \in \RR
| m = A \cup B \in \RR
}}
{{axiom | n = \text {AS} 3
| lc= Closure under Complement Relative to $S$:
| q = \forall A \in \RR
... | Let $S$ be a [[Definition:Set|set]].
Let $\RR$ be an [[Definition:Algebra of Sets|algebra of sets]] on $S$.
Then:
:$S \in \RR$ | By [[Definition:Algebra of Sets/Definition 1|definition $1$ of algebra of sets]], we have that:
{{begin-axiom}}
{{axiom | n = \text {AS} 2
| lc= [[Definition:Closed Algebraic Structure|Closure]] under [[Definition:Set Union|Union]]:
| q = \forall A, B \in \RR
| m = A \cup B \in \RR
}}
{{axiom |... | Algebra of Sets contains Underlying Set/Proof 2 | https://proofwiki.org/wiki/Algebra_of_Sets_contains_Underlying_Set | https://proofwiki.org/wiki/Algebra_of_Sets_contains_Underlying_Set/Proof_2 | [
"Algebra of Sets contains Underlying Set",
"Algebras of Sets"
] | [
"Definition:Set",
"Definition:Algebra of Sets"
] | [
"Definition:Algebra of Sets/Definition 1",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Set Union",
"Definition:Closed under Mapping",
"Definition:Relative Complement",
"Union with Relative Complement"
] |
proofwiki-20348 | Equivalence of Definitions of Generator of Module | Let $R$ be a ring.
Let $M$ be an $R$-module.
Let $S \subseteq M$ be a subset of $M$.
{{TFAE|def = Generator of Module}} | === Definition 1 implies Definition 2 ===
By definition of generated submodule, it follows that:
:$\ds M := \bigcap \set { M' \subseteq M : S \subseteq M', \textrm {$M'$ is a submodule of $M$} }$
Suppose that $M'$ is a proper submodule of $M$ such that $S \subseteq M'$.
It follows that there exists $x \in M \setminus M... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $M$ be an [[Definition:Module over Ring|$R$-module]].
Let $S \subseteq M$ be a [[Definition:Subset|subset]] of $M$.
{{TFAE|def = Generator of Module}} | === [[Definition:Generator of Module/Definition 1|Definition 1]] implies [[Definition:Generator of Module/Definition 2|Definition 2]] ===
By definition of [[Definition:Generated Submodule|generated submodule]], it follows that:
:$\ds M := \bigcap \set { M' \subseteq M : S \subseteq M', \textrm {$M'$ is a submodule of ... | Equivalence of Definitions of Generator of Module | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Generator_of_Module | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Generator_of_Module | [
"Generators of Modules"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Module over Ring",
"Definition:Subset"
] | [
"Definition:Generator of Module/Definition 1",
"Definition:Generator of Module/Definition 2",
"Definition:Generated Submodule",
"Definition:Submodule/Proper",
"Definition:Set Intersection/Family of Sets",
"Definition:Generator of Module/Definition 2",
"Definition:Generator of Module/Definition 2",
"De... |
proofwiki-20349 | Null Sequence induces Neighborhood Basis of Closed Sets in Metric Space | Let $M = \struct {A, d}$ be a metric space.
Let $a \in A$.
Let $\sequence {x_n}$ be a real null sequence such that:
:$\forall n \in N: x_n > 0$
Let $\map {B^-_\epsilon} a$ denote the closed $\epsilon$-ball of $a$ in $M$.
Then:
:$\NN_{\sequence {x_n} } = \set{\map {B^-_{x_n} } a : n \in \N}$ is a neighborhood basis of $... | From Closed Ball in Metric Space is Closed Neighborhood:
:every element of $\NN_{\sequence {x_n} }$ is an closed neighborhood of $a$.
By definition of closed neighborhood of $a$:
:every element of $\NN_{\sequence {x_n} }$ is an neighborhood of $a$ and a closed set.
Let $U$ be an open neighborhood of $a$.
By definition ... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $a \in A$.
Let $\sequence {x_n}$ be a [[Definition:Real Null Sequence|real null sequence]] such that:
:$\forall n \in N: x_n > 0$
Let $\map {B^-_\epsilon} a$ denote the [[Definition:Closed Ball of Metric Space|closed $\epsilon$-ball of $a$ i... | From [[Closed Ball in Metric Space is Closed Neighborhood]]:
:every [[Definition:Element|element]] of $\NN_{\sequence {x_n} }$ is an [[Definition:Closed Neighborhood|closed neighborhood]] of $a$.
By definition of [[Definition:Closed Neighborhood|closed neighborhood]] of $a$:
:every [[Definition:Element|element]] of $\... | Null Sequence induces Neighborhood Basis of Closed Sets in Metric Space | https://proofwiki.org/wiki/Null_Sequence_induces_Neighborhood_Basis_of_Closed_Sets_in_Metric_Space | https://proofwiki.org/wiki/Null_Sequence_induces_Neighborhood_Basis_of_Closed_Sets_in_Metric_Space | [
"Metric Spaces"
] | [
"Definition:Metric Space",
"Definition:Null Sequence/Real Numbers",
"Definition:Closed Ball/Metric Space",
"Definition:Neighborhood Basis",
"Definition:Closed Set/Metric Space"
] | [
"Closed Ball in Metric Space is Closed Neighborhood",
"Definition:Element",
"Definition:Closed Neighborhood",
"Definition:Closed Neighborhood",
"Definition:Element",
"Definition:Neighborhood (Metric Space)",
"Definition:Closed Set/Metric Space",
"Definition:Open Neighborhood/Point",
"Definition:Open... |
proofwiki-20350 | Interior of Jordan Curve is Subset of Image of Null-Homotopy | Let $f : \closedint 0 1 \to \R^2$ be a Jordan curve.
Let $H : \closedint 0 1 \times \closedint 0 1 \to \R^2$ be a path homotopy between $f$ and a constant loop.
Then:
:$\Int f \subseteq \Img H$
where $\Int f$ denotes the interior of $f$, and $\Img H$ denotes the image of $H$. | Let $\map \Omega {\R^2, \map f 0}$ denote the set of all loops based at $\map f 0$, where $\R^2$ is the real number plane with euclidean topology.
Let $c_{\map f 0}: \closedint 0 1 \to \set { \map f 0 }$ be the constant loop.
That is, $c_{\map f 0}$ is the loop $c_{\map f 0} \in \map \Omega {\R^2, \map f 0}$ such that:... | Let $f : \closedint 0 1 \to \R^2$ be a [[Definition:Jordan Curve|Jordan curve]].
Let $H : \closedint 0 1 \times \closedint 0 1 \to \R^2$ be a [[Definition:Path Homotopy|path homotopy]] between $f$ and a [[Definition:Constant Loop (Topology)|constant loop]].
Then:
:$\Int f \subseteq \Img H$
where $\Int f$ denotes th... | Let $\map \Omega {\R^2, \map f 0}$ denote the [[Definition:Set of All Loops (Topology)|set of all loops]] [[Definition:Base Point of Loop|based]] at $\map f 0$, where $\R^2$ is the [[Definition:Real Number Plane with Euclidean Topology|real number plane with euclidean topology]].
Let $c_{\map f 0}: \closedint 0 1 \to ... | Interior of Jordan Curve is Subset of Image of Null-Homotopy | https://proofwiki.org/wiki/Interior_of_Jordan_Curve_is_Subset_of_Image_of_Null-Homotopy | https://proofwiki.org/wiki/Interior_of_Jordan_Curve_is_Subset_of_Image_of_Null-Homotopy | [
"Jordan Curves"
] | [
"Definition:Jordan Curve",
"Definition:Homotopy/Path/Path Homotopy",
"Definition:Loop (Topology)/Constant Loop",
"Definition:Jordan Curve/Interior",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Definition:Loop (Topology)/Set of All Loops",
"Definition:Loop (Topology)/Base Point",
"Definition:Euclidean Space/Euclidean Topology/Real Number Plane",
"Definition:Loop (Topology)/Constant Loop",
"Definition:Loop (Topology)",
"Definition:Homotopy/Path",
"Definition:Given",
"Definition:Homotopy/Path... |
proofwiki-20351 | Closed Ball is Simply Connected | Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space over $\R$ or $\C$.
Let $d: V \times V \to \R_{\ge 0}$ be the metric induced by the norm $\norm {\,\cdot\,}$ on $V$.
Let $\tau$ be the the topology on $V$ induced by the metric $d$.
Let $v \in V$ and $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon^-} v$ be the... | Normed Vector Space is Hausdorff Topological Vector Space shows that $\struct {V, \tau}$ is a topological vector space.
The result now follows from Closed Ball is Convex Set and Convex Set is Simply Connected.
{{qed}}
Category:Normed Vector Spaces
Category:Closed Balls
Category:Simply Connected Spaces
axs5o7vf121ocwb09... | Let $\struct {V, \norm {\,\cdot\,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\R$ or $\C$.
Let $d: V \times V \to \R_{\ge 0}$ be the [[Definition:Metric Induced by Norm|metric induced]] by the [[Definition:Norm on Vector Space|norm]] $\norm {\,\cdot\,}$ on $V$.
Let $\tau$ be the the [[Defini... | [[Normed Vector Space is Hausdorff Topological Vector Space]] shows that $\struct {V, \tau}$ is a [[Definition:Topological Vector Space|topological vector space]].
The result now follows from [[Closed Ball is Convex Set]] and [[Convex Set is Simply Connected]].
{{qed}}
[[Category:Normed Vector Spaces]]
[[Category:Cl... | Closed Ball is Simply Connected | https://proofwiki.org/wiki/Closed_Ball_is_Simply_Connected | https://proofwiki.org/wiki/Closed_Ball_is_Simply_Connected | [
"Normed Vector Spaces",
"Closed Balls",
"Simply Connected Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Metric Induced by Norm",
"Definition:Norm/Vector Space",
"Definition:Topology Induced by Metric",
"Definition:Closed Ball/Normed Vector Space",
"Definition:Topological Subspace",
"Definition:Simply Connected"
] | [
"Normed Vector Space is Hausdorff Topological Vector Space",
"Definition:Topological Vector Space",
"Closed Ball is Convex Set",
"Convex Set is Simply Connected",
"Category:Normed Vector Spaces",
"Category:Closed Balls",
"Category:Simply Connected Spaces"
] |
proofwiki-20352 | Closed Ball in Metric Space is Closed Neighborhood | Let $M = \struct {A, d}$ be a metric space.
Let $x \in A$.
Let $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon^-} x$ be the closed $\epsilon$-ball of $x$ in $M$.
Then $\map {B_\epsilon^-} x$ is a closed neighborhood of $x$ in $M$. | From Closed Ball contains Smaller Open Ball:
:$\map {B_\epsilon} x \subseteq \map {B_\epsilon^-} x$
where $\map {B_\epsilon} x$ denotes the open $\epsilon$-ball of $x$ in $M$.
Hence $\map {B_\epsilon^-} x$ is a neighborhood of $x$ by definition.
From Closed Ball is Closed in Metric Space:
:$\map {B_\epsilon^-} x$ is cl... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Let $x \in A$.
Let $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon^-} x$ be the [[Definition:Closed Ball|closed $\epsilon$-ball]] of $x$ in $M$.
Then $\map {B_\epsilon^-} x$ is a [[Definition:Closed Neighborhood|closed neighborhood]] of $x$ in $... | From [[Closed Ball contains Smaller Open Ball]]:
:$\map {B_\epsilon} x \subseteq \map {B_\epsilon^-} x$
where $\map {B_\epsilon} x$ denotes the [[Definition:Open Ball|open $\epsilon$-ball]] of $x$ in $M$.
Hence $\map {B_\epsilon^-} x$ is a [[Definition:Neighborhood (Metric Space)|neighborhood]] of $x$ by definition.
... | Closed Ball in Metric Space is Closed Neighborhood | https://proofwiki.org/wiki/Closed_Ball_in_Metric_Space_is_Closed_Neighborhood | https://proofwiki.org/wiki/Closed_Ball_in_Metric_Space_is_Closed_Neighborhood | [
"Closed Sets (Metric Spaces)",
"Neighborhoods"
] | [
"Definition:Metric Space",
"Definition:Closed Ball",
"Definition:Closed Neighborhood"
] | [
"Closed Ball contains Smaller Open Ball",
"Definition:Open Ball",
"Definition:Neighborhood (Metric Space)",
"Closed Ball is Closed/Metric Space",
"Definition:Closed Set/Metric Space",
"Definition:Closed Neighborhood",
"Category:Closed Sets (Metric Spaces)",
"Category:Neighborhoods"
] |
proofwiki-20353 | Koopman Operator is Isometry | Let $\struct {X, \BB, \mu, T}$ be a measure-preserving dynamical system.
Let $\map {L^2_\C} \mu$ be the complex-valued $L^2$ space of $\mu$.
Let $U_T : \map {L^2_\C} \mu \to \map {L^2_\C} \mu$ of $T$ be the Koopman operator.
Let $\innerprod \cdot \cdot$ denote the inner product on $\map {L^2_\C} \mu$, i.e.
:$\ds \foral... | Let $f, g \in \map {L^2_\C} \mu $.
Then:
{{begin-eqn}}
{{eqn | l = \innerprod {U_T f} {U_T g}
| r = \int {\overline {U_T f} } \; {U_T g} \rd \mu
}}
{{eqn | r = \int {\overline {f \circ T} } \; {g \circ T} \rd \mu
| c = {{Defof|Koopman Operator on Complex L-2 Space|Koopman operator}}
}}
{{eqn | r = \int {\o... | Let $\struct {X, \BB, \mu, T}$ be a [[Definition:Measure-Preserving Dynamical System|measure-preserving dynamical system]].
Let $\map {L^2_\C} \mu$ be the [[Definition:Complex-Valued Function|complex-valued]] [[Definition:Lp Space|$L^2$ space]] of $\mu$.
Let $U_T : \map {L^2_\C} \mu \to \map {L^2_\C} \mu$ of $T$ be t... | Let $f, g \in \map {L^2_\C} \mu $.
Then:
{{begin-eqn}}
{{eqn | l = \innerprod {U_T f} {U_T g}
| r = \int {\overline {U_T f} } \; {U_T g} \rd \mu
}}
{{eqn | r = \int {\overline {f \circ T} } \; {g \circ T} \rd \mu
| c = {{Defof|Koopman Operator on Complex L-2 Space|Koopman operator}}
}}
{{eqn | r = \int {\... | Koopman Operator is Isometry | https://proofwiki.org/wiki/Koopman_Operator_is_Isometry | https://proofwiki.org/wiki/Koopman_Operator_is_Isometry | [
"Operator Theory"
] | [
"Definition:Measure-Preserving Dynamical System",
"Definition:Complex-Valued Function",
"Definition:Lp Space",
"Definition:Koopman Operator on Complex L-2 Space",
"Definition:L-2 Inner Product",
"Definition:Complex Conjugate",
"Definition:Isometry (Hilbert Spaces)",
"L-2 Space forms Hilbert Space"
] | [
"Category:Operator Theory"
] |
proofwiki-20354 | Locally Euclidean Space is Locally Connected | Let $M$ be a locally Euclidean space of some dimension $d$.
Then $M$ is locally connected. | Follows from:
* Locally Euclidean Space is Locally Path-Connected
* Locally Path-Connected Space is Locally Connected
{{qed}}
Category:Locally Euclidean Spaces
ku9zokev8ccp0o5fjm382r1uxhhdij5 | Let $M$ be a [[Definition:Locally Euclidean Space|locally Euclidean space]] of some dimension $d$.
Then $M$ is [[Definition:Locally Connected Space|locally connected]]. | Follows from:
* [[Locally Euclidean Space is Locally Path-Connected]]
* [[Locally Path-Connected Space is Locally Connected]]
{{qed}}
[[Category:Locally Euclidean Spaces]]
ku9zokev8ccp0o5fjm382r1uxhhdij5 | Locally Euclidean Space is Locally Connected | https://proofwiki.org/wiki/Locally_Euclidean_Space_is_Locally_Connected | https://proofwiki.org/wiki/Locally_Euclidean_Space_is_Locally_Connected | [
"Locally Euclidean Spaces"
] | [
"Definition:Locally Euclidean Space",
"Definition:Locally Connected Space"
] | [
"Locally Euclidean Space is Locally Path-Connected",
"Locally Path-Connected Space is Locally Connected",
"Category:Locally Euclidean Spaces"
] |
proofwiki-20355 | Open Ball in Normed Vector Space is Path-Connected | Let $V$ be a normed vector space with norm $\norm {\,\cdot\,}$ over $\R$ or $\C$.
An open ball in the metric induced by $\norm {\,\cdot\,}$ is path-connected. | Follows from:
* Open Ball is Convex Set
* Normed Vector Space is Hausdorff Topological Vector Space
* Convex Set is Path-Connected
{{qed}}
Category:Normed Vector Spaces
Category:Open Balls
Category:Path-Connected Spaces
5ide26bnfbtclec4vej8504r9z9jdg6 | Let $V$ be a [[Definition:Normed Vector Space|normed vector space]] with [[Definition:Norm on Vector Space|norm]] $\norm {\,\cdot\,}$ over $\R$ or $\C$.
An [[Definition:Open Ball in Normed Vector Space|open ball]] in the [[Definition:Metric Induced by Norm|metric induced by $\norm {\,\cdot\,}$]] is [[Definition:Path-... | Follows from:
* [[Open Ball is Convex Set]]
* [[Normed Vector Space is Hausdorff Topological Vector Space]]
* [[Convex Set is Path-Connected]]
{{qed}}
[[Category:Normed Vector Spaces]]
[[Category:Open Balls]]
[[Category:Path-Connected Spaces]]
5ide26bnfbtclec4vej8504r9z9jdg6 | Open Ball in Normed Vector Space is Path-Connected | https://proofwiki.org/wiki/Open_Ball_in_Normed_Vector_Space_is_Path-Connected | https://proofwiki.org/wiki/Open_Ball_in_Normed_Vector_Space_is_Path-Connected | [
"Normed Vector Spaces",
"Open Balls",
"Path-Connected Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Norm/Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Metric Induced by Norm",
"Definition:Path-Connected/Metric Space/Subset"
] | [
"Open Ball is Convex Set",
"Normed Vector Space is Hausdorff Topological Vector Space",
"Convex Set is Path-Connected",
"Category:Normed Vector Spaces",
"Category:Open Balls",
"Category:Path-Connected Spaces"
] |
proofwiki-20356 | Closed Ball is Path-Connected | Let $V$ be a normed vector space with norm $\norm {\,\cdot\,}$ over $\R$ or $\C$.
A closed ball in the metric induced by $\norm {\,\cdot\,}$ is path-connected. | Follows from:
* Closed Ball is Convex Set
* Normed Vector Space is Hausdorff Topological Vector Space
* Convex Set is Path-Connected
{{qed}}
Category:Normed Vector Spaces
Category:Closed Balls
Category:Path-Connected Metric Spaces
8pek86aqhq85ffzxuuvbnyel1hy3rvq | Let $V$ be a [[Definition:Normed Vector Space|normed vector space]] with [[Definition:Norm on Vector Space|norm]] $\norm {\,\cdot\,}$ over $\R$ or $\C$.
A [[Definition:Closed Ball in Normed Vector Space|closed ball]] in the [[Definition:Metric Induced by Norm|metric induced by $\norm {\,\cdot\,}$]] is [[Definition:Pat... | Follows from:
* [[Closed Ball is Convex Set]]
* [[Normed Vector Space is Hausdorff Topological Vector Space]]
* [[Convex Set is Path-Connected]]
{{qed}}
[[Category:Normed Vector Spaces]]
[[Category:Closed Balls]]
[[Category:Path-Connected Metric Spaces]]
8pek86aqhq85ffzxuuvbnyel1hy3rvq | Closed Ball is Path-Connected | https://proofwiki.org/wiki/Closed_Ball_is_Path-Connected | https://proofwiki.org/wiki/Closed_Ball_is_Path-Connected | [
"Normed Vector Spaces",
"Closed Balls",
"Path-Connected Metric Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Norm/Vector Space",
"Definition:Closed Ball/Normed Vector Space",
"Definition:Metric Induced by Norm",
"Definition:Path-Connected/Metric Space/Subset"
] | [
"Closed Ball is Convex Set",
"Normed Vector Space is Hausdorff Topological Vector Space",
"Convex Set is Path-Connected",
"Category:Normed Vector Spaces",
"Category:Closed Balls",
"Category:Path-Connected Metric Spaces"
] |
proofwiki-20357 | Open Ball in Normed Vector Space is Connected | Let $V$ be a normed vector space with norm $\norm {\,\cdot\,}$ over $\R$ or $\C$.
An open ball in the metric induced by $\norm {\,\cdot\,}$ is connected. | Follows from:
:Open Ball in Normed Vector Space is Path-Connected
:Path-Connected Space is Connected.
{{qed}}
Category:Open Balls
Category:Connected Topological Spaces
Category:Normed Vector Spaces
t90k3juviw6h8incconezi6gs4n82xp | Let $V$ be a [[Definition:Normed Vector Space|normed vector space]] with [[Definition:Norm on Vector Space|norm]] $\norm {\,\cdot\,}$ over $\R$ or $\C$.
An [[Definition:Open Ball in Normed Vector Space|open ball]] in the [[Definition:Metric Induced by Norm|metric induced by $\norm {\,\cdot\,}$]] is [[Definition:Connec... | Follows from:
:[[Open Ball in Normed Vector Space is Path-Connected]]
:[[Path-Connected Space is Connected]].
{{qed}}
[[Category:Open Balls]]
[[Category:Connected Topological Spaces]]
[[Category:Normed Vector Spaces]]
t90k3juviw6h8incconezi6gs4n82xp | Open Ball in Normed Vector Space is Connected | https://proofwiki.org/wiki/Open_Ball_in_Normed_Vector_Space_is_Connected | https://proofwiki.org/wiki/Open_Ball_in_Normed_Vector_Space_is_Connected | [
"Open Balls",
"Connected Topological Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Norm/Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Metric Induced by Norm",
"Definition:Connected Set (Topology)"
] | [
"Open Ball in Normed Vector Space is Path-Connected",
"Path-Connected Space is Connected",
"Category:Open Balls",
"Category:Connected Topological Spaces",
"Category:Normed Vector Spaces"
] |
proofwiki-20358 | Closed Ball is Connected | Let $V$ be a normed vector space with norm $\norm {\,\cdot\,}$ over $\R$ or $\C$.
A closed ball in the metric induced by $\norm {\,\cdot\,}$ is connected. | Follows from:
* Closed Ball is Path-Connected
* Path-Connected Space is Connected
{{qed}}
Category:Closed Balls
Category:Connected Topological Spaces
Category:Normed Vector Spaces
5jupmxqpzk0kxg1xwz3jkxk79c8u3xo | Let $V$ be a [[Definition:Normed Vector Space|normed vector space]] with [[Definition:Norm on Vector Space|norm]] $\norm {\,\cdot\,}$ over $\R$ or $\C$.
A [[Definition:Closed Ball in Normed Vector Space|closed ball]] in the [[Definition:Metric Induced by Norm|metric induced by $\norm {\,\cdot\,}$]] is [[Definition:Con... | Follows from:
* [[Closed Ball is Path-Connected]]
* [[Path-Connected Space is Connected]]
{{qed}}
[[Category:Closed Balls]]
[[Category:Connected Topological Spaces]]
[[Category:Normed Vector Spaces]]
5jupmxqpzk0kxg1xwz3jkxk79c8u3xo | Closed Ball is Connected | https://proofwiki.org/wiki/Closed_Ball_is_Connected | https://proofwiki.org/wiki/Closed_Ball_is_Connected | [
"Closed Balls",
"Connected Topological Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Norm/Vector Space",
"Definition:Closed Ball/Normed Vector Space",
"Definition:Metric Induced by Norm",
"Definition:Connected Set (Topology)"
] | [
"Closed Ball is Path-Connected",
"Path-Connected Space is Connected",
"Category:Closed Balls",
"Category:Connected Topological Spaces",
"Category:Normed Vector Spaces"
] |
proofwiki-20359 | Topological Manifold is Locally Connected | Let $M$ be a topological manifold.
Then $M$ is a locally connected space. | By definition of manifold:
:$M$ is a locally Euclidean space
The result follows from Locally Euclidean Space is Locally Connected
{{qed}}
Category:Topological Manifolds
Category:Locally Connected Spaces
cj89v0cgwqy7p3ezbnqjx60klfmgtbe | Let $M$ be a [[Definition:Topological Manifold|topological manifold]].
Then $M$ is a [[Definition:Locally Connected Space|locally connected space]]. | By definition of [[Definition:Topological Manifold|manifold]]:
:$M$ is a [[Definition:Locally Euclidean Space|locally Euclidean space]]
The result follows from [[Locally Euclidean Space is Locally Connected]]
{{qed}}
[[Category:Topological Manifolds]]
[[Category:Locally Connected Spaces]]
cj89v0cgwqy7p3ezbnqjx60klfm... | Topological Manifold is Locally Connected | https://proofwiki.org/wiki/Topological_Manifold_is_Locally_Connected | https://proofwiki.org/wiki/Topological_Manifold_is_Locally_Connected | [
"Topological Manifolds",
"Locally Connected Spaces"
] | [
"Definition:Topological Manifold",
"Definition:Locally Connected Space"
] | [
"Definition:Topological Manifold",
"Definition:Locally Euclidean Space",
"Locally Euclidean Space is Locally Connected",
"Category:Topological Manifolds",
"Category:Locally Connected Spaces"
] |
proofwiki-20360 | Topological Manifold is Locally Compact | Let $M$ be a topological manifold.
Then $M$ is a locally compact space. | By definition of manifold:
:$M$ is a locally Euclidean space.
The result follows from Locally Euclidean Space is Locally Compact.
{{qed}}
Category:Topological Manifolds
Category:Locally Compact Spaces
7gi7qvjnqywoyzelaubwtao4f7ts6or | Let $M$ be a [[Definition:Topological Manifold|topological manifold]].
Then $M$ is a [[Definition:Locally Compact Space|locally compact space]]. | By definition of [[Definition:Topological Manifold|manifold]]:
:$M$ is a [[Definition:Locally Euclidean Space|locally Euclidean space]].
The result follows from [[Locally Euclidean Space is Locally Compact]].
{{qed}}
[[Category:Topological Manifolds]]
[[Category:Locally Compact Spaces]]
7gi7qvjnqywoyzelaubwtao4f7ts6o... | Topological Manifold is Locally Compact | https://proofwiki.org/wiki/Topological_Manifold_is_Locally_Compact | https://proofwiki.org/wiki/Topological_Manifold_is_Locally_Compact | [
"Topological Manifolds",
"Locally Compact Spaces"
] | [
"Definition:Topological Manifold",
"Definition:Locally Compact Space"
] | [
"Definition:Topological Manifold",
"Definition:Locally Euclidean Space",
"Locally Euclidean Space is Locally Compact",
"Category:Topological Manifolds",
"Category:Locally Compact Spaces"
] |
proofwiki-20361 | Napoleon's Theorem | Let $\triangle ABC$ be an arbitrary triangle.
Let $\triangle ABF$, $\triangle BCD$ and $\triangle ACE$ be equilateral triangles constructed on $AB$, $BC$ and $AC$ respectively on the exterior of $\triangle ABC$.
Let $O_1$, $O_2$ and $O_3$ be the incenters of $\triangle ABF$, $\triangle BCD$ and $\triangle ACE$.
Then $\... | For simplicity of notation, we relabel the incenters $O_1, O_2, O_3$ as $P, Q, R$.
400px
By Line from Vertex of Triangle to Incenter is Angle Bisector:
:$CQ$ bisects $\angle ACE$
:$CP$ bisects $\angle BCD$
Given:
:$\angle ACE = \angle BCD = 60^{\circ}$
Since $\triangle CBD$ and $\triangle ACE$ are both equilateral, the... | Let $\triangle ABC$ be an arbitrary [[Definition:Triangle (Geometry)|triangle]].
Let $\triangle ABF$, $\triangle BCD$ and $\triangle ACE$ be [[Definition:Equilateral Triangle|equilateral triangles]] constructed on $AB$, $BC$ and $AC$ respectively on the exterior of $\triangle ABC$.
Let $O_1$, $O_2$ and $O_3$ be the [... | For simplicity of notation, we relabel the incenters $O_1, O_2, O_3$ as $P, Q, R$.
[[File:Napoleons-Theorem 3.png|400px]]
By [[Line from Vertex of Triangle to Incenter is Angle Bisector]]:
:$CQ$ bisects $\angle ACE$
:$CP$ bisects $\angle BCD$
Given:
:$\angle ACE = \angle BCD = 60^{\circ}$
Since $\triangle CBD$ a... | Napoleon's Theorem/Proof 1 | https://proofwiki.org/wiki/Napoleon's_Theorem | https://proofwiki.org/wiki/Napoleon's_Theorem/Proof_1 | [
"Napoleon's Theorem",
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Incircle of Triangle/Incenter",
"Definition:Triangle (Geometry)/Equilateral",
"File:Napoleons-Theorem.png"
] | [
"File:Napoleons-Theorem 3.png",
"Line from Vertex of Triangle to Incenter is Angle Bisector",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Angle",
"Axiom:Euclid's Common Notion 1",
"Law of Cosines",
"Definition:Polygon/Side",
"Definition:Triangle",
"Definition:Altitude of Triangle",
"... |
proofwiki-20362 | Napoleon's Theorem | Let $\triangle ABC$ be an arbitrary triangle.
Let $\triangle ABF$, $\triangle BCD$ and $\triangle ACE$ be equilateral triangles constructed on $AB$, $BC$ and $AC$ respectively on the exterior of $\triangle ABC$.
Let $O_1$, $O_2$ and $O_3$ be the incenters of $\triangle ABF$, $\triangle BCD$ and $\triangle ACE$.
Then $\... | === {{Lemma|Napoleon's Theorem|1}} ===
{{:Napoleon's Theorem/Lemma 1}}{{qed|lemma}}
=== {{Lemma|Napoleon's Theorem|2}} ===
{{:Napoleon's Theorem/Lemma 2}}{{qed|lemma}}
==== Vectors ====
Side $a = BC$ lies opposite vertex $A$ of $\triangle ABC$.
Let the vector $\mathbf{a}$ have magnitude $\dfrac 1 3 a$ in the direction ... | Let $\triangle ABC$ be an arbitrary [[Definition:Triangle (Geometry)|triangle]].
Let $\triangle ABF$, $\triangle BCD$ and $\triangle ACE$ be [[Definition:Equilateral Triangle|equilateral triangles]] constructed on $AB$, $BC$ and $AC$ respectively on the exterior of $\triangle ABC$.
Let $O_1$, $O_2$ and $O_3$ be the [... | === {{Lemma|Napoleon's Theorem|1}} ===
{{:Napoleon's Theorem/Lemma 1}}{{qed|lemma}}
=== {{Lemma|Napoleon's Theorem|2}} ===
{{:Napoleon's Theorem/Lemma 2}}{{qed|lemma}}
==== [[Definition:Vector (Real Euclidean Space)|Vectors]] ====
Side $a = BC$ lies [[Definition:Triangle (Geometry)|opposite]] [[Definition:Vertex o... | Napoleon's Theorem/Proof 2 | https://proofwiki.org/wiki/Napoleon's_Theorem | https://proofwiki.org/wiki/Napoleon's_Theorem/Proof_2 | [
"Napoleon's Theorem",
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Incircle of Triangle/Incenter",
"Definition:Triangle (Geometry)/Equilateral",
"File:Napoleons-Theorem.png"
] | [
"Definition:Vector/Real Euclidean Space",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Vertex",
"Definition:Vector",
"Definition:Magnitude",
"Definition:Incircle of Triangle/Incenter",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)",
"Definition:Polygon/Vertex",
"Definition:Ve... |
proofwiki-20363 | Napoleon's Theorem/Variant 1 | Let $\triangle ABC$ be an arbitrary triangle.
Let $\triangle ABF$, $\triangle BCD$ and $\triangle ACE$ be equilateral triangles constructed on $AB$, $BC$ and $AC$ respectively toward the interior of $\triangle ABC$.
Let $O_1$, $O_2$ and $O_3$ be the incenters of $\triangle ABF$, $\triangle ACE$ and $\triangle BCD$ resp... | === {{Lemma|Napoleon's Theorem|1}} ===
{{:Napoleon's Theorem/Lemma 1}}{{qed|lemma}} | Let $\triangle ABC$ be an arbitrary [[Definition:Triangle (Geometry)|triangle]].
Let $\triangle ABF$, $\triangle BCD$ and $\triangle ACE$ be [[Definition:Equilateral Triangle|equilateral triangles]] constructed on $AB$, $BC$ and $AC$ respectively toward the interior of $\triangle ABC$.
Let $O_1$, $O_2$ and $O_3$ be t... | === {{Lemma|Napoleon's Theorem|1}} ===
{{:Napoleon's Theorem/Lemma 1}}{{qed|lemma}} | Napoleon's Theorem/Variant 1 | https://proofwiki.org/wiki/Napoleon's_Theorem/Variant_1 | https://proofwiki.org/wiki/Napoleon's_Theorem/Variant_1 | [
"Napoleon's Theorem",
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Incircle of Triangle/Incenter",
"Definition:Triangle (Geometry)/Equilateral",
"File:Napoleons-Theorem-Variant.png"
] | [] |
proofwiki-20364 | Suffix of String is Substring | Let $S$ be a string.
Let $T$ be a suffix of $S$.
Then $T$ is a substring of $S$. | By definition of substring, there exists a string $T'$ such that:
:$S = T'T$
Hence $S$ is the concatenation of the null string, $T'$, and $T$.
Thus by definition of substring, $T$ is a substring of $S$.
{{qed}}
Category:Collations
m7zl93m6ij10z0sagrr55bea0q06p37 | Let $S$ be a [[Definition:String|string]].
Let $T$ be a [[Definition:Suffix|suffix]] of $S$.
Then $T$ is a [[Definition:Substring|substring]] of $S$. | By definition of [[Definition:Substring|substring]], there exists a [[Definition:String|string]] $T'$ such that:
:$S = T'T$
Hence $S$ is the [[Definition:Concatenation (Formal Systems)|concatenation]] of the [[Definition:Null String|null string]], $T'$, and $T$.
Thus by definition of [[Definition:Substring|substrin... | Suffix of String is Substring | https://proofwiki.org/wiki/Suffix_of_String_is_Substring | https://proofwiki.org/wiki/Suffix_of_String_is_Substring | [
"Collations"
] | [
"Definition:String",
"Definition:Suffix",
"Definition:Substring"
] | [
"Definition:Substring",
"Definition:String",
"Definition:Concatenation (Formal Systems)",
"Definition:Null String",
"Definition:Substring",
"Definition:Substring",
"Category:Collations"
] |
proofwiki-20365 | Null String is Identity Element for Concatenation Operator | Let $\AA$ be an alphabet of symbols.
Let $\WW$ denote the set of words in $\AA$.
Let $\epsilon$ denote the null string.
Let $C: \WW \times \WW \to \WW$ denote the concatenation operator on $\WW$:
:$\forall A, B \in \WW: \map C {A, B} := A B$
Then $\epsilon$ is the identity element for $C$. | As $\epsilon$ is the null string:
:$\map C {\epsilon, A}$
{{begin-eqn}}
{{eqn | q = \forall A \in \WW
| l = \map C {\epsilon, A}
| r = \epsilon A
| c = Definition of $C$
}}
{{eqn | r = A
| c = {{Defof|Null String}}
}}
{{eqn | r = A \epsilon
| c = {{Defof|Null String}}
}}
{{eqn | r = \map C... | Let $\AA$ be an [[Definition:Alphabet of Formal Language|alphabet]] of [[Definition:Symbol|symbols]].
Let $\WW$ denote the [[Definition:Set|set]] of [[Definition:Word (Formal Systems)|words]] in $\AA$.
Let $\epsilon$ denote the [[Definition:Null String|null string]].
Let $C: \WW \times \WW \to \WW$ denote the [[Defi... | As $\epsilon$ is the [[Definition:Null String|null string]]:
:$\map C {\epsilon, A}$
{{begin-eqn}}
{{eqn | q = \forall A \in \WW
| l = \map C {\epsilon, A}
| r = \epsilon A
| c = Definition of $C$
}}
{{eqn | r = A
| c = {{Defof|Null String}}
}}
{{eqn | r = A \epsilon
| c = {{Defof|Null St... | Null String is Identity Element for Concatenation Operator | https://proofwiki.org/wiki/Null_String_is_Identity_Element_for_Concatenation_Operator | https://proofwiki.org/wiki/Null_String_is_Identity_Element_for_Concatenation_Operator | [
"Concatenation (Formal Systems)"
] | [
"Definition:Formal Language/Alphabet",
"Definition:Symbol",
"Definition:Set",
"Definition:Word (Formal Systems)",
"Definition:Null String",
"Definition:Concatenation (Formal Systems)",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Null String",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
proofwiki-20366 | Equivalence of Definitions of Generator of Unitary Module | Let $R$ be a ring with unity.
Let $M$ be a unitary $R$-module.
Let $S \subseteq M$ be a subset.
{{TFAE|def = Generator of Unitary Module}} | === Definition by linear combinations implies Definition by intersection of submodules === | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $M$ be a [[Definition:Unitary Module over Ring|unitary $R$-module]].
Let $S \subseteq M$ be a [[Definition:Subset|subset]].
{{TFAE|def = Generator of Unitary Module}} | === [[Definition:Generator of Unitary Module|Definition by linear combinations]] implies [[Definition:Generator of Module/Definition 1|Definition by intersection of submodules]] === | Equivalence of Definitions of Generator of Unitary Module | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Generator_of_Unitary_Module | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Generator_of_Unitary_Module | [
"Generators of Modules"
] | [
"Definition:Ring with Unity",
"Definition:Unitary Module over Ring",
"Definition:Subset"
] | [
"Definition:Generator of Module/Unitary",
"Definition:Generator of Module/Definition 1",
"Definition:Generator of Module/Definition 1",
"Definition:Generator of Module/Unitary",
"Definition:Generator of Module/Unitary",
"Definition:Generator of Module/Definition 1",
"Definition:Generator of Module/Defin... |
proofwiki-20367 | Convex Set is Path-Connected | Let $V$ be a topological vector space over $\R$ or $\C$.
Every convex subset of $V$ is path-connected. | The result follows from:
:Convex Set is Star Convex Set
:Star Convex Set is Path-Connected
{{qed}}
Category:Topological Vector Spaces
Category:Convex Sets (Vector Spaces)
Category:Path-Connected Spaces
f54ufyli0mhk8a11ngff8ysyi70ewrb | Let $V$ be a [[Definition:Topological Vector Space|topological vector space]] over $\R$ or $\C$.
Every [[Definition:Convex Set (Vector Space)|convex subset]] of $V$ is [[Definition:Path-Connected Metric Subspace|path-connected]]. | The result follows from:
:[[Convex Set is Star Convex Set]]
:[[Star Convex Set is Path-Connected]]
{{qed}}
[[Category:Topological Vector Spaces]]
[[Category:Convex Sets (Vector Spaces)]]
[[Category:Path-Connected Spaces]]
f54ufyli0mhk8a11ngff8ysyi70ewrb | Convex Set is Path-Connected | https://proofwiki.org/wiki/Convex_Set_is_Path-Connected | https://proofwiki.org/wiki/Convex_Set_is_Path-Connected | [
"Topological Vector Spaces",
"Convex Sets (Vector Spaces)",
"Path-Connected Spaces"
] | [
"Definition:Topological Vector Space",
"Definition:Convex Set (Vector Space)",
"Definition:Path-Connected/Metric Space/Subset"
] | [
"Convex Set is Star Convex Set",
"Star Convex Set is Path-Connected",
"Category:Topological Vector Spaces",
"Category:Convex Sets (Vector Spaces)",
"Category:Path-Connected Spaces"
] |
proofwiki-20368 | Jordan Curve Bounding Loop in Euclidean Plane | Let $f : \closedint 0 1 \to \R^2$ be a loop in the Euclidean plane $\R^2$.
Let $\epsilon \in \R_{>0}$.
Then there exists a Jordan curve $\gamma : \closedint 0 1 \to \R^2$ such that $\Img f \subseteq \Int \gamma$, and for all $t \in \closedint 0 1$:
:$\map d {\map \gamma t, \Img f} < \epsilon$
where $\map d {\map \gamma... | === Cast a ray from an extreme point of $\Img f$===
Let $\norm {\map f t}: \closedint 0 1 \to \R$ denote the Euclidean norm of $\map f t$, considered as a function of $t$.
From Norm on Vector Space is Continuous Function and Composite of Continuous Mappings between Metric Spaces is Continuous, it follows that $\norm {\... | Let $f : \closedint 0 1 \to \R^2$ be a [[Definition:Loop (Topology)|loop]] in the [[Definition:Real Euclidean Space|Euclidean plane]] $\R^2$.
Let $\epsilon \in \R_{>0}$.
Then there exists a [[Definition:Jordan Curve|Jordan curve]] $\gamma : \closedint 0 1 \to \R^2$ such that $\Img f \subseteq \Int \gamma$, and for a... | === Cast a ray from an extreme point of $\Img f$===
Let $\norm {\map f t}: \closedint 0 1 \to \R$ denote the [[Definition:Euclidean Norm|Euclidean norm]] of $\map f t$, considered as a [[Definition:Real Function|function]] of $t$.
From [[Norm on Vector Space is Continuous Function]] and [[Composite of Continuous Mapp... | Jordan Curve Bounding Loop in Euclidean Plane | https://proofwiki.org/wiki/Jordan_Curve_Bounding_Loop_in_Euclidean_Plane | https://proofwiki.org/wiki/Jordan_Curve_Bounding_Loop_in_Euclidean_Plane | [
"Loops (Topology)",
"Jordan Curves"
] | [
"Definition:Loop (Topology)",
"Definition:Euclidean Space/Real",
"Definition:Jordan Curve",
"Definition:Euclidean Metric",
"Definition:Distance/Sets/Metric Spaces",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Jordan Curve/Interior"
] | [
"Definition:Euclidean Norm",
"Definition:Real Function",
"Norm on Vector Space is Continuous Function",
"Composite of Continuous Mappings between Metric Spaces is Continuous",
"Definition:Continuous Mapping (Metric Space)",
"Closed Real Interval is Compact Space",
"Definition:Compact Space/Real Analysis... |
proofwiki-20369 | Interior of Jordan Curve is Simply Connected | Let $\gamma : \closedint 0 1 \to \R^2$ be a Jordan curve.
Let $\Int \gamma$ denote the interior of $\gamma$.
Let $\tau_0$ denote the subspace topology on $\Int \gamma$, induced by the Euclidean topology on $\R^2$.
Then $\struct {\Int \gamma, \tau_0}$ is simply connected. | Let $\map {B_1} { \mathbf 0 }$ denote the open ball in $\R^2$ with radius $1$ and center equal to the origin $\bszero$.
We are given $\Int \gamma$ is the interior of $\gamma$.
From the Jordan-Schönflies Theorem, it follows that $\Int \gamma$ and $\map {B_1} { \mathbf 0 }$ are homeomorphic.
From Open Ball is Simply Conn... | Let $\gamma : \closedint 0 1 \to \R^2$ be a [[Definition:Jordan Curve|Jordan curve]].
Let $\Int \gamma$ denote the [[Definition:Interior of Jordan Curve|interior]] of $\gamma$.
Let $\tau_0$ denote the [[Definition:Subspace Topology|subspace topology]] on $\Int \gamma$, induced by the [[Definition:Real Number Plane wi... | Let $\map {B_1} { \mathbf 0 }$ denote the [[Definition:Open Ball in Normed Vector Space|open ball]] in $\R^2$ with [[Definition:Radius of Open Ball|radius]] $1$ and [[Definition:Center of Open Ball|center]] equal to the [[Definition:Origin|origin]] $\bszero$.
We are [[Definition:Given|given]] $\Int \gamma$ is the [[De... | Interior of Jordan Curve is Simply Connected | https://proofwiki.org/wiki/Interior_of_Jordan_Curve_is_Simply_Connected | https://proofwiki.org/wiki/Interior_of_Jordan_Curve_is_Simply_Connected | [
"Jordan Curves",
"Simply Connected Spaces"
] | [
"Definition:Jordan Curve",
"Definition:Jordan Curve/Interior",
"Definition:Topological Subspace",
"Definition:Euclidean Space/Euclidean Topology/Real Number Plane",
"Definition:Simply Connected"
] | [
"Definition:Open Ball/Normed Vector Space",
"Definition:Open Ball/Radius",
"Definition:Open Ball/Center",
"Definition:Coordinate System/Origin",
"Definition:Given",
"Definition:Jordan Curve/Interior",
"Jordan-Schönflies Theorem",
"Definition:Homeomorphism/Metric Spaces",
"Open Ball is Simply Connect... |
proofwiki-20370 | Equivalence of Definitions of Matroid Circuit Axioms/Lemma 2 | Let $X \subseteq S$ and $y \in S \setminus X$.
Then:
:$\map \rho {X \cup \set y} = \map \rho X$ {{iff}} $\exists C \in \mathscr C : y \in C \subseteq X \cup \set y$ | Let $X = \set{x_1, \ldots, x_q}$
We have:
{{begin-eqn}}
{{eqn | l = \map \rho {X \cup \set y}
| r = \map t {x_1, \ldots, x_q, y}
| c = Definition of $\rho$
}}
{{eqn | r = \map t {x_1, \ldots, x_q} + \map \theta {x_1, \ldots, x_q, y}_{q+1}
| c = Definition of $t$
}}
{{eqn | r = \map \rho X + \map \thet... | Let $X \subseteq S$ and $y \in S \setminus X$.
Then:
:$\map \rho {X \cup \set y} = \map \rho X$ {{iff}} $\exists C \in \mathscr C : y \in C \subseteq X \cup \set y$ | Let $X = \set{x_1, \ldots, x_q}$
We have:
{{begin-eqn}}
{{eqn | l = \map \rho {X \cup \set y}
| r = \map t {x_1, \ldots, x_q, y}
| c = Definition of $\rho$
}}
{{eqn | r = \map t {x_1, \ldots, x_q} + \map \theta {x_1, \ldots, x_q, y}_{q+1}
| c = Definition of $t$
}}
{{eqn | r = \map \rho X + \map \th... | Equivalence of Definitions of Matroid Circuit Axioms/Lemma 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Circuit_Axioms/Lemma_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Circuit_Axioms/Lemma_2 | [
"Equivalence of Definitions of Matroid Circuit Axioms"
] | [] | [
"Category:Equivalence of Definitions of Matroid Circuit Axioms"
] |
proofwiki-20371 | Cartesian Product of Intervals is Convex Set | Let $n \in \N$.
For all $k \in \set {1, \ldots, n}$, let $\Bbb I_k$ be a real interval of any of the real interval types.
Then the cartesian product $\Bbb I_1 \times \ldots \times \Bbb I_n$ is a convex set. | Let $\mathbf x, \mathbf y \in \Bbb I_1 \times \ldots \times \Bbb I_n$ with:
{{begin-eqn}}
{{eqn | l = \mathbf x
| r = \tuple {x_1, \ldots, x_n}
}}
{{eqn | l = \mathbf y
| r = \tuple {y_1, \ldots, y_n}
}}
{{end-eqn}}
where $x_k , y_k \in \Bbb I_k$ for all $k \in \set {1, \ldots, n}$.
Let $t \in \closedint 0 ... | Let $n \in \N$.
For all $k \in \set {1, \ldots, n}$, let $\Bbb I_k$ be a [[Definition:Real Interval|real interval]] of any of the [[Definition:Real Interval Types|real interval types]].
Then the [[Definition:Cartesian Product|cartesian product]] $\Bbb I_1 \times \ldots \times \Bbb I_n$ is a [[Definition:Convex Set (... | Let $\mathbf x, \mathbf y \in \Bbb I_1 \times \ldots \times \Bbb I_n$ with:
{{begin-eqn}}
{{eqn | l = \mathbf x
| r = \tuple {x_1, \ldots, x_n}
}}
{{eqn | l = \mathbf y
| r = \tuple {y_1, \ldots, y_n}
}}
{{end-eqn}}
where $x_k , y_k \in \Bbb I_k$ for all $k \in \set {1, \ldots, n}$.
Let $t \in \closedint... | Cartesian Product of Intervals is Convex Set | https://proofwiki.org/wiki/Cartesian_Product_of_Intervals_is_Convex_Set | https://proofwiki.org/wiki/Cartesian_Product_of_Intervals_is_Convex_Set | [
"Convex Sets (Vector Spaces)"
] | [
"Definition:Real Interval",
"Definition:Real Interval Types",
"Definition:Cartesian Product",
"Definition:Convex Set (Vector Space)"
] | [
"Definition:Real Interval",
"Definition:Real Interval",
"Definition:Convex Set (Vector Space)",
"Definition:Convex Set (Vector Space)",
"Category:Convex Sets (Vector Spaces)"
] |
proofwiki-20372 | Cartesian Product of Intervals is Simply Connected | Let $n \in \N$.
For all $k \in \set {1, \ldots, n}$, let $\Bbb I_k$ be a real interval of any of the real interval types.
Let $\tau_0$ denote the subspace topology on the cartesian product $\Bbb I_1 \times \ldots \times \Bbb I_n$, induced by the Euclidean topology on $\R^n$.
Then $\struct {\Bbb I_1 \times \ldots \times... | The result follows from Cartesian Product of Intervals is Convex Set and Convex Set is Simply Connected.
{{qed}}
Category:Simply Connected Spaces
8z7eo0mw9kif2mdpq53zqxjrnfruwck | Let $n \in \N$.
For all $k \in \set {1, \ldots, n}$, let $\Bbb I_k$ be a [[Definition:Real Interval|real interval]] of any of the [[Definition:Real Interval Types|real interval types]].
Let $\tau_0$ denote the [[Definition:Subspace Topology|subspace topology]] on the [[Definition:Cartesian Product|cartesian product]]... | The result follows from [[Cartesian Product of Intervals is Convex Set]] and [[Convex Set is Simply Connected]].
{{qed}}
[[Category:Simply Connected Spaces]]
8z7eo0mw9kif2mdpq53zqxjrnfruwck | Cartesian Product of Intervals is Simply Connected | https://proofwiki.org/wiki/Cartesian_Product_of_Intervals_is_Simply_Connected | https://proofwiki.org/wiki/Cartesian_Product_of_Intervals_is_Simply_Connected | [
"Simply Connected Spaces"
] | [
"Definition:Real Interval",
"Definition:Real Interval Types",
"Definition:Topological Subspace",
"Definition:Cartesian Product",
"Definition:Euclidean Space/Euclidean Topology",
"Definition:Simply Connected"
] | [
"Cartesian Product of Intervals is Convex Set",
"Convex Set is Simply Connected",
"Category:Simply Connected Spaces"
] |
proofwiki-20373 | Characterization of Ergodicity in terms of Koopman Operator | Let $\struct {X, \BB, \mu}$ be a probability space.
Let $T: X \to X$ be a measure-preserving transformation.
Let $\map \MM {X, \R}$ be the set of $\BB$-measurable functions.
Let $\map {\LL^2} \mu$ denote the Lebesgue $2$-space.
Let $U_T : \map \MM {X, \R} \to \map \MM {X, \R}$ be the Koopman operator:
:$U_T: f \mapsto ... | === $(1) \implies (3)$ ===
This is clear, since $(3)$ is exactly the definition of ergodicity.
{{qed|lemma}} | Let $\struct {X, \BB, \mu}$ be a [[Definition:Probability Space|probability space]].
Let $T: X \to X$ be a [[Definition:Measure-Preserving Transformation|measure-preserving transformation]].
Let $\map \MM {X, \R}$ be the [[Definition:Set|set]] of $\BB$-[[Definition:Measurable Real-Valued Function|measurable functions... | === $(1) \implies (3)$ ===
This is clear, since $(3)$ is exactly the definition of [[Definition:Ergodic Measure-Preserving Transformation|ergodicity]].
{{qed|lemma}} | Characterization of Ergodicity in terms of Koopman Operator | https://proofwiki.org/wiki/Characterization_of_Ergodicity_in_terms_of_Koopman_Operator | https://proofwiki.org/wiki/Characterization_of_Ergodicity_in_terms_of_Koopman_Operator | [
"Ergodic Measure-Preserving Transformations"
] | [
"Definition:Probability Space",
"Definition:Measure-Preserving Transformation",
"Definition:Set",
"Definition:Measurable Function/Real-Valued Function",
"Definition:Lebesgue Space",
"Definition:Koopman Operator",
"Definition:Ergodic Measure-Preserving Transformation",
"Definition:Constant Mapping",
... | [
"Definition:Ergodic Measure-Preserving Transformation"
] |
proofwiki-20374 | Uniform Prisms are Countably Infinite | There are countably infinite different varieties of uniform prisms. | {{Recall|Uniform Prism}}
{{:Definition:Uniform Prism}}
Hence a uniform prism is made of:
:$2$ bases which are regular polygons
:as many lateral faces as there are sides of one of the bases.
Hence for each type of regular polygon there exists a corresponding uniform prism.
There exists a type of regular polygon for each... | There are [[Definition:Countably Infinite Set|countably infinite]] different varieties of [[Definition:Uniform Prism|uniform prisms]]. | {{Recall|Uniform Prism}}
{{:Definition:Uniform Prism}}
Hence a [[Definition:Uniform Prism|uniform prism]] is made of:
:$2$ [[Definition:Base of Prism|bases]] which are [[Definition:Regular Polygon|regular polygons]]
:as many [[Definition:Lateral Face of Prism|lateral faces]] as there are [[Definition:Side of Polygon|s... | Uniform Prisms are Countably Infinite | https://proofwiki.org/wiki/Uniform_Prisms_are_Countably_Infinite | https://proofwiki.org/wiki/Uniform_Prisms_are_Countably_Infinite | [
"Uniform Prisms",
"Countably Infinite Sets"
] | [
"Definition:Countably Infinite/Set",
"Definition:Uniform Prism"
] | [
"Definition:Uniform Prism",
"Definition:Prism/Base",
"Definition:Polygon/Regular",
"Definition:Prism/Lateral Face",
"Definition:Polygon/Side",
"Definition:Prism/Base",
"Definition:Polygon/Regular",
"Definition:Uniform Prism",
"Definition:Polygon/Regular",
"Definition:Natural Numbers",
"Definitio... |
proofwiki-20375 | Connected Graph is Tree iff Removal of One Edge makes it Disconnected | Let $G = \struct {V, E}$ be a connected simple graph.
Then $G$ is a tree {{iff}}:
:for all edges $e$ of $G$, the edge deletion $G \setminus \set e$ is disconnected. | === Sufficient Condition ===
{{:Connected Graph is Tree iff Removal of One Edge makes it Disconnected/Sufficient Condition/Proof 1}}{{qed|lemma}} | Let $G = \struct {V, E}$ be a [[Definition:Connected Graph|connected]] [[Definition:Simple Graph|simple graph]].
Then $G$ is a [[Definition:Tree (Graph Theory)|tree]] {{iff}}:
:for all [[Definition:Edge of Graph|edges]] $e$ of $G$, the [[Definition:Edge Deletion|edge deletion]] $G \setminus \set e$ is [[Definition:Dis... | === [[Connected Graph is Tree iff Removal of One Edge makes it Disconnected/Sufficient Condition|Sufficient Condition]] ===
{{:Connected Graph is Tree iff Removal of One Edge makes it Disconnected/Sufficient Condition/Proof 1}}{{qed|lemma}} | Connected Graph is Tree iff Removal of One Edge makes it Disconnected | https://proofwiki.org/wiki/Connected_Graph_is_Tree_iff_Removal_of_One_Edge_makes_it_Disconnected | https://proofwiki.org/wiki/Connected_Graph_is_Tree_iff_Removal_of_One_Edge_makes_it_Disconnected | [
"Connected Graph is Tree iff Removal of One Edge makes it Disconnected",
"Connected Graphs",
"Tree Theory"
] | [
"Definition:Connected (Graph Theory)/Graph",
"Definition:Simple Graph",
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Edge Deletion",
"Definition:Connected (Graph Theory)/Graph/Disconnected"
] | [
"Connected Graph is Tree iff Removal of One Edge makes it Disconnected/Sufficient Condition"
] |
proofwiki-20376 | Connected Graph is Tree iff Removal of One Edge makes it Disconnected | Let $G = \struct {V, E}$ be a connected simple graph.
Then $G$ is a tree {{iff}}:
:for all edges $e$ of $G$, the edge deletion $G \setminus \set e$ is disconnected. | Let $G$ be a tree.
Then by definition $G$ has no circuits.
From Condition for Edge to be Bridge, every edge of $G$ is a bridge.
Thus by definition of bridge, removing any edge of $G$ will disconnect $G$. | Let $G = \struct {V, E}$ be a [[Definition:Connected Graph|connected]] [[Definition:Simple Graph|simple graph]].
Then $G$ is a [[Definition:Tree (Graph Theory)|tree]] {{iff}}:
:for all [[Definition:Edge of Graph|edges]] $e$ of $G$, the [[Definition:Edge Deletion|edge deletion]] $G \setminus \set e$ is [[Definition:Dis... | Let $G$ be a [[Definition:Tree (Graph Theory)|tree]].
Then by definition $G$ has no [[Definition:Circuit (Graph Theory)|circuits]].
From [[Condition for Edge to be Bridge]], every [[Definition:Edge of Graph|edge]] of $G$ is a [[Definition:Bridge (Graph Theory)|bridge]].
Thus by definition of [[Definition:Bridge (Gra... | Connected Graph is Tree iff Removal of One Edge makes it Disconnected/Sufficient Condition/Proof 1 | https://proofwiki.org/wiki/Connected_Graph_is_Tree_iff_Removal_of_One_Edge_makes_it_Disconnected | https://proofwiki.org/wiki/Connected_Graph_is_Tree_iff_Removal_of_One_Edge_makes_it_Disconnected/Sufficient_Condition/Proof_1 | [
"Connected Graph is Tree iff Removal of One Edge makes it Disconnected",
"Connected Graphs",
"Tree Theory"
] | [
"Definition:Connected (Graph Theory)/Graph",
"Definition:Simple Graph",
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Edge Deletion",
"Definition:Connected (Graph Theory)/Graph/Disconnected"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Circuit (Graph Theory)",
"Condition for Edge to be Bridge",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Bridge (Graph Theory)",
"Definition:Bridge (Graph Theory)",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Connected (Graph Theory)/Graph/D... |
proofwiki-20377 | Connected Graph is Tree iff Removal of One Edge makes it Disconnected | Let $G = \struct {V, E}$ be a connected simple graph.
Then $G$ is a tree {{iff}}:
:for all edges $e$ of $G$, the edge deletion $G \setminus \set e$ is disconnected. | Let $G$ be a tree.
Hence {{afortiori}} $G$ has no cycles.
Let $v, v' \in V$.
Let the edge $\set {v, v'}$ be removed.
{{AimForCont}} $G$ is still connected.
Then {{apriori}} $v$ and $v'$ are connected.
By If Vertices are Connected then Path Exists between them, there is a path $\tuple {v, v_1, \ldots, v'}$ of length $2$... | Let $G = \struct {V, E}$ be a [[Definition:Connected Graph|connected]] [[Definition:Simple Graph|simple graph]].
Then $G$ is a [[Definition:Tree (Graph Theory)|tree]] {{iff}}:
:for all [[Definition:Edge of Graph|edges]] $e$ of $G$, the [[Definition:Edge Deletion|edge deletion]] $G \setminus \set e$ is [[Definition:Dis... | Let $G$ be a [[Definition:Tree (Graph Theory)|tree]].
Hence {{afortiori}} $G$ has no [[Definition:Cycle (Graph Theory)|cycles]].
Let $v, v' \in V$.
Let the [[Definition:Edge of Graph|edge]] $\set {v, v'}$ be removed.
{{AimForCont}} $G$ is still [[Definition:Connected Graph|connected]].
Then {{apriori}} $v$ and $v'... | Connected Graph is Tree iff Removal of One Edge makes it Disconnected/Sufficient Condition/Proof 2 | https://proofwiki.org/wiki/Connected_Graph_is_Tree_iff_Removal_of_One_Edge_makes_it_Disconnected | https://proofwiki.org/wiki/Connected_Graph_is_Tree_iff_Removal_of_One_Edge_makes_it_Disconnected/Sufficient_Condition/Proof_2 | [
"Connected Graph is Tree iff Removal of One Edge makes it Disconnected",
"Connected Graphs",
"Tree Theory"
] | [
"Definition:Connected (Graph Theory)/Graph",
"Definition:Simple Graph",
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Edge Deletion",
"Definition:Connected (Graph Theory)/Graph/Disconnected"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Cycle (Graph Theory)",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Connected (Graph Theory)/Vertices",
"If Vertices are Connected then Path Exists between them",
"Definition:Path (Graph Theory)",
"Defini... |
proofwiki-20378 | Reverse Triangle Inequality/Seminormed Vector Space | Let $\struct {K, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_K$.
Let $X$ be a vector space over $K$.
Let $p$ be a seminorm on $X$.
Then:
:$\forall x, y \in X : \size {\map p x - \map p y} \le \map p {x - y}$
{{explain|$\size {\map p x - \map p y}$}} | We have:
{{begin-eqn}}
{{eqn | l = \map p x - \map p y
| r = \map p {x - y + y} - \map p y
}}
{{eqn | o = \le
| r = \map p {x - y} + \map p y - \map p y
| c = {{SeminormAxiom|3}}
}}
{{eqn | r = \map p {x - y}
}}
{{end-eqn}}
We also have:
{{begin-eqn}}
{{eqn | l = \map p y - \map p x
| r = \map p... | Let $\struct {K, +, \circ}$ be a [[Definition:Division Ring|division ring]] with [[Definition:Norm on Division Ring|norm]] $\norm {\,\cdot\,}_K$.
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $p$ be a [[Definition:Seminorm|seminorm]] on $X$.
Then:
:$\forall x, y \in X : \size {\map p x - \map ... | We have:
{{begin-eqn}}
{{eqn | l = \map p x - \map p y
| r = \map p {x - y + y} - \map p y
}}
{{eqn | o = \le
| r = \map p {x - y} + \map p y - \map p y
| c = {{SeminormAxiom|3}}
}}
{{eqn | r = \map p {x - y}
}}
{{end-eqn}}
We also have:
{{begin-eqn}}
{{eqn | l = \map p y - \map p x
| r = \ma... | Reverse Triangle Inequality/Seminormed Vector Space | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Seminormed_Vector_Space | https://proofwiki.org/wiki/Reverse_Triangle_Inequality/Seminormed_Vector_Space | [
"Seminorms",
"Triangle Inequality",
"Seminorms"
] | [
"Definition:Division Ring",
"Definition:Norm/Division Ring",
"Definition:Vector Space",
"Definition:Seminorm"
] | [
"Properties of Norm on Division Ring/Norm of Negative of Unity",
"Category:Triangle Inequality",
"Category:Seminorms"
] |
proofwiki-20379 | Connected Graph is Tree iff Removal of One Edge makes it Disconnected/Necessary Condition | Let $G = \struct {V, E}$ be a connected simple graph such that:
:for all edges $e$ of $G$, the edge deletion $G \setminus \set e$ is disconnected.
Then $G$ is a tree. | Let $G$ be a connected simple graph such that for all edges $e$ of $G$, the edge deletion $G \setminus \set e$ is disconnected.
Hence, by definition, every edge of $G$ must be a bridge.
So by Condition for Edge to be Bridge, $G$ has no circuits.
Hence $G$ is a tree by definition. | Let $G = \struct {V, E}$ be a [[Definition:Connected Graph|connected]] [[Definition:Simple Graph|simple graph]] such that:
:for all [[Definition:Edge of Graph|edges]] $e$ of $G$, the [[Definition:Edge Deletion|edge deletion]] $G \setminus \set e$ is [[Definition:Disconnected Graph|disconnected]].
Then $G$ is a [[Def... | Let $G$ be a [[Definition:Connected Graph|connected]] [[Definition:Simple Graph|simple graph]] such that for all [[Definition:Edge of Graph|edges]] $e$ of $G$, the [[Definition:Edge Deletion|edge deletion]] $G \setminus \set e$ is [[Definition:Disconnected Graph|disconnected]].
Hence, by definition, every [[Definition... | Connected Graph is Tree iff Removal of One Edge makes it Disconnected/Necessary Condition | https://proofwiki.org/wiki/Connected_Graph_is_Tree_iff_Removal_of_One_Edge_makes_it_Disconnected/Necessary_Condition | https://proofwiki.org/wiki/Connected_Graph_is_Tree_iff_Removal_of_One_Edge_makes_it_Disconnected/Necessary_Condition | [
"Connected Graph is Tree iff Removal of One Edge makes it Disconnected"
] | [
"Definition:Connected (Graph Theory)/Graph",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Edge Deletion",
"Definition:Connected (Graph Theory)/Graph/Disconnected",
"Definition:Tree (Graph Theory)"
] | [
"Definition:Connected (Graph Theory)/Graph",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Edge Deletion",
"Definition:Connected (Graph Theory)/Graph/Disconnected",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Bridge (Graph Theory)",
"Condition for Edge to be Br... |
proofwiki-20380 | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Lemma | :$G$ has at least one leaf node. | Let $G = \struct {V, E}$ be a non-edgeless connected finite simple graph.
Let us select $v_1 \in V$ and some $v_2 \in V$ which is adjacent to $v_1$.
Such will always exist because $G$ is connected and not edgeless.
For $k \ge 2$, either $v_k$ is adjacent to $v_{k - 1}$ and no other, or it is adjacent to $v_{k - 1}$ and... | :$G$ has at least one [[Definition:Leaf Node|leaf node]]. | Let $G = \struct {V, E}$ be a non-[[Definition:Edgeless Graph|edgeless]] [[Definition:Connected Graph|connected]] [[Definition:Finite Graph|finite]] [[Definition:Simple Graph|simple graph]].
Let us select $v_1 \in V$ and some $v_2 \in V$ which is [[Definition:Adjacent Vertices of Graph|adjacent]] to $v_1$.
Such will ... | Finite Connected Simple Graph is Tree iff Size is One Less than Order/Lemma | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Lemma | https://proofwiki.org/wiki/Finite_Connected_Simple_Graph_is_Tree_iff_Size_is_One_Less_than_Order/Lemma | [
"Finite Connected Simple Graph is Tree iff Size is One Less than Order"
] | [
"Definition:Tree (Graph Theory)/Leaf Node"
] | [
"Definition:Edgeless Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Finite Graph",
"Definition:Simple Graph",
"Definition:Adjacent (Graph Theory)/Vertices",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Edgeless Graph",
"Definition:Adjacent (Graph Theory)/Vertices",
"Def... |
proofwiki-20381 | Sequence of Natural Powers of Right Shift Operator in 2-Sequence Space Converges in Weak Operator Topology | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the $2$-sequence normed vector space.
Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the continuous linear transformation space.
Let $R \in \map {CL} {\ell^2}$ be the right shift operator over $\ell^2$.
Let $\sequence {R^n}_{n \mathop \in \N}$ be a sequence.
L... | By Representation Theorem:
:$\ds \forall \phi \in \map {CL} {\ell^2, \C} : \exists \mathbf x_\phi = \sequence {\map {\mathbf x_\phi} k}_{k \mathop \in \N} \in \ell^2 : \forall \mathbf a = \sequence {\map {\mathbf a} k}_{k \mathop \in \N} \in \ell^2 : \map \phi {\mathbf a} = \sum_{k \mathop = 1}^\infty \map {\mathbf a} ... | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the [[P-Sequence Space with P-Norm forms Normed Vector Space|$2$-sequence normed vector space]].
Let $\map {CL} {\ell^2} := \map {CL} {\ell^2, \ell^2}$ be the [[Definition:Continuous Linear Transformation Space|continuous linear transformation space]].
Let $R \in \map ... | By [[Representation Theorem]]:
:$\ds \forall \phi \in \map {CL} {\ell^2, \C} : \exists \mathbf x_\phi = \sequence {\map {\mathbf x_\phi} k}_{k \mathop \in \N} \in \ell^2 : \forall \mathbf a = \sequence {\map {\mathbf a} k}_{k \mathop \in \N} \in \ell^2 : \map \phi {\mathbf a} = \sum_{k \mathop = 1}^\infty \map {\mathb... | Sequence of Natural Powers of Right Shift Operator in 2-Sequence Space Converges in Weak Operator Topology | https://proofwiki.org/wiki/Sequence_of_Natural_Powers_of_Right_Shift_Operator_in_2-Sequence_Space_Converges_in_Weak_Operator_Topology | https://proofwiki.org/wiki/Sequence_of_Natural_Powers_of_Right_Shift_Operator_in_2-Sequence_Space_Converges_in_Weak_Operator_Topology | [
"Convergence",
"Operator Theory"
] | [
"P-Sequence Space with P-Norm forms Normed Vector Space",
"Definition:Continuous Linear Transformation Space",
"Definition:Right Shift Operator",
"Definition:Sequence",
"Definition:Zero Mapping/Vector Space",
"Definition:Convergent Sequence in Weak Operator Topology",
"Definition:Weak Operator Topology"... | [
"Representation Theorem",
"Definition:Complex Conjugate/Complex Conjugation",
"Cauchy-Bunyakovsky-Schwarz Inequality",
"Combination Theorem for Sequences/Real/Multiple Rule",
"Exchange of Limits",
"Terms in Convergent Series Converge to Zero",
"Definition:Convergent Sequence in Weak Operator Topology",
... |
proofwiki-20382 | Desargues' Theorem/Converse | Let $\triangle ABC$ and $\triangle A'B'C'$ be triangles lying in the same or different planes.
Let:
:$BC$ meet $B'C'$ in $L$
:$CA$ meet $C'A'$ in $M$
:$AB$ meet $A'B'$ in $N$
where $L, M, N$ are collinear.
Then the lines $AA'$, $BB'$ and $CC'$ intersect in the point $O$. | :500px
Let $L$, $M$ and $N$ be collinear {{hypothesis}}.
Then $\triangle BB'N$ and $\triangle CC'M$ are perspective with center $L$ ($L = BC \cap B'C' \cap MN$)
From Desargues' Theorem:
:$O = BB' \cap CC'$
:$A = BN \cap CM$
:$A' = C'M \cap B'N$
are collinear.
Thus:
:$AA' \cap BB' \cap CC' = O$
Hence $\triangle ABC$ and... | Let $\triangle ABC$ and $\triangle A'B'C'$ be [[Definition:Triangle (Geometry)|triangles]] lying in the same or different [[Definition:Plane|planes]].
Let:
:$BC$ meet $B'C'$ in $L$
:$CA$ meet $C'A'$ in $M$
:$AB$ meet $A'B'$ in $N$
where $L, M, N$ are [[Definition:Collinear Points|collinear]].
Then the [[Definition:... | :[[File:DesarguesTheorem.png|500px]]
Let $L$, $M$ and $N$ be [[Definition:Collinear Points|collinear]] {{hypothesis}}.
Then $\triangle BB'N$ and $\triangle CC'M$ are perspective with center $L$ ($L = BC \cap B'C' \cap MN$)
From [[Desargues' Theorem]]:
:$O = BB' \cap CC'$
:$A = BN \cap CM$
:$A' = C'M \cap B'N$
are [... | Desargues' Theorem/Converse | https://proofwiki.org/wiki/Desargues'_Theorem/Converse | https://proofwiki.org/wiki/Desargues'_Theorem/Converse | [
"Desargues' Theorem"
] | [
"Definition:Triangle (Geometry)",
"Definition:Plane Surface",
"Definition:Collinear/Points",
"Definition:Line/Straight Line",
"Definition:Intersection (Geometry)",
"Definition:Point"
] | [
"File:DesarguesTheorem.png",
"Definition:Collinear/Points",
"Desargues' Theorem",
"Definition:Collinear/Points"
] |
proofwiki-20383 | Circuits of Matroid iff Matroid Circuit Axioms | Let $S$ be a finite set.
Let $\mathscr C$ be a non-empty set of subsets of $S$.
Then:
:$\mathscr C$ is the set of circuits of a matroid on $S$
{{iff}}
:$\mathscr C$ satisfies the circuit axioms | From Equivalence of Definitions of Matroid Circuit Axioms it is sufficient to show:
:$(\text a)\quad$if $\mathscr C$ is the set of circuits of a matroid then $\mathscr C$ satisfies circuit axioms (formulation 1)
and
:$(\text b)\quad$if $\mathscr C$ satisfies circuit axioms (formulation 2) then $\mathscr C$ is the set o... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr C$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Then:
:$\mathscr C$ is the [[Definition:Set|set]] of [[Definition:Circuit (Matroid)|circuits]] of a [[Definition:Matroid|matroid]] on $S$
{{i... | From [[Equivalence of Definitions of Matroid Circuit Axioms]] it is sufficient to show:
:$(\text a)\quad$if $\mathscr C$ is the [[Definition:Set|set]] of [[Definition:Circuit (Matroid)|circuits]] of a [[Definition:Matroid|matroid]] then $\mathscr C$ satisfies [[Axiom:Circuit Axioms (Matroid)/Formulation 1|circuit axiom... | Circuits of Matroid iff Matroid Circuit Axioms | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms | [
"Matroid Circuits",
"Circuits of Matroid iff Matroid Circuit Axioms"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Definition:Set",
"Definition:Circuit (Matroid)",
"Definition:Matroid",
"Axiom:Circuit Axioms (Matroid)"
] | [
"Equivalence of Definitions of Matroid Circuit Axioms",
"Definition:Set",
"Definition:Circuit (Matroid)",
"Definition:Matroid",
"Axiom:Circuit Axioms (Matroid)/Formulation 1",
"Axiom:Circuit Axioms (Matroid)/Formulation 2",
"Definition:Set",
"Definition:Circuit (Matroid)",
"Definition:Matroid",
"D... |
proofwiki-20384 | Circuits of Matroid iff Matroid Circuit Axioms/Formulation 2 implies Circuits of Matroid | Let $S$ be a finite set.
Let $\mathscr C$ be a non-empty set of subsets of $S$ that satisfies the circuit axioms:
{{:Axiom:Circuit Axioms (Matroid)/Formulation 2}}
Then:
:$\mathscr C$ is the set of circuits of a matroid $M = \struct{S, \mathscr I}$ on $S$ | We will define a mapping $\rho$ associated with $\mathscr C$.
It will be shown that $\rho$ is the rank function of a matroid $M$ which has $\mathscr C$ as the set of circuits.
For any ordered tuple $\tuple{x_1, \ldots, x_q}$ of elements of $S$, let $\map \theta {x_1, \ldots, x_q}$ be the ordered tuple defined by:
:$\fo... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr C$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$ that satisfies the [[Axiom:Circuit Axioms (Matroid)/Formulation 2|circuit axioms]]:
{{:Axiom:Circuit Axioms (Matroid)/Formulation 2}}
Then:
:$\... | We will define a [[Definition:Mapping|mapping]] $\rho$ associated with $\mathscr C$.
It will be shown that $\rho$ is the [[Definition:Rank Function (Matroid)|rank function]] of a [[Definition:Matroid|matroid]] $M$ which has $\mathscr C$ as the set of [[Definition:Circuit (Matroid)|circuits]].
For any [[Definition:Or... | Circuits of Matroid iff Matroid Circuit Axioms/Formulation 2 implies Circuits of Matroid | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Formulation_2_implies_Circuits_of_Matroid | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Formulation_2_implies_Circuits_of_Matroid | [
"Circuits of Matroid iff Matroid Circuit Axioms"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Axiom:Circuit Axioms (Matroid)/Formulation 2",
"Definition:Set",
"Definition:Circuit (Matroid)",
"Definition:Matroid"
] | [
"Definition:Mapping",
"Definition:Rank Function (Matroid)",
"Definition:Matroid",
"Definition:Circuit (Matroid)",
"Definition:Ordered Tuple",
"Definition:Element",
"Definition:Ordered Tuple",
"Definition:Mapping",
"Definition:Set",
"Definition:Ordered Tuple",
"Circuits of Matroid iff Matroid Cir... |
proofwiki-20385 | Circuits of Matroid iff Matroid Circuit Axioms/Circuits of Matroid implies Formulation 1 | Let $S$ be a finite set.
Let $\mathscr C$ be a non-empty set of subsets of $S$.
Let $\mathscr C$ be the set of circuits of a matroid $M = \struct{S, \mathscr I}$ on $S$
Then:
:$\mathscr C$ satisfies the circuit axioms:
{{:Axiom:Circuit Axioms (Matroid)/Formulation 1}} | ==== $\mathscr C$ satisfies $(\text C 1)$ ====
By definition of circuit of a matroid:
:for all $C \in \mathscr C: C$ is a dependent subset
By definition of a dependent subset:
:for all $C \in \mathscr C$, $C \notin \mathscr I$
By definition of a matroid:
:$\O \in \mathscr I$
Hence:
:$\O \notin \mathscr C$
It follows th... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr C$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Let $\mathscr C$ be the [[Definition:Set|set]] of [[Definition:Circuit (Matroid)|circuits]] of a [[Definition:Matroid|matroid]] $M = \struct{... | ==== $\mathscr C$ satisfies $(\text C 1)$ ====
By definition of [[Definition:Circuit (Matroid)|circuit]] of a [[Definition:Matroid|matroid]]:
:for all $C \in \mathscr C: C$ is a [[Definition:Dependent Subset (Matroid)|dependent subset]]
By definition of a [[Definition:Dependent Subset (Matroid)|dependent subset]]:
:fo... | Circuits of Matroid iff Matroid Circuit Axioms/Circuits of Matroid implies Formulation 1 | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Circuits_of_Matroid_implies_Formulation_1 | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Circuits_of_Matroid_implies_Formulation_1 | [
"Circuits of Matroid iff Matroid Circuit Axioms"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Definition:Set",
"Definition:Circuit (Matroid)",
"Definition:Matroid",
"Axiom:Circuit Axioms (Matroid)/Formulation 1"
] | [
"Definition:Circuit (Matroid)",
"Definition:Matroid",
"Definition:Matroid/Dependent Set",
"Definition:Matroid/Dependent Set",
"Definition:Matroid",
"Axiom:Circuit Axioms (Matroid)/Formulation 1",
"Definition:Circuit (Matroid)",
"Definition:Matroid",
"Definition:Matroid/Dependent Set",
"Definition:... |
proofwiki-20386 | Equivalence of Definitions of Matroid Circuit Axioms/Formulation 1 Implies Formulation 2 | Let $S$ be a finite set.
Let $\mathscr C$ be a non-empty set of subsets of $S$.
Let $\mathscr C$ satisfy the circuit axioms (formulation 1):
{{:Axiom:Circuit Axioms (Matroid)/Formulation 1}}
Then:
:$\mathscr C$ satisfies the circuit axioms (formulation 2):
{{:Axiom:Circuit Axioms (Matroid)/Formulation 2}} | Let $\mathscr C$ satisfy the circuit axioms $(\text C 1)$, $(\text C 2)$ and $(\text C 3)$.
It has only to be shown that circuit axiom $(\text C 4)$ is satisfied by $\mathscr C$.
Let:
{{begin-eqn}}
{{eqn | l = F = \leftset{\tuple{C, D, x, y} }
| o = :
| r = C, D \in \mathscr C \land C \neq D
}}
{{eqn | o =... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr C$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Let $\mathscr C$ satisfy the [[Axiom:Circuit Axioms (Matroid)/Formulation 1|circuit axioms (formulation 1)]]:
{{:Axiom:Circuit Axioms (Matro... | Let $\mathscr C$ satisfy the [[Axiom:Circuit Axioms (Matroid)/Formulation 1|circuit axioms $(\text C 1)$, $(\text C 2)$ and $(\text C 3)$]].
It has only to be shown that [[Axiom:Circuit Axioms (Matroid)/Formulation 2|circuit axiom $(\text C 4)$]] is satisfied by $\mathscr C$.
Let:
{{begin-eqn}}
{{eqn | l = F = \left... | Equivalence of Definitions of Matroid Circuit Axioms/Formulation 1 Implies Formulation 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Circuit_Axioms/Formulation_1_Implies_Formulation_2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Circuit_Axioms/Formulation_1_Implies_Formulation_2 | [
"Equivalence of Definitions of Matroid Circuit Axioms"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Axiom:Circuit Axioms (Matroid)/Formulation 1",
"Axiom:Circuit Axioms (Matroid)/Formulation 2"
] | [
"Axiom:Circuit Axioms (Matroid)/Formulation 1",
"Axiom:Circuit Axioms (Matroid)/Formulation 2",
"Axiom:Circuit Axioms (Matroid)/Formulation 2",
"Axiom:Circuit Axioms (Matroid)/Formulation 1",
"Axiom:Circuit Axioms (Matroid)/Formulation 1",
"Set Difference and Intersection form Partition",
"Set is Subset... |
proofwiki-20387 | Equivalence of Definitions of Matroid Circuit Axioms/Formulation 1 Implies Formulation 3 | Let $S$ be a finite set.
Let $\mathscr C$ be a non-empty set of subsets of $S$.
Let $\mathscr C$ satisfy the circuit axioms (formulation 1):
{{:Axiom:Circuit Axioms (Matroid)/Formulation 1}}
Then:
:$\mathscr C$ satisfies the circuit axioms (formulation 3):
{{:Axiom:Circuit Axioms (Matroid)/Formulation 3}} | Let $\mathscr C$ satisfy the circuit axioms $(\text C 1)$, $(\text C 2)$ and $(\text C 3)$.
We need to show that $\mathscr C$ satisfies circuit axiom:
{{begin-axiom}}
{{axiom | n = \text C 5
| q = \forall X \subseteq S \land \forall x \in S
| mr = \paren {\forall C \in \mathscr C : C \nsubseteq X} \imp... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr C$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Let $\mathscr C$ satisfy the [[Axiom:Circuit Axioms (Matroid)/Formulation 1|circuit axioms (formulation 1)]]:
{{:Axiom:Circuit Axioms (Matroi... | Let $\mathscr C$ satisfy the [[Axiom:Circuit Axioms (Matroid)/Formulation 1|circuit axioms $(\text C 1)$, $(\text C 2)$ and $(\text C 3)$]].
We need to show that $\mathscr C$ satisfies [[Axiom:Circuit Axioms (Matroid)/Formulation 3|circuit axiom]]:
{{begin-axiom}}
{{axiom | n = \text C 5
| q = \forall X \subse... | Equivalence of Definitions of Matroid Circuit Axioms/Formulation 1 Implies Formulation 3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Circuit_Axioms/Formulation_1_Implies_Formulation_3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Circuit_Axioms/Formulation_1_Implies_Formulation_3 | [
"Equivalence of Definitions of Matroid Circuit Axioms"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Axiom:Circuit Axioms (Matroid)/Formulation 1",
"Axiom:Circuit Axioms (Matroid)/Formulation 3"
] | [
"Axiom:Circuit Axioms (Matroid)/Formulation 1",
"Axiom:Circuit Axioms (Matroid)/Formulation 3",
"Axiom:Circuit Axioms (Matroid)/Formulation 1",
"Definition:Contradiction",
"Axiom:Circuit Axioms (Matroid)/Formulation 3"
] |
proofwiki-20388 | Equivalence of Definitions of Matroid Circuit Axioms/Formulation 3 Implies Formulation 1 | Let $S$ be a finite set.
Let $\mathscr C$ be a non-empty set of subsets of $S$.
Let $\mathscr C$ satisfy the circuit axioms (formulation 3):
{{:Axiom:Circuit Axioms (Matroid)/Formulation 3}}
Then:
:$\mathscr C$ satisfies the circuit axioms (formulation 1):
{{:Axiom:Circuit Axioms (Matroid)/Formulation 1}} | Let $\mathscr C$ satisfy the circuit axioms $(\text C 1)$, $(\text C 2)$ and $(\text C 3'')$.
We need to show that $\mathscr C$ satisfies circuit axiom:
{{begin-axiom}}
{{axiom | n = \text C 3
| q = \forall C_1, C_2 \in \mathscr C
| mr = C_1 \ne C_2 \land z \in C_1 \cap C_2 \implies \exists C_3 \in \mat... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr C$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Let $\mathscr C$ satisfy the [[Axiom:Circuit Axioms (Matroid)/Formulation 3|circuit axioms (formulation 3)]]:
{{:Axiom:Circuit Axioms (Matroi... | Let $\mathscr C$ satisfy the [[Axiom:Circuit Axioms (Matroid)/Formulation 3|circuit axioms $(\text C 1)$, $(\text C 2)$ and $(\text C 3'')$]].
We need to show that $\mathscr C$ satisfies [[Axiom:Circuit Axioms (Matroid)/Formulation 3|circuit axiom]]:
{{begin-axiom}}
{{axiom | n = \text C 3
| q = \forall C_1, C... | Equivalence of Definitions of Matroid Circuit Axioms/Formulation 3 Implies Formulation 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Circuit_Axioms/Formulation_3_Implies_Formulation_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Circuit_Axioms/Formulation_3_Implies_Formulation_1 | [
"Equivalence of Definitions of Matroid Circuit Axioms"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Axiom:Circuit Axioms (Matroid)/Formulation 3",
"Axiom:Circuit Axioms (Matroid)/Formulation 1"
] | [
"Axiom:Circuit Axioms (Matroid)/Formulation 3",
"Axiom:Circuit Axioms (Matroid)/Formulation 3",
"Definition:Contrapositive Statement",
"Axiom:Circuit Axioms (Matroid)/Formulation 3",
"Set is Subset of Union",
"Axiom:Circuit Axioms (Matroid)/Formulation 1"
] |
proofwiki-20389 | Equivalence of Definitions of Matroid Circuit Axioms/Formulation 2 Implies Formulation 1 | Let $S$ be a finite set.
Let $\mathscr C$ be a non-empty set of subsets of $S$.
Let $\mathscr C$ satisfy the circuit axioms (formulation 2):
{{:Axiom:Circuit Axioms (Matroid)/Formulation 2}}
Then:
:$\mathscr C$ satisfies the circuit axioms (formulation 1):
{{:Axiom:Circuit Axioms (Matroid)/Formulation 1}} | Let $\mathscr C$ satisfy the circuit axioms $(\text C 1)$, $(\text C 2)$ and $(\text C 4)$.
We need to show that $\mathscr C$ satisfies circuit axiom:
{{begin-axiom}}
{{axiom | n = \text C 3
| q = \forall C_1, C_2 \in \mathscr C
| mr= C_1 \ne C_2 \land z \in C_1 \cap C_2 \implies \exists C_3 \in \mathsc... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr C$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Let $\mathscr C$ satisfy the [[Axiom:Circuit Axioms (Matroid)/Formulation 2|circuit axioms (formulation 2)]]:
{{:Axiom:Circuit Axioms (Matroi... | Let $\mathscr C$ satisfy the [[Axiom:Circuit Axioms (Matroid)/Formulation 2|circuit axioms $(\text C 1)$, $(\text C 2)$ and $(\text C 4)$]].
We need to show that $\mathscr C$ satisfies [[Axiom:Circuit Axioms (Matroid)/Formulation 1|circuit axiom]]:
{{begin-axiom}}
{{axiom | n = \text C 3
| q = \forall C_1, C_2... | Equivalence of Definitions of Matroid Circuit Axioms/Formulation 2 Implies Formulation 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Circuit_Axioms/Formulation_2_Implies_Formulation_1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Circuit_Axioms/Formulation_2_Implies_Formulation_1 | [
"Equivalence of Definitions of Matroid Circuit Axioms"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Axiom:Circuit Axioms (Matroid)/Formulation 2",
"Axiom:Circuit Axioms (Matroid)/Formulation 1"
] | [
"Axiom:Circuit Axioms (Matroid)/Formulation 2",
"Axiom:Circuit Axioms (Matroid)/Formulation 1",
"Axiom:Circuit Axioms (Matroid)/Formulation 2",
"Definition:Subset",
"Definition:Set Difference",
"Axiom:Circuit Axioms (Matroid)/Formulation 2",
"Axiom:Circuit Axioms (Matroid)/Formulation 1"
] |
proofwiki-20390 | Path Component of Locally Path-Connected Space is Closed | Let $T = \struct {S, \tau}$ be a locally path-connected topological space.
Let $G$ be a path component of $T$.
Then $G$ is open in $T$. | Let $x \in \partial G$, where $\partial G$ denotes the boundary of $G$.
By {{Open-set-axiom|3}}, it follows that $S \in \tau$.
As $x \in S$, it follows that $S$ is a neighborhood of $x$.
By definition of locally path-connected space, it follows that there exists a path-connected neighborhood $N$ of $x$ such that $N \su... | Let $T = \struct {S, \tau}$ be a [[Definition:Locally Path-Connected Space|locally path-connected]] [[Definition:Topological Space|topological space]].
Let $G$ be a [[Definition:Path Component|path component]] of $T$.
Then $G$ is [[Definition:Open Set (Topology)|open]] in $T$. | Let $x \in \partial G$, where $\partial G$ denotes the [[Definition:Boundary (Topology)|boundary]] of $G$.
By {{Open-set-axiom|3}}, it follows that $S \in \tau$.
As $x \in S$, it follows that $S$ is a [[Definition:Neighborhood (Topology)|neighborhood]] of $x$.
By definition of [[Definition:Locally Path-Connected Spa... | Path Component of Locally Path-Connected Space is Closed | https://proofwiki.org/wiki/Path_Component_of_Locally_Path-Connected_Space_is_Closed | https://proofwiki.org/wiki/Path_Component_of_Locally_Path-Connected_Space_is_Closed | [
"Path Components",
"Locally Path-Connected Spaces"
] | [
"Definition:Locally Path-Connected Space",
"Definition:Topological Space",
"Definition:Path Component",
"Definition:Open Set/Topology"
] | [
"Definition:Boundary (Topology)",
"Definition:Neighborhood (Topology)",
"Definition:Locally Path-Connected Space",
"Definition:Path-Connected/Set",
"Definition:Neighborhood (Topology)",
"Definition:Boundary (Topology)/Definition 2",
"Definition:Path-Connected/Set",
"Definition:Path (Topology)",
"Def... |
proofwiki-20391 | Circuits of Matroid iff Matroid Circuit Axioms/Lemma 1 | Let $\tuple{x_1, \ldots, x_q}$ be any ordered tuple of elements of $S$.
Let $\pi$ be any permutation of $\tuple{x_1, \ldots, x_q}$.
Then:
:$\map t {x_1, \ldots, x_q} = \map t {x_{\map \pi 1}, \ldots, x_{\map \pi q}}$ | It is sufficient to show that:
:$\forall 1 \le i \le q-1 : \map t {x_1, \ldots, x_i, x_{i + 1}, \ldots, x_q} = \map t {x_1, \ldots, x_{i + 1}, x_i, \ldots, x_q}$
By definition of $t$, we have:
:$\map t {x_1, \ldots, x_i, x_{i + 1}, \ldots, x_q} = \sum_{j = 1}^{i - 1} \map \theta {x_1, \ldots, x_q}_j + \map \theta {x_1,... | Let $\tuple{x_1, \ldots, x_q}$ be any [[Definition:Ordered Tuple|ordered tuple]] of [[Definition:Element|elements]] of $S$.
Let $\pi$ be any [[Definition:Permutation|permutation]] of $\tuple{x_1, \ldots, x_q}$.
Then:
:$\map t {x_1, \ldots, x_q} = \map t {x_{\map \pi 1}, \ldots, x_{\map \pi q}}$ | It is sufficient to show that:
:$\forall 1 \le i \le q-1 : \map t {x_1, \ldots, x_i, x_{i + 1}, \ldots, x_q} = \map t {x_1, \ldots, x_{i + 1}, x_i, \ldots, x_q}$
By definition of $t$, we have:
:$\map t {x_1, \ldots, x_i, x_{i + 1}, \ldots, x_q} = \sum_{j = 1}^{i - 1} \map \theta {x_1, \ldots, x_q}_j + \map \theta {x_... | Circuits of Matroid iff Matroid Circuit Axioms/Lemma 1 | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Lemma_1 | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Lemma_1 | [
"Circuits of Matroid iff Matroid Circuit Axioms"
] | [
"Definition:Ordered Tuple",
"Definition:Element",
"Definition:Permutation"
] | [
"Definition:Ordered Tuple",
"Definition:Ordered Tuple"
] |
proofwiki-20392 | Circuits of Matroid iff Matroid Circuit Axioms/Lemma 2 | Let $X \subseteq S$ and $y \in S \setminus X$.
Then:
:$\map \rho {X \cup \set y} = \map \rho X$ {{iff}} $\exists C \in \mathscr C : y \in C \subseteq X \cup \set y$ | Let $X = \set{x_1, \ldots, x_q}$
We have:
{{begin-eqn}}
{{eqn | l = \map \rho {X \cup \set y}
| r = \map t {x_1, \ldots, x_q, y}
| c = Definition of $\rho$
}}
{{eqn | r = \map t {x_1, \ldots, x_q} + \map \theta {x_1, \ldots, x_q, y}_{q+1}
| c = Definition of $t$
}}
{{eqn | r = \map \rho X + \map \thet... | Let $X \subseteq S$ and $y \in S \setminus X$.
Then:
:$\map \rho {X \cup \set y} = \map \rho X$ {{iff}} $\exists C \in \mathscr C : y \in C \subseteq X \cup \set y$ | Let $X = \set{x_1, \ldots, x_q}$
We have:
{{begin-eqn}}
{{eqn | l = \map \rho {X \cup \set y}
| r = \map t {x_1, \ldots, x_q, y}
| c = Definition of $\rho$
}}
{{eqn | r = \map t {x_1, \ldots, x_q} + \map \theta {x_1, \ldots, x_q, y}_{q+1}
| c = Definition of $t$
}}
{{eqn | r = \map \rho X + \map \th... | Circuits of Matroid iff Matroid Circuit Axioms/Lemma 2 | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Lemma_2 | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Lemma_2 | [
"Circuits of Matroid iff Matroid Circuit Axioms"
] | [] | [
"Category:Circuits of Matroid iff Matroid Circuit Axioms"
] |
proofwiki-20393 | Circuits of Matroid iff Matroid Circuit Axioms/Lemma 3 | :$\mathscr C$ is the set of circuits of $M$. | Let $\mathscr C_M$ be the set of circuit of the matroid $M$. | :$\mathscr C$ is the set of [[Definition:Circuit (Matroid)|circuits]] of $M$. | Let $\mathscr C_M$ be the set of [[Definition:Circuit (Matroid)|circuit]] of the [[Definition:Matroid|matroid]] $M$. | Circuits of Matroid iff Matroid Circuit Axioms/Lemma 3 | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Lemma_3 | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Lemma_3 | [
"Circuits of Matroid iff Matroid Circuit Axioms"
] | [
"Definition:Circuit (Matroid)"
] | [
"Definition:Circuit (Matroid)",
"Definition:Matroid",
"Definition:Circuit (Matroid)"
] |
proofwiki-20394 | Circuits of Matroid iff Matroid Circuit Axioms/Lemma 4 | :$\forall C \in \mathscr C_M : \exists C' \in \mathscr C : C' \subseteq C$ | Let $C \in \mathscr C_M$.
By definition of circuit:
:$C$ is dependent
By matroid axiom $(\text I 1)$:
:$C \neq \O$
Let $x \in C$.
From Proper Subset of Matroid Circuit is Independent:
:$C \setminus \set x$ is independent
We have:
{{begin-eqn}}
{{eqn | l = \map \rho C
| r = \card C - 1
| c = Rank of Matroid... | :$\forall C \in \mathscr C_M : \exists C' \in \mathscr C : C' \subseteq C$ | Let $C \in \mathscr C_M$.
By definition of [[Definition:Circuit (Matroid)|circuit]]:
:$C$ is [[Definition:Dependent Subset (Matroid)|dependent]]
By [[Axiom:Matroid Axioms|matroid axiom $(\text I 1)$]]:
:$C \neq \O$
Let $x \in C$.
From [[Proper Subset of Matroid Circuit is Independent]]:
:$C \setminus \set x$ is ... | Circuits of Matroid iff Matroid Circuit Axioms/Lemma 4 | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Lemma_4 | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Lemma_4 | [
"Circuits of Matroid iff Matroid Circuit Axioms"
] | [] | [
"Definition:Circuit (Matroid)",
"Definition:Matroid/Dependent Set",
"Axiom:Matroid Axioms",
"Proper Subset of Matroid Circuit is Independent",
"Definition:Matroid/Independent Set",
"Rank of Matroid Circuit is One Less Than Cardinality",
"Cardinality of Singleton",
"Cardinality of Set Difference",
"R... |
proofwiki-20395 | Circuits of Matroid iff Matroid Circuit Axioms/Lemma 5 | :$\forall C \in \mathscr C : \exists C' \in \mathscr C_M : C' \subseteq C$ | Let $C \in \mathscr C$.
From circuit axiom $(\text C 1)$:
:$\exists y \in C$
We have:
:$C \subseteq C = \paren{C \setminus \set y} \cup \set y$ | :$\forall C \in \mathscr C : \exists C' \in \mathscr C_M : C' \subseteq C$ | Let $C \in \mathscr C$.
From [[Axiom:Circuit Axioms (Matroid)/Formulation 2|circuit axiom $(\text C 1)$]]:
:$\exists y \in C$
We have:
:$C \subseteq C = \paren{C \setminus \set y} \cup \set y$ | Circuits of Matroid iff Matroid Circuit Axioms/Lemma 5 | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Lemma_5 | https://proofwiki.org/wiki/Circuits_of_Matroid_iff_Matroid_Circuit_Axioms/Lemma_5 | [
"Circuits of Matroid iff Matroid Circuit Axioms"
] | [] | [
"Axiom:Circuit Axioms (Matroid)/Formulation 2"
] |
proofwiki-20396 | Set of Finite Strings is Countably Infinite | Let $\Sigma$ be a finite alphabet.
Let $\Sigma^*$ be the set of finite strings of $\Sigma$.
Then $\Sigma^*$ is a countably infinite set. | Let $b = \size \Sigma$, that is, the number of symbols. | Let $\Sigma$ be a [[Definition:Finite Set|finite]] [[Definition:Alphabet of Formal Language|alphabet]].
Let $\Sigma^*$ be the [[Definition:Set of Finite Strings|set of finite strings]] of $\Sigma$.
Then $\Sigma^*$ is a [[Definition:Countably Infinite Set|countably infinite set]]. | Let $b = \size \Sigma$, that is, the number of [[Definition:Symbol|symbols]]. | Set of Finite Strings is Countably Infinite | https://proofwiki.org/wiki/Set_of_Finite_Strings_is_Countably_Infinite | https://proofwiki.org/wiki/Set_of_Finite_Strings_is_Countably_Infinite | [
"Sets of Finite Strings",
"Infinite Sets",
"Countable Sets"
] | [
"Definition:Finite Set",
"Definition:Formal Language/Alphabet",
"Definition:Set of Finite Strings",
"Definition:Countably Infinite/Set"
] | [
"Definition:Symbol",
"Definition:Symbol",
"Definition:Symbol",
"Definition:Symbol"
] |
proofwiki-20397 | Power Set of Set of Finite Strings is Uncountable | Let $\Sigma$ be a finite alphabet.
Let $\Sigma^*$ be the set of finite strings of $\Sigma$.
Let $\powerset {\Sigma^*}$ be the power set of $\Sigma^*$
Then $\powerset {\Sigma^*}$ is an uncountable set. | From Set of Finite Strings is Countably Infinite, $\Sigma^*$ is a countably infinite set.
The result follows from Power Set of Countably Infinite Set is Uncountable.
{{qed}} | Let $\Sigma$ be a [[Definition:Finite Set|finite]] [[Definition:Alphabet of Formal Language|alphabet]].
Let $\Sigma^*$ be the [[Definition:Set of Finite Strings|set of finite strings]] of $\Sigma$.
Let $\powerset {\Sigma^*}$ be the [[Definition:Power Set|power set]] of $\Sigma^*$
Then $\powerset {\Sigma^*}$ is an [... | From [[Set of Finite Strings is Countably Infinite]], $\Sigma^*$ is a [[Definition:Countably Infinite Set|countably infinite set]].
The result follows from [[Power Set of Countably Infinite Set is Uncountable]].
{{qed}} | Power Set of Set of Finite Strings is Uncountable | https://proofwiki.org/wiki/Power_Set_of_Set_of_Finite_Strings_is_Uncountable | https://proofwiki.org/wiki/Power_Set_of_Set_of_Finite_Strings_is_Uncountable | [
"Sets of Finite Strings",
"Uncountable Sets"
] | [
"Definition:Finite Set",
"Definition:Formal Language/Alphabet",
"Definition:Set of Finite Strings",
"Definition:Power Set",
"Definition:Uncountable/Set"
] | [
"Set of Finite Strings is Countably Infinite",
"Definition:Countably Infinite/Set",
"Power Set of Countably Infinite Set is Uncountable"
] |
proofwiki-20398 | Quotient Norm is Norm | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\Bbb F$.
Let $N$ be a closed linear subspace of $X$.
Let $X/N$ be the quotient vector space of $X$ modulo $N$.
Let $\norm {\, \cdot \,}_{X/N}$ be the quotient norm on $X/N$.
Then $\norm {\, \cdot \,}_{X/N}$ is indee... | === Norm is Well-Defined and Finite ===
Let $\pi$ be the quotient map associated with $X/N$.
We show that if $x, x' \in X$ have $\map \pi x = \map \pi {x'}$, then:
:$\ds \inf_{z \mathop \in N} \norm {x - z} = \inf_{z \mathop \in N} \norm {x' - z}$
From Quotient Mapping is Linear Transformation:
:$\map \pi {x' - x} = 0... | Let $\Bbb F \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $\Bbb F$.
Let $N$ be a [[Definition:Closed Linear Subspace|closed linear subspace]] of $X$.
Let $X/N$ be the [[Definition:Quotient Vector Space|quotient vector space of $X$ modul... | === Norm is Well-Defined and Finite ===
Let $\pi$ be the [[Definition:Quotient Mapping|quotient map]] associated with $X/N$.
We show that if $x, x' \in X$ have $\map \pi x = \map \pi {x'}$, then:
:$\ds \inf_{z \mathop \in N} \norm {x - z} = \inf_{z \mathop \in N} \norm {x' - z}$
From [[Quotient Mapping is Linear T... | Quotient Norm is Norm | https://proofwiki.org/wiki/Quotient_Norm_is_Norm | https://proofwiki.org/wiki/Quotient_Norm_is_Norm | [
"Examples of Norms",
"Quotient Norms"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Linear Subspace",
"Definition:Quotient Vector Space",
"Definition:Quotient Norm",
"Definition:Norm/Vector Space"
] | [
"Definition:Quotient Mapping",
"Quotient Mapping is Linear Transformation",
"Kernel of Quotient Mapping",
"Definition:Linear Subspace",
"Definition:Quotient Norm",
"Kernel of Quotient Mapping",
"Quotient Mapping is Linear Transformation",
"Definition:Linear Subspace",
"Quotient Mapping is Linear Tra... |
proofwiki-20399 | Quotient Vector Space is Vector Space | Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $N$ be a linear subspace of $X$.
Define:
:$X/N = \set {x + N : x \in X}$
where $x + N$ is the Minkowski sum of $x$ and $N$.
Define:
:$\paren {x + N} +_{X/N} \paren {y + N} = \paren {x + y} + N$
for $x, y \in X$, and:
:$\alpha \circ_{X/N} {x + N} = \paren {\... | === Lemma ===
{{:Quotient Vector Space is Vector Space/Lemma}}{{qed|lemma}} | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $X$ be a [[Definition:Vector Space|vector space]] over $K$.
Let $N$ be a [[Definition:Linear Subspace|linear subspace]] of $X$.
Define:
:$X/N = \set {x + N : x \in X}$
where $x + N$ is the [[Definition:Minkowski Sum|Minkowski sum]] of $x$ and $N$.
... | === [[Quotient Vector Space is Vector Space/Lemma|Lemma]] ===
{{:Quotient Vector Space is Vector Space/Lemma}}{{qed|lemma}} | Quotient Vector Space is Vector Space | https://proofwiki.org/wiki/Quotient_Vector_Space_is_Vector_Space | https://proofwiki.org/wiki/Quotient_Vector_Space_is_Vector_Space | [
"Quotient Vector Space is Vector Space",
"Quotient Vector Spaces"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Vector Space",
"Definition:Linear Subspace",
"Definition:Minkowski Sum",
"Definition:Vector Space"
] | [
"Quotient Vector Space is Vector Space/Lemma",
"Quotient Vector Space is Vector Space/Lemma",
"Quotient Vector Space is Vector Space/Lemma"
] |
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